### The

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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