Maths trivia

[pgc01]

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 ).

 

[Domski]

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

 

[Joe4]

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.

 

[RobMatthews]

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:

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

 

[Domski]

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

 

[ZVI]

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

 

[ZVI]

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.

 

[erik.van.geit]

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.

 

[Andrew Fergus]

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!

 

[pgc01]

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.