# Maths trivia



## SydneyGeek (May 17, 2010)

Just got reminded of something today...

1^3 + 2^3 + 3^3 = (1 + 2 + 3)^2
Keeps working as you extend the series. 

Anyone else got a tidbit?

Denis


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## Joe4 (May 17, 2010)

One of my favorites is that is you add up the digits of any number, and that sum is divisible by 9, then that number is divisible by 9, i.e.

1485
1 + 4 + 8 + 5 = 18
18 / 9 = 2

1485 / 9 = 165


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## sulakvea (May 17, 2010)

Take any integer, sum up ODD digits, and then EVEN digits. subtract one from another. if the result is divisible by 11, then the original number is divisible by 11 as well. say - 121. sum of odds =1+1=2. sum of evens =2. 2-2=0, thus 121 is divisible by 11.

313010487215450416255248 - sum of odds =35, sum of evens =46. divisible by 11


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## Fazza (May 17, 2010)

A simple pattern that I like, especially I think because even young children can 'see it'.

1 * 1 = 1
11 * 11 = 121
111 * 111 = 12321
1111 * 1111 = 1234321
11111 * 11111 = 123454321
111111 * 111111 = 12345654321

It is great to then ask about 111111111 * 111111111
Even youngsters can respond with 12345678987654321


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## MorganO (May 18, 2010)

As a youth learning math I liked this concept with the 9's:

1 * 9 = 09 | 0 + 9 = 9
2 * 9 = 18 | 1 + 8 = 9
3 * 9 = 27 | 2 + 7 = 9
.
.
.
9 * 9 = 81 | 8 + 1 = 9
10 * 9 = 90 | 9 + 0 = 9
11 * 9 = 99 | 9 + 9 = 18 | 1 + 8 = 9


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## Sandeep Warrier (May 18, 2010)

The easiest way to get the square of any number ending with 5...

25^2 => take the square of the last digit i.e. 5 and store it. 5^2 = 25.

Now take the 1st digit and multiply it with the number obtained by incrementing the digit by 1 => 2 * (2+1) => 2*3 => 6. Now concatenate this with 25 => 625 = 25^2

Similarly for 35^2 = (3*4) 25 => 1225

etc.
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## Gerald Higgins (May 18, 2010)

Joe4 - you can extend the 9 example.

If the digits sum to 3, 6 or 9, then the number is divisible by 3.
If the digits sume to 3, and the original number is even, then the number is divisble by 6.
There are probably more of these.


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## pgc01 (May 19, 2010)

One thing I remember from highschool that I find interesting is that given 2 numbers, if you know their sum S and their product P, you can find them by solving the equation:

x<sup>2</sup> - Sx + P = 0


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## Oaktree (May 19, 2010)

The one that still makes my head hurt is that there are more integers than rational numbers.


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## Joe4 (May 19, 2010)

> The one that still makes my head hurt is that there are more integers than rational numbers.


What?  How can that be?  I need some clarification on this one!

Isn't every integer a rational number and not every rational number is an integer?

Or is it the argument that integers are infinite, so how can there be anything more than infinity?  If that is case, you might be able to make the argument that there are NOT more rational numbers than integers, but I don't think you can say that there are more integers than rational numbers.


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## pgc01 (May 19, 2010)

Oaktree said:


> The one that still makes my head hurt is that there are more integers than rational numbers.


 
Hi

I think you mean that there are as many integers as rational numbers (since they are both infinite sets this means that you can establish a 1 to 1 relationship between the 2 sets).


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## Domski (May 19, 2010)

Speaking of infinity...

It's not really maths trivia, other than I made my work colleagues' heads hurt when I explained it to them (or tried to):

http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

Dom


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## Oaktree (May 19, 2010)

pgc01 said:


> Hi
> 
> I think you mean that there are as many integers as rational numbers (since they are both infinite sets this means that you can establish a 1 to 1 relationship between the 2 sets).



Yeah, my head would have not exploded if it was just that infinity = infinity, we say they're the same size, and we're done with it.  

But, the different sizes of infinity come in to play (just as the "more real numbers than rational numbers" proof is fairly common), and that's where my head hurts.

Expanding on your suggestion...You can establish a 1:1 relationship of every a/b quotient (rational number) to an integer... but, the fact that e.g. rational numbers 4/1 and 8/2 (equivalent values) would be mapped to separate integers in this approach suggests that you have a 1:many relationship of rationals:integers, which inherently shows that the set of integers is larger.


