Math As Language: Understanding the Equals Sign

It’s easy to forget math is a language for communicating ideas. As words, “two and three is equal to five” is cumbersome. Replacing numbers and operations with symbols helps: “2 + 3 is equal to 5″.

But we can do better. In 1557, Robert Recorde invented the equals sign, written with two parallel lines (=), because “noe 2 thynges, can be moare equalle”.

“2 + 3 = 5″ is much easier to read. Unfortuantely, the meaning of “equals” changes with the context — just ask programmers who have to distinguish =, == and ===.

A “equals” B is a generic conclusion: what specific relationship are we trying to convey?

Simplification

I see “2 + 3 = 5″ as “2 + 3 can be simplified to 5″. The equals sign transitions a complex form on the left to an equivalent, simpler form on the right.

Temporary Assignment

Statements like “speed = 50″ mean “the speed is 50, for this scenario”. It indicates that we decided this equivalence. We could have picked any value, but chose one useful for the problem at hand.

Fundamental Connection

Consider a mathematical truth like a^2 + b^2 = c^2, were a, b, and c are the sides of a right triangle.

I read this equals sign as “must always be equal to” or “can be seen as” because it states a permanent relationship, not a coincidence. The arithmetic of 3^2 + 4^2 = 5^2 is a simplification; the geometry of a^2 + b^2 = c^2 is a deep mathematical truth.

The formula to add 1 to 100 “can be seen as” geometric rearrangement, combinatorics, averaging, or even list-making.

Factual Definition

Statements like

$\displaystyle{e = \lim_{n\to\infty} \left( 1 + \frac{100\%}{n} \right)^n}$

are definitions of our choosing; the left hand side is a shortcut for the right hand side. It’s similar to temporary assignment, but reserved for “facts” that won’t change between scenarios (e always has the same value in every equation, but “speed” can change).

Constraints

Here’s a tricky one. We might write

x + y = 5

x – y = 3

which indicates conditions we want to be true. I read this as “x + y should be 5, if possible” and “x – y should be 3, if possible”. If we satisfy the constraints (x=4, y=1), great!

If we can’t meet both goals (x + y = 5; 2x + 2y = 9) then the “equations” could be true individually but not together.

Example: Demystifying Euler’s Formula

Untangling the equals sign helped me decode Euler’s formula:

$\displaystyle{e^{i \cdot \pi} = -1}$

A strange beast, indeed. What type of “equals” is it?

A pedant might say it’s just a simplification and break out the calulus to show it. This isn’t enlightening: there’s a fundamental relationship to discover.

e^i*pi refers to the same destination as -1. Two fingers pointing at the same moon.

They are both ways to describe “the other side of the unit circle, 180 degrees away”. -1 walks there, trodding straight through the grass, while e^i*pi takes the scenic route and rotates through the imaginary dimension. This works for any point on the circle: rotate there, or move in straight lines.

Two paths with the same destination: that’s what their equality means. Move beyond a generic equals and find the deeper, specific connection (“simplifies to”, “has been chosen to be”, “refers to the same concept as”).

Happy math.

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19 thoughts on “Math As Language: Understanding the Equals Sign”

1. I would guess that one of the things we can take from this is that the “equals sign” gets its meaning from content. After fighting the == === = for the last few days this post couldn’t have come at a better time for me. Keep it up!

2. This is a tremendous post. The connection to two fingers pointing at the same moon has a real chance to enlighten.

What’s missing, to me, is the reading of = that I use most often, which is “is”. As in, 2+3=5 read as “two plus three is five.” The idea that the two distinct seeming things on opposite sides of the equal sign are actually the same thing is a very powerful one. Many students I encounter don’t recognize the gravitas.

3. @mark: Heh, nice coincidence! Yep, the contents being compared impact the equals sign for sure. Even in math, you’re not supposed to compare vectors and scalars (1 vs [1]).

@Harold: Glad it clicked. I find “is” to be a little too generic for my tastes, so maybe “is another representation of” or similar. But good point about the importance of this, it’s the same central concept referred to by different names.

4. i always thought of it as ‘x = y’ means ‘is is the same thing as y’. for the ‘set x to be y’ type of this, i usually add ‘let x = y’ or ‘if x = y’ when i’m doing maths. and isn’t the fundamental thingy meant to use the triple parralel lines ‘identity’ sign?

5. @Anonymous: Math has had to come up with different types of equals (like triple equals, etc.) but they’re not well known.

The general-purpose X “is the same thing as” Y might be true, but isn’t helpful in getting to the subtleties (2 + 2 = 4 has a different meaning from the Pythagorean theorem, even though both use the equal sign, a^2 + b^2 isn’t really “the same thing as” c^2, it’s more “they are entirely different things which have the same magnitude”).

