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What is the difference between an identity, an equation and a conditional equation?

Thank you?

EuYu
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LosAngelesMath
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2 Answers2

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Equation means equality. They are both related to the word equal. If such an equality is true for all values of the variable, it is called an identity, e.g., $\sin^2x+\cos^2x=1$ is true for all x. If however the equation in question only holds for some values, which one is supposed to determine, then it's called conditional, and its variable is termed an unknown.

Lucian
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    An identity may also contain no variables, such as $e^{i\pi}+1=0$. – zz20s May 25 '16 at 02:15
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    Sometimes identities have restrictions on the allowed values. Maybe that's poor form on the part of textbook authors though. E.g. $\cos x \tan x = \sin x$ is usually called an identity even though the left and right sides could be argued to have distinct domains. Of course, this is going by the language in a precalc book. – jdods May 25 '16 at 02:19
  • You could essentially view conditional equations as a label for equations used in algebra. Either way, equations and identities assert equality, only identities assert that both sides represent the same mathematical object, which implies that for every valuable of the variable on both sides the equality holds. – lmn32 Jul 18 '20 at 21:19
  • @Lucian Shouldn't we also specify the allowed values of the variable $x$? I mean for real numbers $x\cdot y= y \cdot x$ but is not an identity for matrices. I know I am nitpicky here but if we let $x$ to be a member of the set "real numbers-matrices" that is the set that contains all real numbers and all matrices then the equation $x \cdot y= y \cdot x$ would not be an identity. – user599310 Sep 01 '20 at 20:45
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    @user599310: Within the set of real matrices, the product is usually not commutative; as such, over that particular set, the aforementioned relation does not constitute an identity; however, over a restricted (infinite) subset, it might. By all variables, I meant all variables within a certain specified domain. – Lucian Sep 01 '20 at 22:05
  • @Lucian Thanks for the clarification. – user599310 Sep 07 '20 at 21:27
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Equation is a mathematical description which is equal only for one variable. But identity is a mathematical description which is always equal for any variable.

atreya
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    This answer is quite inaccurate. – projectilemotion May 31 '17 at 13:38
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    Perhaps you mean “value of a variable” in place of “variable”? That would be closer. Although equations and identities can both have several variables, and equations can have several solutions. – Matthew Leingang Mar 01 '19 at 13:42
  • You could've used "relationship" instead of mathematical description, which would still allow an equation to be a description of a real life scenario – lmn32 Jul 18 '20 at 21:16