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Topic: Equations vs Identities vs Formulae (Read 491 times) |
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Sir Col
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Equations vs Identities vs Formulae
« on: Oct 5th, 2007, 11:53am » |
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I think I know the answer to this, but research on all of these terms lead to several contradictory "definitions". So I'd appreciate your take on the meaning of words "equation", "identity", and "formula" in the context of algebra in mathematics, but especially "formula".
My understanding...
An expression is a collection of terms made up of functions, variables, and/or numerical values.
In an identity the expression on the left hand side is ALWAYS equal to the expression on the right hand side; for example, 3+4=7, 2(3x+1)=6x+2, sin2(x)+cos2(x)=1.
In an equation the expression on the left hand side is NOT always equal to the expression on the right hand side; for example, x+4=7, 2(3x+1)=5x+3, sin(x)+cos(x)=1.
Solving an equation is determining when the two expressions are equal.
I have seen some sources disagreeing with the above, arguing that two expressions separated by an equal sign are an equation by definition and identities are special case equations, for which the LHS and RHS are always equal. Others, like my own understanding, argue that equations and identities are mutually exclusive; cf. oblongs vs squares.
Then we have formulae... we would all agree that C=pi*d, s=ut+at2/2, A=a*b*sin(C)/2 are formulae.
So what is a formula?
Is it an identity? Or is it simply an expression for which the equal sign is merely used to say, "the value of the following expression can be represented by the letter, __"?
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TenaliRaman
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Re: Equations vs Identities vs Formulae
« Reply #1 on: Oct 5th, 2007, 1:48pm » |
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I have always "solved the equation" and "proved the identity", but never "solved an identity" and "proved an equation".
For formula, i stick to the definition given here,
(mathematics) a standard procedure for solving a class of mathematical problems (ref)
So you have a formula that solves the quadratic equation, a formula that solves the cubic, so on and so forth.
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towr
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Re: Equations vs Identities vs Formulae
« Reply #2 on: Oct 6th, 2007, 1:14pm » |
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An equation is the result of equating things, an identity of identifying things (with eachother), and a formula is the result of a formulation (formal notation).
I'd subscribe to the idea that an identity is a special case of an equation, where the left and right hand side are not just incidentally equal, but always. Hence for an equation it makes sense to find out the condition under which it is true, while for an identity it is true under all conditions (which is still something that may need to be proven).
A formula for, say, a function would be an identity, telling you exactly what it is equal to for it's whole domain. But e.g. inequalities are just as much formulas, imo.
Basicly I'd say that a formula is a mathematical (or logical, or in any case, formal) expression.
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