# What is the difference between “$=$” and “$\equiv$”?

I was recently thinking about some of my past math classes, and depending on the context I recall my professors would sometimes use the "$$\equiv$$" symbol in places where I'd feel "$$=$$" to be more appropriate. For example, since this would often be the case in my classes on differential equations and Fourier series, we would have (for $$n \in \Bbb N, k \in \Bbb Z$$)

$$(-1)^{2n+1} \equiv -1$$ $$\sin(k\pi) \equiv 0$$

Is there a particular reason in this context why we would say "$$\equiv$$" instead of "$$=$$"? The latter feels more natural in this context, which makes me think that there's some reason my professors would use the former.

I'm familiar with the notion of the "$$\equiv$$" symbol in the context of, say, elementary number theory (specifically modular arithmetic) where we might say

$$10 \equiv 1 \pmod 3$$

which isn't saying "$$10$$ equals $$1$$", just that "$$10$$ is like $$1$$ in this context." But that doesn't seem to fit the case as with the first two statements - because I don't believe it is that $$(-1)^{2n+1}$$ is like $$-1$$, or that $$\sin(k \pi)$$ is like $$0$$, they are $$-1$$ and $$0$$ respectively.

Am I just mistaken on this latter fact? Is there something I'm missing? What, precisely, is the difference between the two notations?

• I think people usually use $\equiv$ for equality of functions. For your examples, the functions $f: \mathbb{Z} \to \mathbb{R}$ and $g: \mathbb{Z} \to \mathbb{R}$ given by $f(n) = (-1)^{2n+1}$ and $g(k) = \sin(k \pi)$ are identically the functions $-1$ and $0$, so we write $f \equiv -1$ and $g \equiv 0$. – mathworker21 Jan 10 at 3:39
• when speaking about functions, we would interpret $f(x) = 0$ as the statement that there exists an $x$ where $f(x) = 0$, and interpret $f(x) \equiv 0$ as the statement that for all $x$, $f(x) = 0$. The first statement read aloud says $f$ is equal to 0 at $x$, the second is $f$ is identically 0, that is, it is equal to the 0 function. – staedtlerr Jan 10 at 3:40
• @staedtlerr nobody interprets "$f(x) = 0$" as "there exists an $x$ where $f(x) = 0$". people would assume the speaker is referring to some specific $x$. – mathworker21 Jan 10 at 3:41
• It is usually used as denoting “identically equal to”. For example, $\sin(k \pi)=0$ could be read as an equation in $k$, but $\sin(k \pi) \equiv 0$ means that this is always 0 (for integer k). Just like $f(x) \equiv 0$ would mean the function is identically 0 – Fede Poncio Jan 10 at 3:41
• To piggy-back on mathworker21's comment, $(-1)^{2n+1}$ is a number, depending on a variable $n$, whereas $\mathbb{N} \to \mathbb{Z} : n \mapsto (-1)^{2n+1}$ is a function. To say $(-1)^{2n+1} \equiv -1$ is shorthand for saying the above function is equal to the constant function $1$, or in other words,$$n \mapsto (-1)^{2n+1} = n \mapsto -1.$$ – Theo Bendit Jan 10 at 3:42

I'll give an example of each.

$$2x=x+1$$

This holds when $$x=1$$ only, and so the equality symbol is appropriate. In short, we use an $$=$$ when specific values solve the expression.

On the contrary, we have:

$$2x\equiv x +x$$ Whatever the value of $$x$$, this holds. This is an algebraically obvious one, but another might be $$\sin^2 x + \cos^2 x \equiv 1$$

The identity symbol $$\equiv$$ is used when an equality holds for all values in the domain specified (e.g. $$\Bbb R$$).

• +1. I like to think of "$\equiv$" as "$=$" with an emphatic underscore. :) – Blue Jan 10 at 4:11
• Oooh, nice analogy, thanks for that Blue. – Eevee Trainer Jan 10 at 4:17

The equal sign "$$=$$" is used for equalities ,equations, identities and definitions of functions.

Examples of equalities are $$3=2+1$$ or $$12^2 = 144$$ where two numbers are the same.

Examples of equations are $$3x+1=10$$ or $$x^2-4=0$$ where for some values of the variable both sides result in the same value.

Examples of identities are $$\sin ^2 x + \cos ^2 x =1$$ or $$(x+y)^2 = x^2+y^2+2xy$$ where both sides are identical for every value of variables.

For identities sometimes $$\equiv$$ is used instead of $$=$$ for example we may use $$e^{i\theta} \equiv \cos \theta + i \sin \theta$$ or $$\sin ^2 x + \cos ^2 x \equiv 1$$ or $$(x+y)^2 \equiv x^2+y^2+2xy$$ to emphasize that this is an identity not an equation.

For functions we sometimes use $$f(x) \equiv c$$ to emphasize that the given value $$c$$ is for all values of $$x$$