document.write( "Question 510131: whats the difference between definition and theorem?
\n" ); document.write( "example: definition of mid point vs mid point theorem? \n" ); document.write( "

Algebra.Com's Answer #341804 by richard1234(7193) $\"\"$ $\"About$

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Definitions do not need to be proven; they are simply definitions. A theorem is a statement that requires a proof, in which the proof is often based on previously proven theorems or axioms. It is important you know the difference between a definition and a theorem; when you have to prove something you need to make clear definitions and label your steps sequentially. Here are several examples of definitions/theorems from a wide variety of subjects:\r
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\n" ); document.write( "\n" ); document.write( "Definition: for a given line segment there exists a midpoint.
\n" ); document.write( "Theorem: the midpoint divides the segments into two equal segments.\r
\n" ); document.write( "\n" ); document.write( "---------------------\r
\n" ); document.write( "\n" ); document.write( "Definition: we can define a complex number in the form a + bi where i is the imaginary unit.
\n" ); document.write( "Theorem:

(Euler's formula)\r
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\n" ); document.write( "Definition: given arbitrary non-negative real numbers

, ...,

and

, ...,

, we may assume

for

and

for

.\r
\n" ); document.write( "\n" ); document.write( "Theorem: The sum

is maximized with

and minimized with

(Rearrangement inequality).\r
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\n" ); document.write( "Definition: the altitudes of a triangle meet at a common point called the orthocenter.
\n" ); document.write( "Theorem: there exists such a point (this is a tricky one and the proof is not often covered in geometry classes).\r
\n" ); document.write( "\n" ); document.write( "---------------------\r
\n" ); document.write( "\n" ); document.write( "Definition: z is a root of the polynomial

if f(z) = 0.
\n" ); document.write( "Theorem: Any n-degree polynomial contains exactly n complex roots, including multiple roots (fundamental theorem of algebra). \n" ); document.write( "

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