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## sulakvea (May 19, 2010)

there are EQUAL number of integers and rational numbers.

however, there are MORE irrational numbers (like pi(), sqrt(2), etc) than either integers or rationals. the proof is actually pretty simple, google Cardinal Numbers


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## Fazza (May 19, 2010)

Reminds me of the initially surprising result that average waiting times are longer when fuel rationing is put in place. Along the lines that only cars with licence plates ending in odd numbers can be served on odd numbered days and only those ending in even numbers on even numbered days. With such restrictions in place, delays are greater than if any car can get fuel on any day. Such restrictions have been put in place to reduce delays and the opposite occurs.


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## pgc01 (May 19, 2010)

Oaktree said:


> Expanding on your suggestion...You can establish a 1:1 relationship of every a/b quotient (rational number) to an integer... but, the fact that e.g. rational numbers 4/1 and 8/2 (equivalent values) would be mapped to separate integers in this approach suggests that you have a 1:many relationship of rationals:integers, which inherently shows that the set of integers is larger.


 
What you say is logic, but that's a logic that applies to finite sets. For infinite sets you cannot use that kind of logic or you can easily get contradictory results.

For ex., since your reasoning seems to imply that there are more integers that rationals, let me propose another one. Let's define the relationship between the integers and the rationals n->1/n (for all n <>0). For n=0 the corresponding value is 0. Here we see that all the integers correspond to a rational between -1 and 1. Since there are rationals outside that interval (and also rationals inside that interval that cannot be reduced to a fraction with numerator equal to 1), the conclusion is that there are many more rationals that integers.

These 2 reasonings seem ok, but they lead to contradictory conclusions. That's why in mathematics, when you are working with infinite sets you must use other rules. 

You say that 2 infinite sets have the same cardinality if there is one bijective function that relates them. If you can define a 1 to 1 relationship between them, then you say they have the same cardinality.

For ex., let's take the natural numbers (1,2,3,4,5,...)

and the integers (... ,-5,-4,-3,-2,-1,0,+1,+2,+3,+4,+5, ...)

We could define a relationship like

1,2,3,4,5,... <-> +1,+2,+3,+4,+5,...

and it would seem that there are more integers that natural numbers (all the negatives and zero are left out), but we can find one relationship 1 to 1:

1,2,3,4,5,6,7,8,9,10,11... <-> 0,+1,-1,+2,-2,+3,-3,+4,-4,+5,-5,...

Since we can find this 1 to 1 relationship we say they have the same cardinality.

The same happens in the previous example with the integers and the rationals. We can build a relationship 1 to 1, so they also have the same cardinality (which is the similar to say they have the same number of elements in the case of finite sets).


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## pgc01 (May 19, 2010)

sulakvea said:


> there are EQUAL number of integers and rational numbers.
> 
> however, there are MORE irrational numbers (like pi(), sqrt(2), etc) than either integers or rationals. the proof is actually pretty simple, google Cardinal Numbers


 
Hi Sulakvea

If you are talking about the cardinality of infinite sets then rational/irrational is not a good division for the real numbers, it makes more sense algebraic/transcendent numbers.

In fact the algebraic numbers also have the same cardinality as the natural numbers, but they include irrationals like the sqrt(2) that you mentioned. Pi, on the other hand, is a transcendent number, and the cardinality of the transcendent numbers is indeed superior to the cardinality of the natural numbers.


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## Joe4 (May 19, 2010)

pgc,

I think I was thinking along the same lines as you.  For an infinite set, I don't see how there can be more integers for rationals.  For every integer other than 1 and 0, there are at least distinct rational numbers, i.e.

Working through any integers "n" except 1 and 0, if you take 
*1/n* 
and 
*(n-1)/n*
Those rational numbers will never be repeated (that is, you can come up with two unique rational numbers for each integers not equal to 1 and 0).

This would seem to suggest that there are more rational numbers than integers.


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## pgc01 (May 19, 2010)

Joe

the number of the rationals is, in fact, the same as the one of the integers (in the sense of the cardinality of infinite sets).

To prove that we have to find a relationship 1 to 1 between the 2 sets.

This is a proof based on the work of Georg Cantor that shows such a relationship between the set of natural numbers and the positive rationals.

