Over the years, my daughter, Elizabeth, has shared with me an interest in mathematics. Here is a republishing of a discovery of hers. Mathematics is not just about solving problems set by other people. It is also about discovery and invention.

On March 11, 2002, Elizabeth discovered that the difference between two [consecutive] square numbers is the sum of their square roots.

Another way of looking at this is:

(x + 1)

^{2}= x^{2}+ 2x + 1 = x^{2}+ (x + (x + 1))This can be generalized:

(x + k)

^{2}= x^{2}+ 2kx + k^{2}= x^{2}+ k(x + (x + k))In particular, if x=48 and k=4, we get:

52

^{2}= (48 + 4)^{2}= 48^{2}+ 4(48 + (48 + 4)) = 2304 + 4(100) = 2704Another way of looking at this is geometrically:

As a practical example, we know that 20^{2} is 400.

So, what is 25^{2}?

By Elizabeth's law, the difference between the two squares must be 5 (that is, 25-20) times their sum.

In other words, 25 squared must be 400 + 5(20 + 25).

Regrouping, this is 400 + 5 x 20 + 5 x 25.

So, 25^{2} is 400 + 100 + 125 = 625.

Fun, huh?

Just noticed that another way of writing this is: 400 + (25-20)(25+20), which has a kind of pleasing symmetry.

## 2 comments:

I'll probably have to read that through a few times before I actually get it.

How smart and mathematical of Elizabeth to figure that out :)

Once I understand it fully, I will also have to help Josie to understand (and remember) it. It looks like a helpful thing to know. :)

Wow, Elizabeth! And she was only 15 when she figured that out. Way to go. Also not from Josie.

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