Correct Usage of Equal and Equivalent in Math Statements

Date: 02/20/2008 at 11:43:16
From: Valerie
Subject: when do I use equal and when do I use equivalent properly?

Is there a mathematical difference between equal and equivalent and if
so, what is the difference?  For example:

Are two algebraic expressions (with the same solution set) equivalent
or equal? 

Are two numeric expressions (with the same value) equal or equivalent?

For example: is 10^4 equal to OR equivalent to 10 x 10 x 10 x 10?  

We would write an equal sign between the two numeric expressions given
above, but would we "say" they were equivalent or equal?  Would we
write a question properly that reads:  Find an expression that is
equal to OR equivalent to 10^4. ?

Date: 02/20/2008 at 23:45:39
From: Doctor Peterson
Subject: Re: when do I use equal and when do I use equivalent properly?

Hi, Valerie.

Sometimes I think the terms "equal" and "equivalent" are, well,
equivalent; their meanings overlap, and the precise distinction can
vary from one case to another.  But in general,

  equal means that two entities are the same entity

  equivalent means that two entities have the same EFFECT,
  in some sense

Two expressions can be called equal for some specific value of the
variable, because they yield the same number; they are equivalent when
they are ALWAYS equal--true for any value of the variable.  So here
"equivalent" is a stronger term.

Two equations are equivalent when they have the same solution set.
(Note: an expression doesn't have a solution set!)  You would never
say that two equations are equal, because that would get very confusing!

Two sets are equivalent when they have the same cardinality (number of
elements); they are equal when they are actually the same set
(consisting of the same elements).  Here "equal" would seem to be the
stronger term!

Two fractions are equivalent when their values are equal; here I think
the only reason to use the word "equivalent" rather than "equal" is to
emphasize that they are two DIFFERENT ways to express the SAME value.

In your question about numerical expressions, I think "equivalent"
would be the better choice, because, as with fractions, the emphasis
is on an expression that looks different but has the same numerical
value after you evaluate it.  It wouldn't be wrong to say "equal", but
this just feels a little more right.

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


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