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Algebra in Pictures

Today, let’s do something amazing to see how ordinary formal calculations can be replaced with pictures, or diagrams.

Antonio Salieri, an Italian classical composer, proposed to “check harmony by algebra.” But can we do the opposite and put algebra itself to a test? What if we attempt to put equations into pictures? This is certainly possible (and, no, we won’t be covering our calculations in rosebuds).

To get into the swing of things, let's start with simple equations. Before you know it, we'll get to the complex and unexpected ones. These equations are hard to believe. In fact, they are not easily proven even with the use of algebra. You may notice that all of our pictorial proofs are quite short. Rather than giving you lengthy text peppered throughout with tedious calculations, these proofs feature essentially just one phrase: “carefully examine the following diagram…”

commutative property of multiplication

Let’s begin with the simplest example.

This is something that we have learned early on in school – the commutative property of multiplication. We are so accustomed to this property that most us never question why, in fact, this is so. a × b = b × a (that is, a + a + ... + a (b times) equals b + b + ... + b (a times)) Meanwhile, this property can be easily demonstrated using geometry, as shown.


Many of you will recognize which equations the next set of pictures refer to!

Here, a large square is divided into two smaller squares and two rectangles. The area of the large square equals to the combined areas of the four smaller shapes. If we calculate the areas of these shapes, then we will arrive at the special product equation: (a + b)^2 = a^2 + 2ab + b^2.


In the next two diagrams, we see two more equations familiar to us from school.

(a - b)^2 = а^2 - 2ab + b^2 and a^2 - b^2 = (a + b) × (a - b)


This one provides us an easy way of finding an equation for the sum of the cubes

Don’t you think that all of this is quite elegant? We have examined a mathematical equation using two completely different approaches and reached the same solution. Now we are ready for the more difficult, and hence more interesting, problems. It is time to draw some more!

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