"Beauty of Mathematics" Topic

Here's something to make you think…

link

Pete

That is pretty

and pretty amazing

That is downright elegant.

Tom

Not Danica? link

I was way too old to finally understand this for the first time, but I tried to get my kids to memorize the 9 x tables by realizing that the answer to the "9's" is simply taking one number less than the number you are dividing by 9, and adding what's left to equal 9. For example 9 x 7 is 6 + 3, or, 63.

I am always impressed by the symmetry of number progressions. It really does look like "someone" is trying to speak to us through mathematics. Too bad for me, though, as I am functionally illiterate when it comes to maths….

Danica is an attractive young woman, but mathematics will still be beautiful long after we are all dust.

Also sprach Streitax

Cool.

But numbers are scary. Too many secrets of the universe are hidden in them. Some day, some scientist will be doing advanced math, and Cthulhu will pop out and eat us all.

Mathematics is just another language. Beautiful in the hands of a poet, dangerous in the hands of a politician.

I wonder if the last sample, where the results are numeric palindromes, would work in any other system than base 10? I'm too lazy to check myself.

Ralph

"any other system than base 10?"

Yes, but only up to one short of the base number.

Octal:
1111111 ^ 2 = 1234567654321

Hex:
111111111111111 ^ 2 = 123456789ABCDEFEDCBA987654321

Huh? Danica McKellar I understand, not them number things.

Math stuff can be pretty cool sometimes. My kid just discovered number bases, and decided that a base 36 code would be cool. He uses the "hexadecimal" convention of using letters for digits higher than 9, so base 36 holds all the digits and letters in each "place". So he can write words as numbers in base36, then convert them to base10 and then they're "in code".

(This is what I get for teaching him FORTH – you can change the number base that the system works in trivially, and you're not stuck with the usual base10, base8 and base16 choices.)

GreekGreekGreekGreekGreekGreekGreek….

[Huh? Danica McKellar I understand, not them number things.]

You understand women? Wow.

"any other system than base 10?"

Yes, but only up to one short of the base number.

Octal:
1111111 ^ 2 = 1234567654321

Hex:
111111111111111 ^ 2 = 123456789ABCDEFEDCBA987654321

Thanks for that. Now back to base 10 for a question. The palindrome peaks at the number of 1's in the multipliers. So 1111 X 1111 (four 1's) = 1234321 and
1111111111 X 1111111111 (ten 1's)= 1234567890987654321, right?

So how do I calculate 1111 X 1111 in base 9. or in base 11?

Ralph

"1111111111 X 1111111111 (ten 1's)= 1234567890987654321, right"

Nope. The pattern breaks when the number of 1's is greather then or equal to the base. When you reach the base, you have to start carrying the ones (1234567900987654321)

"So how do I calculate 1111 X 1111 in base 9. or in base 11?"

It's the same. Base 5 or more will return the same palindrome for 1111^2. Though the value of the palindrome is different (1234321 means different things in base 8, 9, 10, 16 …).

The beauty of the woman is understandable, not the mind, let's be clear….

The nice thing about arithmatic (apparently professors make the distinction- it's maths when you don't have any digits as values! That really is all Greek.) is the internal consistency. As I pointed out on the language board once
Compare the plurals of
House and Mouse
and the singular of
Dice and Lice.

Maths on the other hand always works the same way. This is great for me, not only do I have a 'number wired' brain, I know the tricks to speed things up.
for instance
any 2 digit number x 11
add the digits together. place result between original digits. If 10+ then put the units in between and add 1 to the first digit
24 x 11.
2+4 = 6
hence 264

75 x 11
7 + 5 = 12
add the 1 to the 7 = 8
=>825

All numbers that divide by three the digits add up to a multiple of 3. If still not sure on that result keep adding until you have 1 digit of either 3 6 or 9

eg
does 40626 divide by 3
4+0+6+2+6 = 18
1+8 = 9
Thus 40626 does divide by 3.
Now I chose a number at random, and multiplied by 3 for this example, to make sure I got a good example. Serendipidously I got a result that also divides by 6. Not only does it divide by three, it is also even. Thus is divides by 6.

PLUS it also divides by 9. This works the same way as '3', but the final result must divide by 9 (or work down to a single 9). So when 40626 added up to 18, I knew it was a multiple of 9- and 1+8 = 9!

It does NOT divide by 4- the final two columns must divide by 4 for this. (If you're not sure, the number must be even, AND if the Tens is even the units must be 0 4 or 8, if Tens column is odd then Units must be 2 or 6. Alternatively just keep taking 20 away until you have 20 or less, and that is the first 5 of the 4xTable. So 94: -80=14. 14 does not divide by 4, thus 94 doesn't. Or 194. Or 273783631812681294 etc etc.)

Thanks, ADub74

It still looks elegant as 12345678900987654321. And it is still a palindrome, too.

Ralph

Base 36 is pretty cool. Not as cool as base 60, but still pretty cool.

Here's another one since some of you seem to like these sorts of things:

The sum of the odd numbers is the series of squares of integers. e.g.

1 (or 1^2) = 1

4 (or 2^2) = 1 + 3

9 (or 3^2) = 1 + 3 + 5

This little relation is also what gives us (indirectly) the shape of the periodic table of elements.