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Squares and square roots

Published: April 9th, 2011 (wednesday), 21:28. Updated: November 11th, 2011 (friday), 18:16

Have you ever wondered why, when we raise something to the second power, we say that we "square" it?

Square numbers

Pythagoreans believed that everything is a number. In a sense that everything in our Universe could be represented and described by numbers. They also related numbers to shapes and geometric figures. So when they spoke about square of some number, they spoke of literally the square as a geometric shape – they were constructing a square with the lenght of its side equal to that number, ant then the area of this square were the result of raising the number to the second power.

Hovewer, besides squares and cubes of numbers, they also knew triangular numbers, pentagonal numbers, hexagonal numbers etc. It is an interesting topic and we'll get back to it time to time. In this lecture nevertheless, we'll deal only with squares for now.

Here's how the firs few perfect squares (those which sides are whole numbers) look like:

First few perfect squares

Red numbers below each of the squares are equal to their areas. They come from multiplying the side of each square by itself. The side of a square we call its square root (or simply just root). And so we have geometrically explained the relation between a second power (square) and a square root! :-)

Thus, raising to the 2nd power means multiplying a number represented by the side of a square by itself (its own multiple) and geometrically it always makes a shape of a square, which area is the result:

Formula (1)
Raising to the second power (squaring).

On the other hand, taking a square root means the reverse operation: finding the side of a known square, having a known area.

Formula (2)
Taking a square root.

Differences of squares

Here it's worth noting a very interesting property of square numbers: If you take any two subsequent squares and subtract from each other, the difference will be always a subsequent odd number! Amazing! How does this mindblowing trick work and how?

Even if you succeeded to proove it algebraically, such a proof probably wouldn't get you anywhere near to understanding, because it's hard to understand it without geometry. Geometry, as always, allows to spot and understand the pattern. So here it is:

Differences of squares are odd numbers

As you see on the animation nearby, each subsequent square is built by attaching a new L-shaped "layer" to the existing square (Pythagoreans called this shape a gnomon, which means "he who knows" – what do you think: why such a name? :->). Every such a shape consists of two parts which are the sides of the original square extended to the outside by one unit (thus 2a), plus one little part – a unit square (1), which together gives us always an odd number (2a + 1), because although these two extended sides are always paired together (which gives us always an even number), this one additional unit square breaks that symmetry ;-)

Such a technique of extending shapes layer by layer will come in handy shortly, when we'll be talking about the differential calculus, so make sure you understand this trick well. But for now, let's get back to our squares.

Short multiplication formulas

Of course nothing stops us from calculating the difference between any two arbitrary sqares, also with non-integer sides. Numbers will not always be odd then, because they even won't be necessarily integers, but some patterns will still remain righteous. We call these patterns short multiplication formulas, because they're "shortcuts" in the process of calculating squares.

Square of a sum

Square of a sum geometrically

Square of a difference

...to be continued...

Difference of squares

...to be continued...


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