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| − | <pre> 7MM?
| + | '''Absolute value''' is a [[function]] measuring a number's distance from zero. The absolute value of <math>x</math> is written <math>|x|</math>. |
| − | MMMMM7MM
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| − | MMM+ $MMM MMMMD ,M MMMMM
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| − | M MMM MM MMM MMMMM MM
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| − | OM MO MM MMN MMM MM
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| − | ,M MM, MM MMI MMM M
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| − | M MMM MMM MM? MM MM
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| − | M NM MM MM MM MM
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| − | M MMM MM MMM M MM
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| − | M NM MMM MM M MM
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| − | 7 MM MMM MM M MMMMMM
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| − | M MM MMM :MM M MMMMI7M
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| − | M M MMM MM + MMM MM
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| − | M MMMM= M +MM ,MM MMM MMM+
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| − | M 8M MMMMMM NM MMM MM MMM, MMMIMM
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| − | ID MM MMMMM M M+ NMM MMD MM MM
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| − | M MM MMMM M MM MM NMM MN MM NMM
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| − | M: M MM M OM MMMMMMMMMMM, ,MM
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| − | M $ MI MM MM MMMMMMMMM ~MM
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| − | M MM M MM OM OMMMN MMM
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| − | MMMMMZ M MM, MN MMMMMMM MMM
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| − | =MM MM MM NMMMMMM DMMMMM
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| − | MZ M?NMMM MMM MM +MM MM
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| − | MM ~MM MMMMM MM MMMMM,
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| − | M MMM MMM M ZMM MMM
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| − | MM ,M: MM MM
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| − | MMMMMM MM MM
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| − | M+ M MM NMMMM
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| − | M+ M MM MMMM
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| − | MM MMMM MMM
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| − | MM 7MMM MM
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| − | MM MO MM
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| − | MM MM MM
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| − | MM MM MM
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| − | +MZ ZM MM
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| − | MM M MM MM
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| − | MM $MMM MM
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| − | MM, MM MM
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| − | MM M ?M$ MM
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| − | NM~ M MI MMM
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| − | MM MM M M MMM
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| − | MMMM M M MM MMMM
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| − | :MM IM M MMMMMMMMM
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| − | MM ,MMMMM M MMZ DMMO
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| − | ?M MM+ M$ MMM MM
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| − | ZMMMMM M MMMMMMMM M
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| − | $M $ MMM N M:
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| − | MM M MMMM MD
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| − | MM M M MMMZ M8
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| − | MM ? MMM MMMM M
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| − | MM M OMMM8 M MM MMMMMMN +M
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| − | 8M MMMMM, MMMMMMMM7 MM
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| − | M MM~ MM
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| − | M MMN
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| − | MM MMM
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| − | MM +MM~
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| − | 7MM MMM$
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| − | MMM8 MMMZ
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| − | DMMM MMMM
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| − | MN MMMMMN
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| − | MMM7 MMMMMI
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| − | MMMMMMMMMMMMMMMM: MMM
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| − | :MMMMD ~ MMMM
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| − | MMMMMMMM$ MMMMMMM D :MMMM
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| − | MO MMM DMMMMMMMMMMMO~MMMMMMMM 7MMMMMM
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| − | MM MMN MMMMMMMMM
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| − | 8MMMMMMMM MM MM~MMMM=
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| − | ,MMMMMM: 7MM MM MM
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| − | MMM7 :~?I 8MM MM MM
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| − | MMMMMMMMMMMMMMM MMM +MMM, MM
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| − | MMD MMM, MMMMMMM MM+ ?MMM= ZMM
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| − | MM NMMMMMI +MMM M MMMM MMM
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| − | MM MMMM MD8M MMM MMMMM MM
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| − | MM MMMN MMZ M MMMMMMMMMM MMMM8 MM
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| − | M,MM M$ MM MN MMMMMMMM MM
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| − | M+ MMM M MM DMM
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| − | MZ MM MM $
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| − | MM MM
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| − | MMMM
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| − | NM=
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| − | </pre>
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| − | The '''absolute value''' of a number is a measure of the size of that number. The absolute value of <math>x</math> is written <math>|x|</math>.
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| | | | |
| − | :If <math>x</math> is a positive number, then <math>|x| = x</math>.
| + | *If <math>x</math> is a positive number, then <math>|x| = x</math>. |
| | + | *If <math>x</math> is a negative number, then <math>|x| = -x</math>. |
| | + | *If <math>x=0</math> then <math>|x| = 0</math>. |
| | | | |
| − | :If <math>x</math> is a negative number, then <math>|x| = x</math>.
| + | For example, <math>|-3| = 3</math>, and <math>|5| = 5</math>. Notice that <math>|x|</math> is never negative. |
| | | | |
| − | :If <math>x=0</math> then <math>|x| = 0</math>.
| + | Absolute value has several useful properties. One is the ''multiplicative'' property. If <math>x</math> and <math>y</math> are two numbers, then <math>|xy| = |x| \times |y|</math>. Another is the ''triangle inequality'', which is the fact that <math>|x+y| \leq |x| + |y|</math>. For example, if <math>x = 3</math> and <math>y = -5</math>, then <math>|x+y| = |3 + (-5)| = |3 - 5| = |-2| = 2</math>, while <math>|x| + |y| = |-5| + |3| = 5 + 3 = 8</math>. In this case, the triangle inequality is the fact that 2 is not more than 8. |
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| − | Absolute value has several useful properties. One is the ''multiplicative'' property. If <math>x</math> and <math>y</math> are two numbers, then <math>|xy| = |x| \times |y|</math>. Another is the ''triangle inequality'', which is the fact that <math>|x+y| \leq |x| + |y|</math>. For example, if <math>x = 3</math> and <math>y = -5</math>, then <math>|x+y| = |3 + (-5)| = |3 - 5| = |-2| = 2</math>, while <math>|x| + |y| = |-5| + |3| = 5 + 3 = 8</math>. In this case, the triangle inequality is the fact that 2 is not more than 8.
| + | If <math>x</math> is a [[real number]], then <math>\sqrt{x^2} = |x|</math>. The absolute value is necessary because the principal [[square root]] is, by definition, nonnegative. |
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| − | [[Complex number]]s also have an absolute value. If <math>z = x+iy</math> is a complex number with real part <math>x</math> and imaginary part <math>y</math>, then <math>|z| = \sqrt{x^2 + y^2}</math>. If we represent <math>z</math> as a point in the complex plane with coordinates <math>(x,y)</math>, then <math>|z|</math> is the distance from this point to the origin. The absolute value of complex numbers also has the multiplicative property and satisfies the triangle inequality. | + | [[Complex number]]s also have an absolute value (sometimes called the modulus). If <math>z = x+iy</math> is a complex number with real part <math>x</math> and imaginary part <math>y</math>, then <math>|z| = \sqrt{x^2 + y^2}</math>. If we represent <math>z</math> as a point in the complex plane with coordinates <math>(x,y)</math>, then <math>|z|</math> is the distance from this point to the origin. The absolute value of complex numbers also has the multiplicative property and satisfies the triangle inequality. |
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| | [[Category:Algebra Terms]] | | [[Category:Algebra Terms]] |
| − | [[Category:Algebra]]
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| − | [[Category:Mathematics]]
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