Step 1 -
We start by building a table with all the rationals

- in each row the fractions have the denominator equal to the row index
- in each column the fractions have the numerator equal to the column index

As an example, these would be the first rows and column of the table:


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BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px" width=30>B</TH><TH style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px" width=30>C</TH><TH style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px" width=30>D</TH><TH style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px" width=30>E</TH><TH style="BORDER-BOTTOM-COLOR: #888888; 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PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*2*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1/2</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; 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BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">3/3</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">4/3</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">5/3</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">6/3</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*4*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1/4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">2/4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">3/4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">4/4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">5/4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">6/4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*5*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1/5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">2/5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">3/5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">4/5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">5/5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">6/5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*6*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1/6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">2/6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">3/6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">4/6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">5/6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">6/6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*7*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="PADDING-LEFT: 1em; BACKGROUND: #9cf" colSpan=8>[Book1]Sheet1</TD></TR></TBODY></TABLE>

Notice that although the table has all the positive rationals, we have duplicates, since 1/2=2/4=4/8=8/16 ...


Step 2 
We get rid of duplicates. To do that we simply exclude all fractions that are not simplified.


<TABLE style="BORDER-BOTTOM-COLOR: #cccccc; BORDER-RIGHT-WIDTH: 2px; BORDER-TOP-COLOR: #cccccc; BORDER-COLLAPSE: collapse; FONT-FAMILY: Arial,Arial; BACKGROUND: #fff; BORDER-TOP-WIDTH: 2px; BORDER-BOTTOM-WIDTH: 2px; BORDER-RIGHT-COLOR: #cccccc; FONT-SIZE: 10pt; BORDER-LEFT-COLOR: #cccccc; BORDER-LEFT-WIDTH: 2px" border=1 cellPadding=1><TBODY><TR><TH style="BORDER-BOTTOM-COLOR: #888888; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px"></TH><TH style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px" width=30>A</TH><TH style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; BACKGROUND: #9cf; 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TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px" width=30>F</TH><TH style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px" width=30>G</TH></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*1*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">2</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">3</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*2*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1/2</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #ff0000; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">2/2</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">3/2</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #ff0000; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">4/2</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">5/2</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #ff0000; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">6/2</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*3*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1/3</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">2/3</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #ff0000; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">3/3</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">4/3</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">5/3</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #ff0000; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">6/3</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*4*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1/4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #ff0000; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">2/4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">3/4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #ff0000; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">4/4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">5/4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #ff0000; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">6/4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*5*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1/5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">2/5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">3/5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">4/5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #ff0000; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">5/5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">6/5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*6*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1/6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #ff0000; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">2/6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #ff0000; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">3/6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #ff0000; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">4/6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">5/6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #ff0000; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">6/6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*7*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="PADDING-LEFT: 1em; BACKGROUND: #9cf" colSpan=8>[Book1]Sheet2</TD></TR></TBODY></TABLE>

Now the table has all the rationals without duplicates

Step 3
We make this 2D list into a 1D list. We do this going through the 45 degrees diagonals, from right to left, starting at the top left corner.

We write those values in column B and in column A we write the natural numbers

We get:


<TABLE style="BORDER-BOTTOM-COLOR: #cccccc; BORDER-RIGHT-WIDTH: 2px; BORDER-TOP-COLOR: #cccccc; BORDER-COLLAPSE: collapse; FONT-FAMILY: Arial,Arial; BACKGROUND: #fff; BORDER-TOP-WIDTH: 2px; BORDER-BOTTOM-WIDTH: 2px; BORDER-RIGHT-COLOR: #cccccc; FONT-SIZE: 10pt; BORDER-LEFT-COLOR: #cccccc; BORDER-LEFT-WIDTH: 2px" border=1 cellPadding=1><TBODY><TR><TH style="BORDER-BOTTOM-COLOR: #888888; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px"></TH><TH style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px" width=30>A</TH><TH style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px" width=30>B</TH><TH style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px" width=30>C</TH></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*1*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*2*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">2</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">2</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*3*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">3</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1/2</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*4*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">3</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*5*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1/3</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*6*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*7*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">7</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">3/2</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*8*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">8</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">2/3</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*9*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">9</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1/4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*10*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">10</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*11*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">11</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1/5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*12*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">12</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*13*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">13</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">5/2</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*14*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">14</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">4/3</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*15*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">15</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">3/4</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*16*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">16</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">2/5</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*17*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">17</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">1/6</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*18*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: left; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">...</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">...</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="BORDER-BOTTOM-COLOR: #000000; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #000000; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BACKGROUND: #9cf; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #000000; BORDER-LEFT-COLOR: #000000; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em">*19*</TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: center; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD><TD style="BORDER-BOTTOM-COLOR: #888888; TEXT-ALIGN: right; PADDING-BOTTOM: 0.25em; BORDER-RIGHT-WIDTH: 1px; BORDER-TOP-COLOR: #888888; PADDING-LEFT: 0.5em; PADDING-RIGHT: 0.5em; BORDER-TOP-WIDTH: 1px; BORDER-BOTTOM-WIDTH: 1px; BORDER-RIGHT-COLOR: #888888; BORDER-LEFT-COLOR: #888888; BORDER-LEFT-WIDTH: 1px; PADDING-TOP: 0.4em"></TD></TR><TR><TD style="PADDING-LEFT: 1em; BACKGROUND: #9cf" colSpan=4>[Book1]Sheet3</TD></TR></TBODY></TABLE>

