What “=” Means

You’ve seen it before.

Maybe you use it all the time.

Maybe you haven’t touched it since high school or college.

But it is likely that, early in your education, the = sign showed up in some nasty column of homework problems. In that context, “=” seems like a prompt for the answer.

Try it yourself. How do you read these?

3+1 = ___

2+4 = ___

7+3 = ___

Chances are, if you’re not trained in math, you read the = sign as a command to fill in the blank.

Even in higher levels of education, the “=” sign is often used as a shorthand for “get the answer”. For example, sin2(x)+cos2(x) = ________ might be posed as a question on a trigonometry test.

Can you spot the sine wave?

A friend (who is starting to explore higher math) asked me about an equation she didn’t understand. After a few minutes of talking, I realized that my friend never learned any meaning to “=” besides “get the answer.” That is simply not what “=” means. And this mistake is more than a harmless misconception. It actually makes higher math impossible to comprehend.

If, as is the case with many people, you never learned what “=” really means, you will never decipher a sentence like this: “There are no solutions to xn + yn = zn for positive integers x, y, z and n>3.”

Even if you are familiar with algebra, you quickly hit a problem.  Anyone who thinks “=” means “find the answer” will likely turn away. How are you supposed to solve for anything here?

This equation is the topic of an upcoming post. It’s got a great backstory.

Here is where one small idea can open up mathematics for even a self-proclaimed “not math person.” Really, = doesn’t have anything to do with questions or answers at all.

Compare this to an English sentence, like Jennifer ate a hamburger. If Jennifer really ate a hamburger, this sentence is true. If she didn’t, this sentence is false. But either way, it is a grammatically correct sentence.

Now take the phrase, Jennifer. Or, Jennifer ate a. Neither of these are complete sentences.

Imagine I told you what Jennifer ate, then gave you a pop quiz: Jennifer ate a ______. The implied question here is “fill in the blank so that it makes a true sentence.”

No one would think that “ate” means “get the answer.” In this sentence, “ate” is just the verb.

The verb in this simple English sentence is analogous to the role of = in math. Except that = is probably the most powerful verb in the math vocabulary. When two things are =, that means they are the fundamentally the same. If you would replace one with the other, you’d be saying the exact same thing.

So let’s see = in a math statement. If I say “2+2=17,” I am saying, “2+2 and 17 represent the exact same thing.” This, of course, is not true. But my equation is still “grammatically correct.”

There are many uses for the “verb” = in math. One important use is to take statements about equations and try to prove or disprove them. 2+2=17 can be proven untrue in a few simple steps. The proof of the example with xn + yn = zn is not as easy. That particular problem took over 300 years of focused effort before anyone solved it!

We can make beautiful things with mathematical functions. But people have trouble seeing the beauty when they think they’re being asked to solve something.

= is really a friendly symbol. It presents two ideas and offers you a chance to see if they match up. It’s something of a pity how it is used on elementary school tests, because that makes people associate it with “right” or “wrong”—a certain way to make you nervous and maybe even hate math. = isn’t about right and wrong. It’s about true or not true. And that is something a person can enjoy.

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18 thoughts on “What “=” Means

    • tamarlichter says:

      “Wrong” actually has nothing to do with math. The only thing that math studies is true and false. And proving things false is just as important as proving things true.
      If a teacher tells you that you got something wrong, that’s got nothing to do with the math: that’s got to do with the classroom structure where falsehoods are treated as wrong.

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  1. SB says:

    Excellent stuff. This reminded me of two things:

    1) A nice post on math.SE (the question should be asked by every mathematician at some point in his career–its kinda of like “why am I doing what I’m doing?”. A lot of the answers and comments are insightful):
    http://math.stackexchange.com/questions/384861/is-mathematics-one-big-tautology

    2) For EL’s benefit as well, the following quote from the Rambam (Moreh I:2):
    “The intellect which was granted to man as the highest endowment, was bestowed on him before his disobedience. With reference to this gift the Bible states that man was created
    in the “form and likeness of God”. On account of this gift of intellect man was addressed by God, and received His commandments, as it is said: “And the Lord God commanded Adam” (Gen. ii. 16)–for no commandments are given to the brute creation or to those who are devoid of understanding. Through the intellect man distinguishes between the true and the false. This faculty Adam possessed perfectly and completely. The right and the wrong are terms employed in the science of apparent truths (morals), not in that of necessary truths, as, e.g, it is not correct to say, in reference to the proposition ‘the heavens are spherical’, it is ‘good’ or to declare the assertion that ‘the earth is flat’ to be ‘bad’; but we say of the one it is true, of the other it is false. Similarly our language expresses the idea of true and false by the terms ’emet’ and ‘sheker’, and of the morally right and the morally wrong, by ‘tov’ and ‘ra’. Thus it is the function of the intellect to discriminate between the true and the false–a distinction which is applicable to all objects of intellectual perception.

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  2. EF says:

    So maybe some more interesting answers for 3+1=_____ would be “1+2/2+1/0.5” or “the number of colors needed to color any graph”.

    Like

  3. Andrew Kircher says:

    Thank you for this post. I am fascinated by how this simple reframing of “=” transforms x=y from being a question of right of wrong to a critical assessment of what = means in the context of this sentence. What allows for equivalence. Really poetic.

    Like

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