Difference between revisions of "Absolute value"

Latest revision as of 23:05, March 1, 2021

Absolute value is a function measuring a number's distance from zero. The absolute value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is written Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x|} .

If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is a positive number, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x| = x} .
If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is a negative number, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x| = -x} .
If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=0} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |x| = 0} .

For example, $|-3|=3$ , and $|5|=5$ . Notice that $|x|$ is never negative.

Absolute value has several useful properties. One is the multiplicative property. If $x$ and $y$ are two numbers, then $|xy|=|x|\times |y|$ . Another is the triangle inequality, which is the fact that $|x+y|\leq |x|+|y|$ . For example, if $x=3$ and $y=-5$ , then $|x+y|=|3+(-5)|=|3-5|=|-2|=2$ , while $|x|+|y|=|-5|+|3|=5+3=8$ . In this case, the triangle inequality is the fact that 2 is not more than 8.

If $x$ is a real number, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sqrt{x^2} = |x|} . The absolute value is necessary because the principal square root is, by definition, nonnegative.

Complex numbers also have an absolute value (sometimes called the modulus). If Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z = x+iy} is a complex number with real part Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} and imaginary part Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z| = \sqrt{x^2 + y^2}} . If we represent Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z} as a point in the complex plane with coordinates Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x,y)} , then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z|} 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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-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>.
+'''Absolute value''' is a [[function]] measuring a number's distance from zero.  The absolute value of <math>x</math> is written <math>|x|</math>.
-: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.
-[[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.
+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.
-==Notes and references==
-<references/>
-[[Category:Algebra]]
-<b>If x is a negative number, then  | x | = − x.</b>
+[[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.
-This is incorrect.  It should read "If x is a negative number, then  | - x | = x."
+[[Category:Algebra Terms]]

Difference between revisions of "Absolute value"

Latest revision as of 23:05, March 1, 2021

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