Now, it's clear that the list in column B will include all the rationals and we were able to establish a relationship 1 to 1 between the set of the natural numbers and the set of the positive rationals. This means that we can say that they have the same cardinality.

I hope this is clear as it's past bed time.


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## Joe4 (May 19, 2010)

I guess its been too long since I was in college... you lost me on step three. The way I see it, you have gone through 6 integers so far, and come up with 17 rational number. So I guess I am not understanding the 1-1 relationship.

I tend to try to approach this things from a mere logical standpoint, and maybe that is what gets me in trouble. Can you spot the flaw in my logic?

Starting with integer number 3, I can come up with two rational numbers for each number, and never repeat one (namely, 1/n and (n-1)/n), i.e.

3: 1/3 and 2/3
4: 1/4 and 3/4
5: 1/5 and 4/5
6: 1/6 and 5/6
7: 1/7 and 6/7
8: 1/8 and 7/8
...

So it seems to me that excluding 0,1, and 2, for any given integer, I can come up with at least two unique rational numbers, which would lead me to believe that there are more rational numbers than integers.

I'm sure that Cantor and you are correct, and I am just missing something - I am just failing to see it. I have to admit that it has been years since I worked out proofs like these and I have pretty much forgotten what things like "cardinality" mean.

Alas, it is time for me to go to bed also.


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## Joe4 (May 19, 2010)

Just had one thought here before I go to bed.

Is my problem that I am trying to apply concepts on finite set logic to infinite sets?  Dealing with infinite sets kind of changes the rules and logic a bit.


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## Joe4 (May 19, 2010)

Here's one that is more in my "wheelhouse". Many people have a hard-time understanding the Monty Hall Paradox, and the concept that given this scenario, you should always switch your choice (to increase your odds of winning).


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## pgc01 (May 20, 2010)

Joe4 said:


> The way I see it, you have gone through 6 integers so far, and come up with 17 rational number. So I guess I am not understanding the 1-1 relationship.


 
If you imagine extending the table to infinity, both in the number of rows and in the number of columns, you could always

- given a whatever natural number n, there's 1 (and 1 only) corresponding rational number (the one in column B, row n)

- given a whatever rational number n, there's 1 (and 1 only) corresponding natural number (we look up the rational number in column B and get the corresponding natural number in column A)


Notice that it does not mean that by ordering the sets differently you cannot get to other conclusions. In your example it seems that there are more rationals, in Oaktree's example it seems there are more integers. In my other example with Integers and Naturals with 2 different ways of ordering the sets I got 2 different conclusions.

This is usual in infinite sets. If you order them differently you can get contradictory conclusions. The rule is then to consider the sets to have the same cardinality if you can find at least 1 way to order the sets so that they have a bijective relationship, meaning a 1 to 1 relationship in both senses.

So, in this case, since we can find at least 1 way to order the sets (the one I posted in step 3) that establishes a correspondence 1 to 1 in both senses betweeen the 2 sets, we say they have the same cardinality.


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## pgc01 (May 20, 2010)

On a lighter note, when you add odd numbers, starting by 1, you always get squares:

1 = 1<sup>2</sup>
1 + 3 = 2<sup>2</sup>
1 + 3 + 5 = 3<sup>2</sup>
1 + 3 + 5 +7 = 4<sup>2</sup>
1 + 3 + 5 +7 + 9 = 5<sup>2</sup>
...


