a brief biography of the equals sign
Like Roald Dahl and Catherine-Zeta Jones, the equals sign was born in Wales.
It was 1557—not that long ago, in the scheme of things. Just a few years before the birth of Shakespeare. In fact, the Danish prince and the Scottish king captivated the public long before their humble Welsh neighbor reached wide renown.
The early equals sign was a lovely but ungainly thing, a long pair of parallels that its inventor called Gemowe Lines:
Over the centuries, this stilt-legged creature shortened into the compact and tidy symbol we know today.
And before that? Well, mathematicians simply spelled out equalities with the phrase “is equal to.”
10 is equal to 7 + 3.
8 x 9 is equal to 72.
And of course, a2 + b2 is equal to c2.
The equals sign offered a way to avoid the tedious repetition of these words. Or, as Robert Recorde, the father of the symbol, put it: to auoide the tedioufe repetition of thefe woords.
An equals sign, then, is a verb. It’s the mathematical equivalent of “to be”—just as common, just as concise, and just as powerful.
But that’s not what kids see, is it?
To them, ‘equals’ means something other than ‘equals.’
In their arithmetic years, kids almost always encounter equal signs in a single, limited context: to call for the result of an operation. They fill their days with questions like this:
They get so used to statements of the form [number] [operation] [number] = that they’re a little creeped out by statements like this:
And totally deceived by statements like this:
Asked what goes in the blank, kids choose 9, because 4 + 5 = 9.
Or, when asked to perform multiple operations—start with 7, multiply by 5, subtract 9, and divide by 2—I often find my students writing streams of ungrammatical gibberish like this:
Of course, only the last of those equal signs makes any sense. The other two are bald lies. 7 x 5 doesn’t equal 35 – 9, and 35 – 9 doesn’t equal 26 / 2.
Poor old Robert Recorde would hang his head in sorrow.
Luckily, there’s a simple visualization of equality that can brush aside many of these misconceptions in one forceful sweep.
An equation is a statement of balance.
See the two sides? I’ve got different weights on each but the total is the same.
I can take away the same weight—say, 15 pounds—from each side, and they’ll still be equal. They’re not the same as they were before, but still the same as each other.
Similarly, I could add 5 pounds to each side, and they’d still be equal. Or I could double each side. Or halve each side.
So long as I do the same thing to each side, they’ll still be equal.
That familiar mantra—“Do the same thing to both sides of the equation”—is not an arbitrary dictate, cooked up by the Algebraic Rules and Regulations Committee in some air-conditioned boardroom. It’s a simple fact, which I capture in this rhyme:
If two things are equal
then do what you will
to both things at once;
they’ll be equal still.
Other symbols make sense in this light, too. The “>” symbol means “the thing on the left weighs more.”
The “<” symbol means the opposite.
And the “≠” symbol means “these two things aren’t equal, although I’m not telling you which one is bigger.”
I’ve had students ask me whether we can switch the sides of an equation, as if they need to consult the Bylaws of Algebra in some dusty legal library before making such a move. But understood with balance statements, it’s obviously true.
My younger students can mostly solve linear equations in x. But they do so by wordless numerical intuition (“I just knew x had to be 7”) or by blindly executed procedures (“I subtracted 11, then I divided by 2, but I don’t know why that works”). With balances in mind, suddenly it all makes sense:
Weirdly, my school’s textbooks teach “solving for y in terms of x” as a fundamentally different problem than merely “solving for x.” But in this light, they’re virtually identical.
The visualization can offer insight to older students, too. Students often solve simultaneous equations by “adding the two equations.”
Of course, you can’t really “add equations.” That’s nonsense. What you can do is add the same thing to both sides of one equation.
And then, on the left, instead of “52” we write something that’s equal to 52:
I don’t mean to say that this visualization is a magic pill, a cure for all misconceptions and fevers. No single key can unlock every door in mathematics. For that, you need flexible thinking, creativity, a healthy faith in your own abilities and a healthy skepticism of your own results.
Still, thinking properly about the equals sign sure helps.
I’ll confess: moving to the UK has heighted my affection for Wales. Sure, the English gave us a global language, the industrial revolution, and soccer. But the Welsh have gifted our planet a humble little symbol that compresses into two quick pen-strokes the far-reaching idea of equality itself.
Further Reading, for the Curious
FROM THE COMMENTS (i.e., the superior shadow-blog existing just below the surface of mine):
Nevin objects to my objection: “There are so many mathematical terms and symbols that are already overloaded to have multiple meanings, that [this use of the equals sign] really isn’t hurting anything.” The problem, to me is that students are not consciously adopting a different convention. Instead, they’re doing something unconventional, believing it’s conventional, and potentially missing an important concept in the process.
John Cowan points out, sensibly: “I think they use = that way because that’s what a calculator does; its = button means ‘Compute the (possibly intermediate) answer.'”
Howard worries about over-emphasizing notational issues: “It seems to me that there is an insane desire to make stuff look like math and force the mathematical language down like the production of foie gras“