Difference between revisions of "Magnitude"
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| − | A '''magnitude''' is a measure of the size of a mathematical entity. For example, the magnitude of a [[complex number]] is the distance from the number (graphed on the complex plane) to the origin, a measure of the size of a complex number. The magnitude is generally a | + | A '''magnitude''' is a measure of the size of a mathematical entity. For example, the magnitude of a [[complex number]] is the distance from the number (graphed on the complex plane) to the origin, a measure of the size of a complex number. The magnitude is generally a nonnegative real number. |
Formulaically, the magnitude of a real number <math>x</math> is its [[Absolute value | absolute value]] <math>|x|</math>, sometimes written <math>\sqrt{x^2}</math>. The magnitude <math>|z|</math> of a complex number <math>z</math> equals <math>\sqrt{\mathrm {Re}(z)^2 + \mathrm{Im}(z)^2}</math>. Both types of magnitude are bound by a form of the [[Triangle Inequality]] which states that <math>|a| + |b| \geq |a + b|</math>. | Formulaically, the magnitude of a real number <math>x</math> is its [[Absolute value | absolute value]] <math>|x|</math>, sometimes written <math>\sqrt{x^2}</math>. The magnitude <math>|z|</math> of a complex number <math>z</math> equals <math>\sqrt{\mathrm {Re}(z)^2 + \mathrm{Im}(z)^2}</math>. Both types of magnitude are bound by a form of the [[Triangle Inequality]] which states that <math>|a| + |b| \geq |a + b|</math>. | ||
| + | |||
| + | ==Homomorphism property== | ||
| + | For complex numbers <math>z</math> and <math>\omega</math>, we have the identity <math>|z\omega| = |z||\omega|</math>. Because the absolute value of a real number equals its magnitude when treated as a complex number, the identity also holds for absolute values of real numbers. | ||
| + | |||
| + | ===Proof=== | ||
| + | Let <math>z = a + bi</math> and <math>\omega = c + di</math> be complex numbers. | ||
| + | |||
| + | We have <cmath>z\omega = (a + bi)(c + di) = (ac - bd) + (ad + bc)i,</cmath> so | ||
| + | <cmath>\begin{align*} |z\omega| &= \sqrt{(ac - bd)^2 + (ad + bc)^2} \\ | ||
| + | &= \sqrt{(ac)^2 - 2abcd + (bd)^2 + (ad)^2 + 2abcd + (bc)^2} \\ | ||
| + | &= \sqrt{(ac)^2 + (bd)^2 + (ad)^2 + (bc)^2} \\ | ||
| + | &= \sqrt{(a^2 + b^2)(c^2 + d^2)} \\ | ||
| + | &= \sqrt{(a^2 + b^2)}\sqrt{(c^2 + d^2)} \\ | ||
| + | &= |z||\omega|.\\ | ||
| + | \end{align*}</cmath> | ||
{{stub}} | {{stub}} | ||
[[Category:Definition]] | [[Category:Definition]] | ||
Latest revision as of 17:27, 2 March 2023
A magnitude is a measure of the size of a mathematical entity. For example, the magnitude of a complex number is the distance from the number (graphed on the complex plane) to the origin, a measure of the size of a complex number. The magnitude is generally a nonnegative real number.
Formulaically, the magnitude of a real number 
is its absolute value 
, sometimes written 
. The magnitude 
of a complex number 
equals 
. Both types of magnitude are bound by a form of the Triangle Inequality which states that 
.
Homomorphism property
For complex numbers and 
, we have the identity 
. Because the absolute value of a real number equals its magnitude when treated as a complex number, the identity also holds for absolute values of real numbers.
Proof
Let 
and 
be complex numbers.
We have ![\[z\omega = (a + bi)(c + di) = (ac - bd) + (ad + bc)i,\]](http://latex.artofproblemsolving.com/c/2/6/c2630dbca4c13257b2739244f39c5ce7f07d4513.png)
so
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