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## Expiry (May 20, 2010)

pgc01 said:


> On a lighter note, when you add odd numbers, starting by 1, you always get squares:
> 
> 1 = 1<sup>2</sup>
> 1 + 3 = 2<sup>2</sup>
> ...



Another way to always get squares, is to open up a discussion on Maths Trivia. Ho ho ho.


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## schielrn (May 20, 2010)

Expiry said:


> Another way to always get squares, is to open up a discussion on Maths Trivia. Ho ho ho.


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## Joe4 (May 20, 2010)

> Another way to always get squares, is to open up a discussion on Maths Trivia. Ho ho ho.


As Huey Lewis used to say, "its hip to be square"!


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## Oaktree (May 20, 2010)

I have no doubt that you're right, Pedro.  

But this:





> That's why in mathematics, when you are working with infinite sets you must use other rules.


 is when I realized I didn't want to go further down the path of obscurity that is higher level math.  IMO, math divorced from logic leads to crazy people.

The set of rationals includes integers + other things, but the set of rationals is the same size as the set of integers.  That makes more sense


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## Joe4 (May 20, 2010)

> is when I realized I didn't want to go further down the path of obscurity that is higher level math.


Amen to that, brother!  I was a math major myself in college, but when you get into some of that "high" level stuff like dealing with infinite sets, it can really blow your mind!  My "logical" mind doesn't like that.

Of course, it has also been 16 years since I looked at anything like this, so I have probably forgotten most of what I learned anyway!


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## erik.van.geit (May 20, 2010)

*Monty Hall Paradox spreadsheet*

Copy down as much rows as you like.
This result was obtained using 24.000 rows.

```
A   B          C                 D              E      F    G               
 1 car 1st choice remaining goat(s) opened by host switch win? probability win 
 2 2   1          3                 3              2      1    65,81999%       
 3 3   3          12                2              1      0                    
 4 2   1          3                 3              2      1                    
 5 3   1          2                 2              3      1                    
 6 2   2          13                1              3      0                    
 7 2   3          1                 1              2      1                    
 8 2   3          1                 1              2      1                    
 9 1   1          23                3              2      0                    
10 2   1          3                 3              2      1                    
Blad1
[Table-It] version 09 by Erik Van Geit
```


```
RANGE   FORMULA (1st cell)
A2:B10  =RANDBETWEEN(1,3)
C2:C10  =SUBSTITUTE(SUBSTITUTE("123",A2,""),B2,"")
D2:D10  =IF(LEN(C2)=1,C2,MID(C2,RANDBETWEEN(1,2),1))
E2:E10  =--SUBSTITUTE(SUBSTITUTE(123,B2,),D2,"")
F2:F10  =IF(A2=E2,1,0)
G2      =SUM(F:F)/(COUNTA(F:F)-1)
[Table-It] version 09 by Erik Van Geit
```
kind regards,
Erik


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## pgc01 (May 20, 2010)

> is when I realized I didn't want to go further down the path of obscurity that is higher level math. IMO, math divorced from logic leads to crazy people.


 
Oaktree, Joe

I'm sorry, you are right. It's really bad wording, it seems as if I was saying that you should not use logical thought which I agree would make the person crazy.

What I mean is that if you are dealing with infinite sets you must be very careful in the way you test and interpret your tests related to comparisons of "number of elements".

In fact, in plain English, does a sentence like this one make sense?



> ... you have a 1:many relationship of rationals:integers, which inherently shows that the set of integers is larger.


 
We are saying that both sets are infinite. What does it mean one of them is larger? If we lookup "infinite", for ex. in MW: " immeasurably or inconceivably great or extensive ", in plain English it's difficult to understand, or to accept, that an infinite set is bigger or has more elements than another infinite set.


This means that if we really want to make this type of comparisons between infinite sets we must first define what we mean by being larger or having the same number of elements in this context. And that's where the mathematics of infinite sets enters, it's not that it's in someway preverting the usual rules of logic, that would make no sense. It simply lays out some definitions and creates a terminology that allows us to communicate as far as this "size" comparisons are related.

I personally think that the notion of the cardinality is quite simple and down to earth: If we can define a 1:1 relationship (a bijection) between the 2 sets, we say they have the same cardinality. This seems to me a good definition to allow us to say that 2 infinite sets are "the same size". In fact this relationship could also be applied to finite sets, to say that they have the same number of elements.

So when I say that the usual rules don't apply I simply mean that, in this context, an infinite set may be the same size as a part of it.

