Go ahead, try this: ask a high school math teacher for the single student error across all grade levels that's the most infuriating. I'd bet that the most common answer you get is some version of how students incorrectly square the sum of two numbers (as in (a + b)2) to get the sum of the squares of the individual numbers (as in a2 + b2). In fact, though, those expressions are not equal; that's not how squaring works. Indeed, in my experience, students who can absorb the intuition about why they are not equal are much more likely to succeed in future. This lesson will try to make sense out of this inequality:
Before explaining why it isn't true, I think it's important to understand why so many people think it is true. There are several reasons a student might think it's true, and each of those reasons has its own logic.
(1) I know it's true for multiplication:
This is a classic case of a student paying more attention to the notation than to what the symbols actually mean. Squaring means multiplying an expression times itself: (ab)2 means (ab)(ab). But crucially, that's a whole bunch of multiplications. And we all know that you can regroup and rearrange items being multiplied at will (formally, we say that multiplication is associative and commutative). And when you do regroup and rearrange, you see there are two a's and two b's all multiplied together; thus a2b2. But look at what (a + b)2 means: (a + b)(a + b). Where's the possibility for rearranging so simply? It's not there. The mixture of plus and times makes this expression harder to work with than if it's all multiplication.
(2) I know it's legal to do this thing called 'distributing':
That thing is a capital Greek letter 'psi', and I used it to demonstrate how anything can be distributed, even if it looks complicated (we'll need that idea in a minute). But many students don't know that the word 'distribute' is actually a nickname for a much longer description of the process shown above: it's the 'distributive property of multiplication over addition'. Distributing leaflets over a parking lot means that every person in the parking lot gets a leaflet, and distributing multiplication over addition means that every element of the addition (the a and the b) gets multiplied. It's just a fact about our number system that you can choose to add two numbers before you multiply the sum by a third number (the left side of the above equation), or you can multiply each of the two numbers by the third number individually first (the right side), then add the products, and you'll get the same result. In fact, the equation in (1) above is a demonstration of a different distributive property: 'the distributive property of exponents over multiplication'. And that one is true because multiplication is associative and commutative.
But the 'distributive property of exponents over addition' is simply not valid. Various distributive properties are different mathematical processes (a subtlety you might miss if you don't know their complete names), and they don't all have to be valid. In fact, most are not; any one that is valid is special and important.
(3) But...it just seems like they should be equal!
Well...only at first maybe. It's a common misconception that mathematical processes that look right on the page or seem right in your head must be true. Mathematicians approach it in the completely opposite direction—in a way, nothing should be true until you can prove it's true. And equality in this case just...isn't true! Too bad! Out of luck! Oh well, we'll learn to deal with it!
Once you look at this situation very carefully, using examples and various different mathematical interpretations, you'll find that you probably don't really even believe that the two sides could be equal. You already have the knowledge to convince yourself of that. Let's look at some of those examples.
If you're in a candy store scooping candy into a bag, and you buy one-and-a-half pounds of candy that costs $1.50 per pound, how much will you get charged? You might not know the answer off the top of your head, but I'll bet you know that $1.25 is a ridiculous answer. You're buying more than a pound of candy, so the price would just have to be more than $1.50. Yet if you think that the square of a sum is the sum of the individual squares, you'd have to believe that $1.25 is right:
One-and-a-half pounds times one-and-a-half dollars per pound doesn't equal one-and-a-quarter dollars. Note the crucial placement of the not-equals-sign. Be sure you understand the reason for every step of that line of math.
I'll bet you can probably square the number 20 in your head. 2 times 20 is 40, so 20 times 20 is 400. You might not be as quick with squaring 21. The actual answer isn't important, but do you have the intuition that it's not 401? Well that intuition comes from a deep-down intuition that the square of a sum is not the sum of the individual squares:
What does (a + b)2 equal, then? Or, put another way, is there an expression without parentheses that always has the exact same value? The formal algebra is below. It uses the distributive property of multiplication over addition that we saw above. The popular term for the process that you might know is FOILing:
Now, a2 + 2ab + b2 isn't exactly an obvious way to rewrite (a + b)2, but it does have the advantage of being correct! If you try to interpret that in a different way, though (geometrically, for example), it can make much more sense. The natural way to understand the concept of squaring is through looking at the area of a square—which is calculated by squaring. So below is a picture of a square whose sides are each a + b long. To make that more clear, those sides are broken up into their separate a and b parts.
Asking about (a + b)2, then, is just like asking about the area of that whole square. But the whole square is broken up into smaller squares and rectangles, and we know enough information to calculate each of those smaller parts separately. The areas of the two smaller squares are calculated below.
Notice that the areas of the two smaller squares together come nowhere close to totaling the area of the large square. In algebra terms, we'd have to say that (a + b)2 must simply be greater than a2 + b2. Of course that means they can't be equal, which is exactly what we've been trying to understand! This picture actually tells us even more, though. It tells us how much greater. Each of the blue rectangles has a length of a and a width of b, so they each have an area of a times b. And there's two of them. Which means precisely that (a + b)2 = a2 + 2ab + b2, just as we saw in the algebra.
Finally, I'll show one more way to understand the original inequality. This last way requires that you know what the Pythagorean theorem says, and I'll assume here that you do (if not, you can skip this part). Note the square built on the hypotenuse of the triangle below:
The square has an area of c2, which the Pythagorean theorem says is equal to a2 + b2. But that is exactly the right hand side of our original inequality. So it makes sense to ask about the square represented by the left hand side. A square with that area would have to have a side length of a + b. But it's clear from looking at the triangle that a + b has to be bigger than c (walking along the hypotenuse must require fewer steps than walking along the legs of the triangle—technically that's called 'the triangle inequality'). That is, the two sides of the inequality each geometrically represent the area of a square, but those squares can't be the same size, so the two expressions can't be equal.
That is, I hope, enough to convince you.