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