Ex: the set of naturals has the same cardinality ("is the same size") as the multiples of 3.

1,2,3,4,5...
3,6,9,12,15..

We can easily establish the relationship 1:1, in this case n<->3n. This means that we say that they have the same cardinality.

This is an example that shows that the usual rules for finite sets do not apply, we would be tempted to say that the first set is larger than the second one, as it includes all the elements of the second set and some more. But these are infinite sets and we can also easily understand that, in fact, for each element of the first set there's one corresponding element of the second set, so it makes sense (in this context of infinite sets) to say they are "the same size".

Since I cannot hope this is less boring, I hope it is at least clearer (if someone got this far ).


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## Domski (May 20, 2010)

Expiry said:


> Another way to always get squares, is to open up a discussion on Maths Trivia. Ho ho ho.



I got lost on page two when the cardinal became infinitely rational. We all know he was a very bad man and hated the musketeers!!!

Dom


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## Joe4 (May 20, 2010)

pedro,

I think I am beginning to see the light and it is coming back to me.  I think I was trying to impose finite logic to infinite sets by focusing on specific values, one a time. 

In a finite world, like say from 1-100, there are certainly more instances of integers "n" than "3n" in your example.  But in an infinite world, that is not true for the reasons you explained.  Its like asking which set of infinity is larger, when they have the same cardinality.

Thanks for helping me see the light again.  On the surface, it seems illogical and can make your head spin, but if you dive into it deep enough, it begins to make sense.


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## RobMatthews (May 20, 2010)

Joe4 said:


> Here's one that is more in my "wheelhouse". Many people have a hard-time understanding the Monty Hall Paradox, and the concept that given this scenario, you should always switch your choice (to increase your odds of winning).


 
I have nothing to add, except that I love:



			
				Wikipedia said:
			
		

> We also need to assume that winning a car is preferable to winning a goat for the contestant.


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## Domski (May 20, 2010)

RobMatthews said:


> Originally Posted by Wikipedia
> We also need to assume that winning a car is preferable to winning a goat for the contestant.



I'd sooner have a goat at the moment the price fuel is and, bonus, it would save on the lawn cutting.

Dom


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## ZVI (May 20, 2010)

1 * 9 + 2 = 11
12 * 9 + 3 = 111
123 * 9 + 4 = 1111
1234 * 9 + 5 = 11111
12345 * 9 + 6 = 111111
123456 * 9 + 7 = 1111111
1234567 * 9 + 8 = 11111111
12345678 * 9 + 9 = 111111111
123456789 * 9 +10= 1111111111


9 * 9 + 7 = 88
98 * 9 + 6 = 888
987 * 9 + 5 = 8888
9876 * 9 + 4 = 88888
98765 * 9 + 3 = 888888
987654 * 9 + 2 = 8888888
9876543 * 9 + 1 = 88888888
98765432 * 9 + 0 = 888888888 

1 * 8 + 1 = 9
12 * 8 + 2 = 98
123 * 8 + 3 = 987
1234 * 8 + 4 = 9876
12345 * 8 + 5 = 98765
123456 * 8 + 6 = 987654
1234567 * 8 + 7 = 9876543
12345678 * 8 + 8 = 98765432
123456789 * 8 + 9 = 987654321


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## ZVI (May 20, 2010)

Have you ever asked the question why digit 1 is "one", 2 is "two", 3 is "three" and so on?
The logic was initially based on amount of the digit's angles.
Try to count up the amount of each figure's corners in the image below.


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## erik.van.geit (May 21, 2010)

ZVI said:


> Have you ever asked the question why digit 1 is "one", 2 is "two", 3 is "three" and so on?
> The logic was initially based on amount of the digit's angles.
> Try to count up the amount of each figure's corners in the image below.


Heared that before; is there proof for that statement? It seems to me quite "forced" (don't know a better english word for this), especially when looking to 7 and 9.


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## Andrew Fergus (May 21, 2010)

For multiplying 2 digits numbers by 11: split the number into two and insert the sum of the 2 digits into the middle.

For instance: 11 x 34, split 34 into 3__4 and insert the sum of the two digits (3+4=7) into the middle, giving 374.

If the sum of the two digits is greater than 9, then carry the 1 onto the first digit.  For instance: 11 x 78 --> 7__8, inserting 15 becomes add 1 to the front giving 8__8 and insert the 5 giving 858.

