In a previous post, we talked about the difference between numbers and numerals. There are often many numerals that express the same number. For example, the numerals “4” (Hindu-Arabic), “IV” (Roman), “٤” (Arabic), and “四” (Chinese) all express the same number, the number four. But let’s suppose we just use the Hindu-Arabic numeral system, and not any of the others. Then is there anything we can write down other than “4” that expresses the number four?
Yes there is! All of the following express the number four:
All of these sequences of symbols express the number four, and they all are in the Hindu-Arabic numeral system. So we don’t need to go outside the Hindu-Arabic numeral system to find other ways of expressing the number four. However, only the first of these, “4”, is a numeral. “1+3” is not a numeral, it is rather two numerals “1” and “3” with a plus sign “+” in between them. What do we call sequences of symbols like these that express numbers but are not necessarily numerals? We call them expressions. All of the sequences of symbols pictured, even “4”, are expressions, but only “4” is a numeral.
Two expressions may either express the same number or express different numbers. For example, “2+2” and “1+3” express the same number, four, but “1+1” and “2+3” express different numbers, namely two and five respectively. We use the equals sign to state that two expressions express the same number. For example, “2+2=1+3” states that “2+2” expresses the same number as “1+3”. This is true, because the expressions “2+2” and “1+3” do indeed express the same number, the number four. We can take any two expressions from the above list and put an equals sign in between them in order to state the truth that they both express the same number.
Let’s look at bit more at what we are doing here. What is going on in this sequence of symbols?
Is this sequence of symbols a numeral? No, it has many numerals in it, instances of “1”, “2”, and “3”. Is it an expression? No, it has two expressions in it, “2+2” and “1+3”. It is a new kind of thing, an equation. What does this equation mean? It doesn’t express a number like numerals and expressions do. Rather it states a truth, the truth being that the expressions “2+2” and “1+3” express the same number.
We call “2+2=1+3” an equation because by writing it we equate “2+2” and “1+3” — we consider them to be equal. “Equate” and “equation” are not just words that are used in mathematics, they are used in ordinary English as well. An insecure person may “equate criticism with insult”, and if we are talking about this trait of theirs, we might refer to it as “their equation of criticism with insult”. However, this ordinary English usage is not to be confused with the precise mathematical usage about two expressions expressing the same number. I just gave the ordinary English usage so you can see that the words “equate” and “equation” are not exclusive to mathematics, and that their mathematical usage has some bearing on their general usage in English.
What if we put an equals sign between two expressions that don’t express the same number?
This equation is false. The expressions “1+2” and “2+3” don’t express the same number. Nonetheless, it is still an equation. Equations can be true or false. They don’t have to be true in order to be equations. The insecure person mentioned above equated criticism with insult, but and this equation was not correct, but it was an equation nonetheless.
What if we put an equals sign between two instances of the same expression?
This is an equation too. Is it true or false? Well, the expression “1+2” on the left of the equals sign expresses the number three, and the expression “1+2” on the right of the equals sign also expresses the number three. Since the expressions on either side of the equals sign express the same number, the equation is true. In general, we can always make a true equation by writing the same expression on the left and the right of an equals sign.
Please notice that what I am saying about the equals sign is different than what you might have been taught in school. In school, you might have been taught that “1+1=2” means that when you add one and one, you get two. In this interpretation, the equals sign is placed between a procedure, “1+1”, and a result, “2”. Even if your teachers didn’t tell you that an equals sign is put between a procedure and a result, you still might have gleaned this interpretation from how it is used in exercises like “35×24=__” where you are given a procedure to do (multiply thirty-five and twenty-four) and you are asked to write down the result in the blank space. But here I am saying that the equals sign is not about results of procedures at all. It is just about expressions expressing the same thing. Consider the following equation:
This equation wouldn’t make any sense in school’s interpretation. How could the result of the “procedure” “2” be “1+1”? What would that even mean? But in my interpretation (which is the correct one), this makes perfect sense. The expressions “2” and “1+1” express the same number, the number two.
In fact, whenever you have a true equation, (say “1+1=2”), swapping the expressions gives you another true equation (for this case “2=1+1”). The expression on the left side of the equals sign and the expression on the right side of the equals sign are interchangeable. It doesn’t matter which expression you put on which side. This is an important property of equality called symmetry. In school’s interpretation, equality is not symmetric because the right side is distinguished as the “result of the procedure”, so it loses this crucial property.
(Not all equations in English are symmetric. For example, if the insecure person above equates criticism with insult, that does not mean that they also equate insult with criticism. I think that equating insult with criticism would be the mark of a very secure and very naïve person. Here is a clear difference between equations in English and in math, and it shows that we can’t always take our intuitions from English and assume that things work the same way in math. It also serves as an example of a general trend that things tend to be more simple (always symmetric) in mathematics than in English (not always symmetric). I am reminded of the mathematician John von Neumann’s quote “If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.”)
Now that you know this, you can use it to troll your teachers. If they give you a problem like “2+2=?”, you can answer like this:
You would be completely correct, and if your teacher complains it is due to their own miseducation in their schooling and you can refer them to this post so that they can learn what the equals sign really means. Let’s say your teacher gives you a problem you really don’t feel like doing, such as “355×224=__”. You would be entirely correct if you just wrote:
Now you don’t have to do the problem at all, and you’re still correct! If your teacher complains, ask if it is not enough that your answer be correct. Is math not about being correct? What more can they ask for? [sarcasm: math is about creativity, not simply being correct]
(A mathematically sophisticated teacher can get around this by writing something like “express 2+2 in the simplest form”. Then you would have to write “4”, because it is the simplest way to express the number four.)
The equals sign is often misused outside of math, and this really irks me when I see it. Before I got used to it, the meme phrase “mind=blown” really irked me. If your mind is blown, that doesn’t mean it’s equal to blown! Your mind and blown are still two different things! Also, it’s really common in chemistry, if you’re talking about, say, molecular mass, to write something like “CH4=16″ to say that the molecular mass of CH4 (methane) is 16. This is wrong! Methane is not equal to the number 16!
I feel bad being mad about this because it is as if I am a nationalist of mathematics trying to uphold the sanctity of its symbols in the greater world. Of course the equals sign is public property and people are free to make use of it however they like. I just wish that real mathematical understanding was a bigger part of our culture.
Disclaimer: In this post I have just explained one use of the equals sign. There are other uses too, both within math and in other fields, like computer science. Sometimes slightly different symbols, like “≈” and “≡” are used to tell apart these different uses. Equality is also not the only concept of sameness. There are other ways for two things to be the same. I will talk about this stuff in later posts, I just wanted to mention it so that people don’t think that the notion discussed above is the be-all and end-all of the equals sign, equality, or sameness.
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