Playing with math keeps me sharp

[1MileCrash]

"Playing" with math keeps me sharp!

I'm pretty new to math, just taking Calc II and a proof class.

But every once in a while, while doing homework, I get obsessed with looking for patterns in math and expressing them. They are usually pointless, and come out to be kind of "duh" in the end, but man, I feel like I can do anything after doing that. It's fun to do.

Maybe one day, I'll come up with something useful. Here is a walkthrough of what I did tonight.

During Physics homework, I got "reminded" of the idea that a square has the most area per perimeter of any rectangle. Ergo, decreasing one side and increasing the other will give a smaller area. So I looked for a pattern there.

10 x 10 = 100
9 x 11 = 99
8 x 12 = 96
7 x 13 = 91
6 x 14 = 84
5 x 15 = 75
...

The numbers decrease from the next in the pattern of odd numbers, 1, 3, 5, 7, 9, (100 - 99 = 1; 99 - 96 = 3; 96 - 91 = 5)

So I looked for a pattern there, by adding up those numbers. For example, 5 x 15 is (1 + 3 + 5 + 7 + 9 = 25) away from 100. After writing all of those:

10 x 10 = 100; 0
9 x 11 = 99; 1
8 x 12 = 96; 4
7 x 13 = 91; 9
6 x 14 = 84; 16
5 x 15 = 75; 25
...

I quickly realized that those numbers (1, 4, 9, 16) are the difference that either of the terms are from 10, squared. So 8 x 12 is 4 away from 100, and 12 and 8 are 2 away from 10, then square that and you get 4.

So I expanded it to say that those numbers just average to a certain number, in this case 10. So, I then noted that multiplication of two numbers is equal to the average of the two numbers, squared, minus the difference either number is from the average, squared!

Or written out:

http://img28.imageshack.us/img28/1725/conjecture01.png


Tomorrow I'm going to do something with cubes, AKA ABC.





Do you guys "play" with math in this way?

 

[nucl34rgg]

Yes! This can be very fun, and will become useful to help you get a firm intuitive grasp of elementary algebra. (What is the equation "really saying?") It may also be useful for number theory if you plan to continue in math.

Here is something you may find amazing. (I thought it was awesome!) See if you can find an expression for the number of positive divisors of a given positive integer. Hint: the fundamental theorem of arithmetic says we can factor positive integers greater than one into a unique product of primes raised to some powers. What would these exponents have to do with counting the number of divisors for a given number? What properties do the divisors of a number share with that particular number? :D

If you want to check if your expression is right, look up "euler tau function." Also, you would probably enjoy this page as well: http://en.wikipedia.org/wiki/Divisor_function which shows you cool tricks for summing the different powers of divisors of a given number. I hope you enjoy your future adventures in math!

 

[ander]

This is awesome. I started doing this while I was on medical leave from college. I hadn't done any schoolwork in mathematics for years and I just started playing with numbers, writing them down in different ways and trying to find patters. I came across something while I was staring at the standard factored quadratic (x + a)(x + b) = c and multiplied out x^2 + (b+a)x + ab = c, just started plugging in numbers and for the first time realized that it can be used to calculate the product of any two numbers that hover around a common base. For example, 96 and 103:

(96)*(103) = (100 - 4)*(100 + 3) = 10,000 + (-4 + 3)*(100) + (-12)

= 10,000 - 100 -12 = 9888

Like you say now it seems obvious but for years I'd been manipulating x and y and setting it all equal to zero without thinking that it actually could describe real numbers. Playing around with this let's you multiply two or three digit numbers pretty fast in your head.

Another day I was stuck in traffic and started to look at the difference between perfect squares:

1 - 0 = 1
4 - 1 = 3
9 - 4 = 5
16 - 9 = 7
25 - 16 = 9
36 - 25 = 11

. . . and for the first time I realized that it was just the odd numbers 1, 3, 5, 7, 9, 11. A bit like your bit but less developed.

Great stuff, glad you reminded me!

 

[Edgardo]

You both discovered a related theorem:
The sum of the first n odd numbers always yields the square number n².

There is a wonderful, intuitive explanation for this which you can read here.

If you are interested in number theory puzzles have a look at http://nrich.maths.org/public/
Some puzzles:
Number Rules - OK
Number Tracks

 

[silentbob14]

Amazing ;] Sometimes I like to proove some theory or equation with as little maths as possible. As if going back in time and trying to "feel" how some equations were derived for the first time hundreds of years ago.

 

[ander]

You both discovered a related theorem:
The sum of the first n odd numbers always yields the square number n².

There is a wonderful, intuitive explanation for this which you can read here.

If you are interested in number theory puzzles have a look at http://nrich.maths.org/public/
Some puzzles:
Number Rules - OK
Number Tracks

Great! thanks for this.

 

[Curl]

OP, when I read the title I was going to come in and make fun at you but that is some neat stuff, I just want to say props and keep it up. Definitely have a good eye and a nice ability to see numbers in your head.