It's easier for the 2 digits smaller numbers, and pretty logical when you can see what it is doing.

_____________________________________________

Here is an abridged proof as to why there is no maximum prime number:

Multiply all known primes (let's call this set of numbers z) and add 1 the product which gives us either:
a) a new prime number, or
b) a number for which one of it's factors is a prime number greater than the maximum prime number in the set z.

For example, assuming we are new to primes and our universe of prime number extends to 11: 2x3x5x7x11+1 = 2311 which is a prime number - i.e. we found a prime number higher than the highest known primes in our known set (being 11).

If our known universe of primes was extended to 13, 2x3x5x7x11x13+1=30031.  30031 is not a prime number, but one of it's factors is 59, which is a new prime number greater than the largest known prime used of 13.  In short, this is why there is no maximum prime number.

Admittedly once you get beyond a googol, the numbers have very little practical use, if at all!


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## pgc01 (May 21, 2010)

Thanks Joe. What I also find interesting is that we're like extending the meaning of the concept of infinite in plain English giving it a richer meaning by adding like higher degrees of infinite. It does make the head spin.


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## ZVI (May 21, 2010)

erik.van.geit said:


> Heared that before; is there proof for that statement? It seems to me quite "forced" (don't know a better english word for this), especially when looking to 7 and 9.


Hi Erik, 
Agree with you that it seems more amusing than historically proved 
Vladimir


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## pgc01 (May 21, 2010)

One I always found very interesting

We know that a rational to the power of another rational can have a rational result, like

2<SUP>2</SUP> = 4

We also know that an irrational to the power of a rational can have a rational result, like

sqrt(2)<SUP>2</SUP> = 2

But can an irrational to the power of an irrational have a rational result?

Suprisingly the answer is yes, it has been proven that such pairs of irrational numbers exist!


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## Joe4 (May 21, 2010)

> But can an irrational to the power of an irrational have a rational result?
> 
> Suprisingly the answer is yes, it has been proven that such pairs of irrational numbers exist!


Pedro,

That is very interesting!  Do you have any examples, or links to an example?  I would love to see one.


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## pgc01 (May 21, 2010)

Joe

I don't remember any example but I remember the proof, and it's very simple. It's also funny because it proves that such a pair exists but we  don't know which.

Using the known property of power to power (a^b)^c = a^(b*c)

(sqrt(2)^sqrt(2))^sqrt(2) = sqrt(2)^(sqrt(2)*sqrt(2)) = sqrt(2)^2 = 2

Using just the 2 extremes of the expression

(sqrt(2)^sqrt(2))^sqrt(2) = 2

One of 2 things must be true:

- either sqrt(2)^sqrt(2) is rational and then our pair is (sqrt(2),sqrt(2))

- or sqrt(2)^sqrt(2) is irrational and then our pair is (sqrt(2)^sqrt(2),sqrt(2))

either way there's one pair (irrational,irrational) that satisfies irrational^irrational = rational

I'm sure there are works done on this but I never looked for any, I just remember this because the first time I saw it I thought it was an interesting result.


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## baitmaster (May 25, 2010)

amaze your friends with your ability to divide by 7 - I used this trick at a restaurant and everyone was shocked that I could split the bill to infinte decimal places...

1/7 = 0.142857 recurring
2/7 = 0.285714 " "
etc

series shifts and repeats consistently, and is easily remembered by the multiples of 7 within it, and the multiples of 14 that start it...


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## baitmaster (May 25, 2010)

To multiply 2 numbers using only 2s:

1) Write both numbers.
2) Halve one number, and double the other. Ignore remainders
3) Write these under the originals, and repeat step until you reach 1
4) Add up the numbers on the "doubled" side, that correspond to any _odd_ values on the "halved" side, including the original numbers if appropriate

The result is the multiple.

If you stick a 1 next to the numbers you included, and 0 against those you didn't, then read bottom to top, thats the answer in binary


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## Andrew Fergus (May 25, 2010)

baitmaster said:


> amaze your friends with your ability to divide by 7 - I used this trick at a restaurant and everyone was shocked that I could split the bill to infinte decimal places...
> 
> 1/7 = 0.142857 recurring
> 2/7 = 0.285714 " "
> ...


Cool.  This got me thinking......

Check out the sequence for 13ths - there are two different repeating series, and note the pattern as to which series is the relevant series.

And 17ths seems to be like the series for 7ths - just a little harder to remember.


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