Difference between revisions of "Equation"
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A unique aspect to equations is the ability to modify an original equation by performing operations (such as [[addition]], [[subtraction]], [[multiplication]], [[division]], and [[exponential function|powers]]). | A unique aspect to equations is the ability to modify an original equation by performing operations (such as [[addition]], [[subtraction]], [[multiplication]], [[division]], and [[exponential function|powers]]). | ||
| − | It's important to note the distinction between an equation and an | + | It's important to note the distinction between an equation and an ''[[identity]]''. An identity in terms of some [[variable]]s states that two expressions are equal for every value of those variables: for example, |
<math>x^2 - y^2 = (x - y)(x + y)</math> | <math>x^2 - y^2 = (x - y)(x + y)</math> | ||
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is an equation that is true for some particular values of <math>x</math>. | is an equation that is true for some particular values of <math>x</math>. | ||
| + | |||
| + | In other words, one can say that an identity is a [[tautology | tautological]] equation. | ||
==See also== | ==See also== | ||
Revision as of 06:53, 24 April 2008
| This is an AoPSWiki Word of the Week for April 15-22 |
An equation is a relation which states that two expressions are equal, identical, or otherwise the same. Equations are easily identifiable because they are composed of two expressions with an equals sign ('=') between them.
Equations are similar to congruences (which relate geometric figures instead of numbers) and other relationships which fall into the category of equivalence relations.
A unique aspect to equations is the ability to modify an original equation by performing operations (such as addition, subtraction, multiplication, division, and powers).
It's important to note the distinction between an equation and an identity. An identity in terms of some variables states that two expressions are equal for every value of those variables: for example,
is an identity that is true regardless of the values of 
and 
(and indeed holds in a commutative ring). However,
is an equation that is true for some particular values of .
In other words, one can say that an identity is a tautological equation.
See also
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