6. I’d disagree with the equating of math as a foreign language. I think mathematical notation is a foreign language, but math itself is some entity separate from its notation.

7. @anonymous @kalid Oh, I though you were referring to the different equals signs in most computer languages (JS has all three)—didn’t know it originated from a math norm. BTW, is this a weird way to refer to the conversation between you and @anonymous? I thought it looked like nesting.

8. For some reason, the insights pane was not working for me. It just freezes after clicking the “Post Insight” button. So here it is, copied out:

Aha! The insight that helped was:

Details:
Till now I was viewing the Euler’s formula as more of a transformation. But the insight that -1 was just getting there and $e^{iπ}$ was going around the circle. For your article on co-ordinates and some recently acquired trig, I also got the $cos x$ and $i sin x$ part.

Only thing. Where does the $e$ part come from?

9. And thought about adding Live Previews for the comments yet? If that’s too hard, then let us edit comments or at the very least, delete them. But then, that would require some sort of an accounts system which would be hard too. You’d have to extend the Aha! account to BE.

So anyways, just keep it on your to-do list: http://workflowy.com

10. The curveball I remember was in my first computer class on the IBM 1130 studying Fortran IV. The stanza x=10 was explained to mean “replace variable x with value 10″, or something like that. Keep up the good work. Very interesting!

11. “x=x+1″ if you are familiar with programming – certainly all of us are – then you came across this statement in your life. Well I remember the first time I saw this statement I got the meaning very easily and I was astonished since it is mathematically wrong ! but why ? it sees x as a box/container and as a value as well, it says take the box that it is named x and add to what is inside it a “1″ ! .. the first x in the statement means the box named x, while the second after the equal sign means what inside the box, or in another way .. x (now) is the x (of the past) + 1.

12. @Gabr: You bring up a great point. The internal mechanics of how “x = x + 1″ works is actually pretty complicated. Inside a computer, x is really a memory location (or register), so we’re saying “Read the value stored in location x, add 1 to it, and store that back into location x”.

13. these are really very interesting and helpful info. for students,,,,u are doing good job….well don,,,keep it continue

14. I suspect that all of these forms of equality you have listed fall under the umbrella of “they are the same, modulo one’s interpretation”. For instance, $\displaystyle{3+4=7}$ is a useful operation if we say that the original group of seven is being partitioned into a group of three and a group of four, i.e. on the left there are tight rules for admission into each of the two groups, on the right the rules are relaxed to allow all seven members into a single group. The important part, however, is that such partitions are figments of our imagination. When we look around, we partition the space around us into familiar objects; not at all unlike what is happening in a simplification.

However, equality can sometimes be too strong for our purposes. For instance, say we want to reason about all chairs at once. The usual way is to work modulo an equivalence relation, which behaves a like equality but allows more things to be equal. Keeping with the example of chairs, you look up the definition of a chair (check that it agrees with your intuition) then say two objects are equivalent ($\displaystyle{\sim}$) if and only if they share these defining properties. Modulo $\displaystyle{\sim}$ we can pick an arbitrary chair and describe the whole class. Where does this form of reasoning actually come up? Certainly in abstract algebra, where two structures may have the exact same internal structure, but have different names for their constituents (like calling a “window” various other names or using various languages to name the structures comprising a house). This kind of equivalence is called an isomorphism ($\displaystyle{\cong}$) and allows us to pick any “representative” by naming its constituents however we like. A final example is in (elementary) geometry, where it is generally accepted that one is to reason about a whole class of figures by first choosing a representative.

Finally, in the formalities of set-theory, equality has a very strict form: for a given set $\displaystyle{X}$, the relation $\displaystyle{R=\{(x,x):x\in X\}}$ is the equality relation on the elements of $\displaystyle{X}$ (the careful reader will notice that I have used $\displaystyle{=}$ to define $\displaystyle{R}$; this is allowed because we are really using the axiom of extensionality–the elements of $\displaystyle{R}$ are defined to be the same as those in set on the right). Of course the elements of $\displaystyle{X}$ may be interpreted in various ways, which is why one might say $\displaystyle{(1,2-1)\in R}$ if $\displaystyle{X}$ is the set of integers, rational numbers, real numbers, complex numbers, etc. So the concept of equality being the same “up to interpretation” takes a very strict meaning in foundational mathematics.

15. 0^0=? why?

16. Your teaching on understanding “=” and the Harold post on 20120924 referring to which is “is” finally helped me to understand why someone would say, “It depends on what the meaning of the words ‘is’ is.”