Difference between revisions of "Absolute value"
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| − | + | '''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 negative number, then <math>|x| = -x</math>. | ||
| + | *If <math>x=0</math> then <math>|x| = 0</math>. | ||
| − | + | For example, <math>|-3| = 3</math>, and <math>|5| = 5</math>. Notice that <math>|x|</math> is never negative. | |
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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. | 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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| + | 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. | ||
[[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. | [[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. | ||
[[Category:Algebra Terms]] | [[Category:Algebra Terms]] | ||
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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 .
- If is a positive number, then .
- If is a negative number, then .
- If then .
For example, , and . Notice that 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 never negative.
Absolute value has several useful properties. One is the multiplicative property. 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} and 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} are two numbers, 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 |xy| = |x| \times |y|} . Another is the triangle inequality, which is the fact that 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| \leq |x| + |y|} . For example, 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 = 3} and 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 = -5} , 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+y| = |3 + (-5)| = |3 - 5| = |-2| = 2} , while 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| = |-5| + |3| = 5 + 3 = 8} . In this case, the triangle inequality is the fact that 2 is not more than 8.
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 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.