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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 positive real number.  
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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 nonnegative real number.  
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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>.
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==Homomorphism property==
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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.
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===Proof===
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Let <math>z = a + bi</math> and <math>\omega = c + di</math> be complex numbers.
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We have <cmath>z\omega = (a + bi)(c + di) = (ac - bd) + (ad + bc)i,</cmath> so
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<cmath>\begin{align*} |z\omega| &= \sqrt{(ac - bd)^2 + (ad + bc)^2} \\
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&= \sqrt{(ac)^2 - 2abcd + (bd)^2 + (ad)^2 + 2abcd + (bc)^2} \\
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&= \sqrt{(ac)^2 + (bd)^2 + (ad)^2 + (bc)^2} \\
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&= \sqrt{(a^2 + b^2)(c^2 + d^2)} \\
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&= \sqrt{(a^2 + b^2)}\sqrt{(c^2 + d^2)} \\
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&= |z||\omega|.\\
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\end{align*}</cmath>
  
 
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[[Category:Definition]]
 
[[Category:Definition]]

Latest revision as of 18:27, 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 $x$ is its absolute value $|x|$, sometimes written $\sqrt{x^2}$. The magnitude $|z|$ of a complex number $z$ equals $\sqrt{\mathrm {Re}(z)^2 + \mathrm{Im}(z)^2}$. Both types of magnitude are bound by a form of the Triangle Inequality which states that $|a| + |b| \geq |a + b|$.

Homomorphism property

For complex numbers $z$ and $\omega$, we have the identity $|z\omega| = |z||\omega|$. 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 $z = a + bi$ and $\omega = c + di$ be complex numbers.

We have \[z\omega = (a + bi)(c + di) = (ac - bd) + (ad + bc)i,\] so \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*}

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