Identity vs Equation - Acute Angel

Last time on Dragon Ball Z in Remark 2.2, we saw the equation

$\displaystyle x+3=x+3$

for which any number x is a solution. This is because the equation is always true no matter what value we assign to the variable x.

You might think it looks frivolous as both sides are exactly identical, and you have the point! In fact, we call an equality or equation that is always true an identity. For instance, the equation

$\displaystyle (x+3)^2=x^2+6x+9$

is an identity because any number x satisfies the equation, as we will see. On the other hand, the equation

$\displaystyle x^2=16$

is not an identity because the only numbers that satisfy the equation are 4 and –4.

As an aside, we sometimes use the triple bar symbol ≡ to indicate identities instead of the usual = symbol. Let’s try to abide by this notation at least for this post.

In essence, an identity A ≡ B means that A and B are synonyms, i.e. anytime we see A, we can replace it by B, and vice versa. In some cases, A may even look very different from B, but “spiritually” they are the same object.

Let’s investigate a common math identity below.

The binomial identity

The general binomial theorem (or binomial identity) looks like this:

$\displaystyle(x+y)^n = \sum_{k=0}^n {n \choose k} x^{n - k} y^k.$

It’s quite intimidating to the untrained eye, isn’t it? For the purpose of this post, we’ll therefore just focus on the case when n = 2, i.e.

$\displaystyle (x+y)^2=x^2+2xy+y^2.$

We’ll try to prove this identity in two different ways.

Algebraic approach

The goal is simple, i.e. to go from the left hand side of the equality to the right hand side through some algebraic tomfoolery. Here we go!

$\displaystyle \begin{aligned} &(x+y)^2 &\\\\ &=(x+y)(x+y)\qquad&(1) \\\\ &=(x+y)x+(x+y)y\qquad&(2) \\\\ &=x^2+yx+xy+y^2\qquad&(3) \\\\ &=x^2+xy+xy+y^2\qquad&(4) \\\\ &=x^2+2xy+y^2\qquad&(5)\end{aligned}$

Voilà! The identity is proved in just 5 simple steps. Notice that none of the steps above depends on the actual numerical values of x and y. Instead, the validity of each step depends solely on the properties of numbers. In particular,

Step (2) and Step (3) use the left and right distributive properties, respectively;
Step (4) uses the commutative property.

The upshot of the discussion is that this equality is true for any numbers x and y, hence an identity.

Geometric approach

A heuristic way of looking at a product of two objects is to view it as an area of some sort. For example, in our case, terms like x² and xy can be regarded as the areas of some square and rectangle, respectively.

A picture is worth a thousand words, provided that you can draw it out.

We see that (x + y)² is exactly the area of the biggest square, which is the same as the sum of the smaller parts, as shown above. There you go! A simple proof by picture, how intuitive is that?

What’s so important about identities?

No, the question above is not asked from a philosophical point of view, but a mathematical one. You see, proving or even writing down some math identities can be quite painstaking at times (see this example). All these efforts better be worth it and serve good purposes other than the not-so-convincing “iT rEvEaLs tHe bEaUtY oF mAtH” cliché.

Let me try to give two reasons behind their significance.

Conceptual reason

Some identities are pivotal in revealing the beauty of math structural properties of some mathematical objects which would otherwise stay hidden.

For those of you who are familiar with coordinate geometry, you may recall that the graph of the function y = Ax² + Bx + C on the usual xy-plane always takes the shape of a vertical parabola, as long as A is not zero. But have you ever wondered why that’s the case? The key to understanding this is none other than the binomial identity we saw earlier. However, the discussion of this particular topic deserves its own post in future, so let’s now move on to the second reason.

Practical reason

Some identities are invaluable tools to carry out certain mathematical procedures.

If you’ve learned calculus before, you might have heard a nifty trick called trigonometric substitution on the topic of integration. Basically, it involves the trigonometric identity

$\displaystyle \cos^2 x+\sin^2 x\equiv 1$

or some of its variants. And yes, it’s not immediately obvious why this identity is true. To learn more about it, click here.

All right, I hope this article helps you better understand and appreciate identities in mathematics. Stay tuned for more applications of them in future posts. As usual, you can also scroll down for some additional remarks. But until next time, stay acute, stay positive. See you!

Remark 3.1

It’s always crucial to know how things come about in mathematics. Not knowing the roots of something may very well be the root cause of misconceptions and wrong understanding.

To illustrate this, consider again the equality

$\displaystyle (x+y)^2=x^2+2xy+y^2$

which is an identity when x and y are usual numbers. In order for the algebra to work out, we need the following properties of numbers:

Right and left distributive properties
Commutative property

What if now we’re in the world where we don’t get to enjoy either of the said properties? A typical example would be the world of matrices, whereby matrix multiplication is not commutative in general. Let’s say

$\displaystyle X=\begin{pmatrix} 1 & 2 \\ -7 & 3 \end{pmatrix}\qquad\qquad Y=\begin{pmatrix} 8 & -3 \\ 2 & 5 \end{pmatrix}$

(we usually use capital letters to denote matrices). Sparing you the details, we get

$\displaystyle \begin{aligned} (X+Y)^2&=\begin{pmatrix} 86 & -17 \\ -85 & 69 \end{pmatrix}\\\\ X^2+2XY+Y^2&=\begin{pmatrix} 69 & -17 \\ -102 & 86 \end{pmatrix}\end{aligned}$

and therefore

$\displaystyle (X+Y)^2 \neq X^2+2XY+Y^2$

i.e. the binomial identity does not apply to the world of matrices!

The point I’d like to hammer home is this:

Whether or not an equality is an identity not only depends on the equation itself, but also on what the variables can be.

This principle is in the same spirit as the one we saw when we discussed the existence of solutions.

Remark 3.2

If you want to annoy your math teacher, try this:

It’s true that (x + y)² and x² + y² are not always identical. However,

$\displaystyle (x+y)^2 = x^2+y^2$

is still a valid equation to be considered (remember the difference between an identity and a general equation).

Let’s actually investigate when this equation is true. Applying the binomial identity, it can be rewritten and simplified as follows:

$\displaystyle x^2+2xy+y^2 = x^2+y^2\quad\Rightarrow\quad 2xy=0$

Therefore, (x + y)² = x² + y² precisely when 2xy = 0, which means either one of the following needs to happen:

x = 0 or y = 0, which will trivialize the equation to y² = y² or x² = x² , respectively;
2 = 0… Wait, what? There’s no way for this to be true… or is there?

As a matter of fact, there are mathematical “arenas” whereby 2 is treated as 0. In math jargon, we say the “arenas” have characteristic 2, and the equality (x + y)² = x² + y² is actually an identity in those places!

In summary, if the message of Remark 3.1 is

“What seems true may not always be true.”

then the message of Remark 3.2 is

“What seems false may not always be false.”