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’s only good for the problem at hand, and there’s no need to remember this “fact”.

**Fundamental Connection**

Consider a mathematical truth like a^{2} + b^{2} = c^{2}, where 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 n is:

which can be seen as a type of geometric rearrangement, combinatorics, averaging, or even list-making.

**Factual Definition**

Statements like

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:

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.

## Other Posts In This Series

- Developing Your Intuition For Math
- Why Do We Learn Math?
- How to Develop a Mindset for Math
- Learning math? Think like a cartoonist.
- Math As Language: Understanding the Equals Sign
- Avoiding The Adjective Fallacy
- Finding Unity in the Math Wars
- Brevity Is Beautiful
- Learn Difficult Concepts with the ADEPT Method
- Intuition, Details and the Bow/Arrow Metaphor
- Learning To Learn: Intuition Isn't Optional
- Learning To Learn: Embrace Analogies
- Learning To Learn: Pencil, Then Ink
- Learning to Learn: Math Abstraction
- Learning Tip: Fix the Limiting Factor
- Honest and Realistic Guides for Learning
- Empathy-Driven Mathematics
- Studying a Course (Machine Learning) with the ADEPT Method
- Math and Analogies
- Colorized Math Equations
- Analogy: Math and Cooking
- Learning Math (Mega Man vs. Tetris)

mark ptakSeptember 24, 2012 at 6:25 amI 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!

HaroldSeptember 24, 2012 at 6:27 pmThis 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.

kalidSeptember 25, 2012 at 9:27 am@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.

AnonymousSeptember 28, 2012 at 12:23 pmi 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?

kalidSeptember 28, 2012 at 3:58 pm@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”).

Mathematics and Multimedia Blog Carnival 24September 30, 2012 at 12:55 am[…] Kalid Azad of Better Explained explains the different uses of equal sign in Math as a Languge: Understanding the Equal Sign. […]

JaySeptember 30, 2012 at 9:09 amI’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.

YatharthROCKOctober 3, 2012 at 3:59 am@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.

YatharthROCKOctober 3, 2012 at 6:11 amFor 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:

the equal sign in Euler’s formula

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?

YatharthROCKOctober 3, 2012 at 6:14 amAnd 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

PatOctober 3, 2012 at 10:06 amThe 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!

GabrOctober 5, 2012 at 4:50 am“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.

kalidOctober 9, 2012 at 10:17 am@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”.

fareah afzalJanuary 14, 2013 at 11:17 amthese are really very interesting and helpful info. for students,,,,u are doing good job….well don,,,keep it continue

AnonymousJanuary 15, 2013 at 4:45 pmI 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, 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 () if and only if they share these defining properties. Modulo 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 () 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 , the relation is the equality relation on the elements of (the careful reader will notice that I have used to define ; this is allowed because we are really using the axiom of extensionality–the elements of are defined to be the same as those in set on the right). Of course the elements of may be interpreted in various ways, which is why one might say if 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.

sarkan bahmanJanuary 17, 2013 at 11:26 pm0^0=? why?

kalidJanuary 25, 2013 at 2:06 pm@sarkan: check out http://betterexplained.com/articles/understanding-exponents-why-does-00-1/

jbhOctober 30, 2013 at 9:55 amYour 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.”

kalidOctober 30, 2013 at 10:12 am@jbh: Awesome, glad it clicked!

eric nii odoi laryeaJanuary 9, 2014 at 9:18 ami really appreciate the site very well i will very happy because i will really learn hard to my exam in shs till university

eric nii odoi laryeaJanuary 9, 2014 at 9:25 amthank you

Yatharth AgarwalFebruary 27, 2014 at 1:42 amYou could also add this non-mathematical example: when I say “We have the same toy”, I mean both our toys are the same; but when I say “We have the same mother”, I mean that both are mothers point to the same exact object. Important distinction, I think; and one I first realized via programming (when 2 objects have equal values, but are not the same thing is one case; and when 2 names, as in Python where we don’t really have any variables per se, point to the same object in memory, akin to ByRef in more primitive languages, is the second).

kalidFebruary 27, 2014 at 10:27 pmThanks Yatharth, great point. And a funny thought — in the math world, are we all using the same number 17, or different copies of it?

FionaOctober 5, 2014 at 9:17 amI feel uneasy about an area being equal to a different type quantity, e.g., a volume etc., even if numerically equal Are there times in maths where different quantities can be equal?

kalidOctober 5, 2014 at 7:45 pmHi Fiona, great question. In algebra, technically we don’t have any units, so 3, x, x^2, 15x^3 are all of the same.

For example, if x = 3, then x^2 = 9, and they can be compared directly. We might have an equation like x^2 – x = 6 and solve it to see that x=3 (or -2). The various powers can interact.

However, in certain applications, like physics, we might keep track of units (length, area, volume, etc.) and then realize we can’t mix the results.

Ondřej KubůApril 11, 2015 at 9:49 amI do not have much knowledge of programming. Can you explain me the differences between =; ==; and ===? Thanks for your answer.

kalidApril 11, 2015 at 2:03 pm@Ondřej: Sure thing. In programming, `=` is assignement, so `x = 3` assigns the number 3 to the variable x.

`==` is a check for equality, so `x == 3` is checking whether x already equals 3 (and results in true or false).

`===` is a _strict_ check for equality ,where x must be the number 3 and not the character “3”. In some programming scenarios, characters are automatically converted to their numeric equivalent where possible, and `===` is making sure that isn’t happening.

Ondřej KubůApril 12, 2015 at 12:29 amThanks for comment, I thought it should be mentioned in here .

DorjiOctober 1, 2016 at 5:36 amI love the idea that maths is a language. But Richard Feynman struck me with the words that mathematics is language plus logic. Now it seems like it is a tool built on logic of how things work in a situation in a form of notations(language) and that it is database of ideas abstracted for future references.

One of my mottos is to go after aha-momments, and you inspire me to follow that fascination.

AmyFebruary 26, 2017 at 9:23 amThank you, this is really helping me lay a solid foundation before I get back into learning maths for the first time in 20-odd years.

alondraplazaFebruary 6, 2018 at 10:31 amhi i love math