Quote:
Originally Posted by hmmmcurious
Neat trick:
Take your phone number (or any number for that matter) call it X,
re-arrange the digits, call this number Y.
Then max{X,Y}-min{X,Y} is always divisible by 9.
Example,
X=4161234567
Y=1234164576
X-Y= 325229999*9
Try it out!
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I think there's a rather simple argument actually, involving Modular Arithmetic...the hardest part is explaining the terminology.
So first I will summarize:
1) X and Y have the same remainder when dividing by 9 since rearranging digits does not change their remainder (This ONLY works with 9...)
2) The remainders cancel when you compute X - Y
3) Your new number, X - Y has 0 remainder when dividing by 9...which means it's a multiple of 9.
And now I will post the argument / explanation:
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First, when I say the word 'mod' it's an operation that returns the
remainder when dividing by that number. So 6 = 0 mod 3, because 6 / 3 = 2, with 0 remainder...similarly 11 = 2 mod 3, because 8 / 3 is "3 and a little bit" and has 2 remainder.
So the claim is that if you take X, rearrange the digits to make Y, take the difference and divide by 9 you'll get an integer. This means precisely that X and Y have the same remainder when divided by 9...because when you subtract the 2 numbers, "the remainders disappear."
Example: 38 and 47 have the same remainder when divided by 9. That is:
38 = 9 * 4 + 2 (remainder)
47 = 9 * 5 + 2 (remainder)
When I subtract them:
47 - 38 = (9*5 + 2) - (9*4 + 2) = (9 * 5 - 9*4) + (2 - 2) = 9*1 (the 2's 'cancel'), which is a multiple of 9. This works whenever the 2 numbers have the same remainder.
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So if I can argue that rearranging the digits of your phone number does nothing to its remainder when divided by 9, then we're done...because the remainders will cancel like in the example.
Infact it's even better.... You can rearrange the digits of
ANY number, subtract it from the original and get a multiple of 9.
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I won't prove this, but I'll try to show you some neat things which lead to seeing why this works. To find your remainder when dividing by 9:
1) Add up the digits of your number. If your number is less than 9, this is your remainder (ie. if your number is 4, it clearly has remainder 4 when divided by 9).
2) If your number is greater than 9...then look at that number, and add up its digits.
3) Continue this until your number is less than 9.
Example: Suppose your number is 129302392020.
With the aid of a calculator, we can see that 129302392020 = 9 * 1436932446 + 6 (that is, it has 6 remainder when divided by 9)
My algorithm produces the same answer:
1) Add the digits: 1 + 2 + 9 + 3 + 0 + 2 + 3 + 9 + 2 + 0 + 2 + 0 = 33
2) 33 > 9, so add the digits of 33: 3 + 3 = 6.
3) 6 < 9...and is the remainder we want!
Try this out for yourself if you're bored, or take my word for it xD
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Now, why is this algorithm enough to convince you that the remainder doesn't change when you rearrange the numbers?
...
All we're doing is adding the digits...and it doesn't matter what order we add them in! If I add 1 + 2 + 3 + 4, it's the exact same as adding 3 + 2 + 4 + 1. (:
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Quote:
Originally Posted by hmmmcurious
Another math one:
There are infinitely many different sizes of infinity.
This is somewhat involved (maybe Mowicz will explain :wink: ), but the set of natural numbers has a different size than the set of real numbers, although both are infinite.
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Here are some little 'facts' about infinities, that are definitely counter-intuitive:
1) There are exactly the same number of Natural numbers (1,2,3,...) as there are Integers (..., -3, -2, -1, 0, 1, 2, 3,...)
2) We can never count all the (real) numbers between the numbers 1 and 2. Infact, if we try and make any listing, we can construct a new real number which is not in the list (this is how we prove such a thing). But we can make a list of the Natural Numbers...1,2,3 and so on.
This is why there are 'more' Real numbers than natural numbers.
3) There are exactly the same amount of (real) numbers between the numbers 1 and 2, as there are between 1 and 100,000,000 (or even from -infinity to infinity!)
4) There are exactly the same amount of points in the real line as there are in the plane...or 3-D space...or n-D space!
So I've talked about two sizes of infinity...why are there infact Infinitely many? Well, this concept is beautifully explained with pictures...if anyone is curious, come by my office in September and I'll draw it up on the blackboard. (:
Here are some references for the curious:
http://en.wikipedia.org/wiki/Cardinal_number
http://en.wikipedia.org/wiki/Continuum_hypothesis
