The

05Nov06

Kate’s Journal: The Beauty of Mathematics:

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

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

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

Brilliant, isn’t it?
And finally, take a look at this symmetry:

1 x 1 = 1
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
11111 x 11111 = 123454321
111111 x 111111 = 12345654321
1111111 x 1111111 = 1234567654321
11111111 x 11111111 = 123456787654321
111111111 x 111111111=123456789 87654321

I’ve stolen this rather shamelessly, and I apologize to the original author. Surely it took time to type. But while I find this kind of thing interesting and aesthetically beautiful, the real beauty of mathematics comes not from how things look, but why they work. For example, take the last “pyramid” in the above display.

We have

1×1=1
11×11=121
111×111=12321

This isn’t magic, it has to do with the polynomial expansions of these numbers.

1×1=1, obviously.

11×11=(10+1)(10+1) is a quadratic. When we expand it, we get

10(10)+10(1)+1(10)+1(1)

or

10×10 + 2x10x1+1×1

which has a certain symmetry simply because there is a 1 in the 100’s place, a 2 in the 10’s place, and a 1 in the 1’s place.

Looking at 111×111, we have (100+10+1)x(100+10+1), a third-degree polynomial. The expansion, generically, is (a+b+c)*(d+e+f)=a d + b d + c d + a e + b e + c e + a f + b f + c f, so wehave a 1 in the 10,000’s place, a 2 in the 1,000s place, a 3 in the 100’s place, a 2 in the 10’s place and a 1 in the 1’s place, or 12321. In general, we have a polynomial of the form (a_n+a_{n-1}+…+a_1)^2. Each increment of n adds a new unique squared term, and we always have a unique one on the end because a_1 always equals 1. So incrementing n by one adds one of each term from the previous expansion for every all but the largest and smallest terms. Notice:

1
121
12321
1234321

Each term in each expansion is obtained by adding down the previous column. Now that’s beautiful.

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