Associated Topics || Dr. Math Home || Search Dr. Math

### Congruence Symbol

```Date: 05/23/2003 at 04:26:49
From: Marc
Subject: Congruence symbol

How does congruence (3-lined symbol) relate to equality (=)?

When I say two equations are congruent, what am I actually saying?
e.g. when I say two equations are equal, I mean that they represent
the same numerical value, no? Or perhaps that for all possible values
of their variables they are equal?

Thanks.
```

```
Date: 05/23/2003 at 12:51:51
From: Doctor Peterson
Subject: Re: Congruence symbol

Hi, Marc.

The "triple-equal" symbol (which I will represent here with "==") is
actually used in several different ways, one of which you appear to
be using, but with the name given to a different usage.

First, please note that you are referring to the symbol being used
not between two EQUATIONS, but two EXPRESSIONS.  That makes a big
difference here; an equation includes an equals sign.

One use is for an "is defined as":

Defined - Eric Weisstein's World of Mathematics
http://mathworld.wolfram.com/Defined.html

We can say, for example, that

tan(x) == sin(x)/cos(x)

to say that the tangent is defined as the ratio of sine to cosine, if
we choose to define it that way.

A closely related use is for "identical equality," which I think is
what you have in mind:

Identities - AS Guru Maths (BBCi)
http://www.bbc.co.uk/education/asguru/maths/13pure/04trigonometry/
23identities/index.shtml

This indicates an "identity," an equation that is always true, for any
value of its variable(s). This is sometimes denoted in the same way,
and read "is identically equal to":

(a + b)^2 == a^2 + 2ab + b^2

because this is true for all a and b. The important point is that it
is not just its numerical value that happens to be the same in one
instance, but rather the meaning of one expression, or function,
itself is exactly the same as the other.  That makes this a deeper
level of equality.

A third use is for "congruence" in number theory:

Congruence - Eric Weisstein's World of Mathematics
http://mathworld.wolfram.com/Congruence.html

We say that two numbers are "congruent modulo n" or

a == b (mod n)

if a - b is a multiple of n.

Another use, which I am not familiar with, is in logic:

Equivalent - Eric Weisstein's World of Mathematics
http://mathworld.wolfram.com/Equivalent.html

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
```

```Date: 10/14/2004 at 12:51:51
From: Cristian
Subject: Congruence symbol

There is another use of the '==' symbol: in geometry, where two shapes are
congruent, they are numerically (i.e., except for positioning) equal.

-Cristian
```
Associated Topics:
College Definitions
High School Definitions

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search