Addition of Squared Numbers, Its Interesting Facts

Adding of squared numbers has many "side" stories besides the usual Pythagorean Theorem.

Looking at the below pattern of adding 2 square numbers., will tell us that it is our friendly Pythagorean Theorem.

3² + 4² = 5²


How about if we double the equation of the Theorem, we will get the below:

2(3² + 4²) = 2 (9 + 16) = 50

However, the resultant 50 can be expressed into 1 + 49, which is also 1² + 7².

Therefore we can easily say that,

2(3² + 4²) = 1² + 7².

Message: Doubling an addition of 2 squared numbers results in another addition of 2 squared numbers.

Another example:
2(1² + 2²) = 1² + 3²

Why is this so?
What is the underlying principle of the finding?

To answer the questions, we need to understand the below.

What is the actual meaning of "squaring" a number?

It is to find the area within the square region bounded by the number as its side. Therefore the addition of 2 square numbers is simply the sum of 2 square areas.

We also know that addition of 2 square numbers is Pythagorean Theorem at work, and is actually the addition of 2 areas in a square. Pythagorean Theorem also states that this addition of 2 smaller square numbers results in a bigger square area.

Thus, by doubling the resulting bigger square area (as done at the beginning of this post), we still obtain another square area, but only double in size. A square area, no matter what, always has the area format of A x A, with A being its side unit.

By reversing the results of Pythagorean Theorem, starting from the bigger area, we can logically expect another set of square numbers having the same added size. This explains why doubling the sum of 2 squared numbers can form yet another equation of 2 squared numbers added up.

The knowledge presented here, has application in the engineering area in that surface area of an object can be re-sized by doubling or reducing by half and still keeping the same proportion relatively.

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