Absolute Value of a Complex Number and Complex Conjugates

The Geometric Interpretation of Complex Numbers Functions

Absolute Value of a Complex Number and Complex Conjugates

The absolute value (or modulus) of a complex number is its distance from the origin in the complex plane:

The complex conjugate of is , obtained by reflecting across the real axis.

Quick Reference

Concept	Formula	Example
Modulus	\bar{z} = a - bi\overline{3 + 4i} = 3 - 4iz \bar{z}	z
Product of moduli		z/w
Modulus of conjugate	z = x + yi $You can't use 'macro parameter character #' in math modeis:</p> <mjx-container aria-label="\|z\| = \sqrt{x^2 + y^2}" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.564ex;" xmlns="http://www.w3.org/2000/svg" width="14.779ex" height="2.946ex" role="img" focusable="false" viewBox="0 -1052.7 6532.1 1302.2" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mi" transform="translate(278,0)"><use data-c="1D467" xlink:href="#MJX-TEX-I-1D467"></use></g><g data-mml-node="mo" transform="translate(743,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(1298.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="msqrt" transform="translate(2354.6,0)"><g transform="translate(1020,0)"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mn" transform="translate(605,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1230.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="msup" transform="translate(2231,0)"><g data-mml-node="mi"><use data-c="1D466" xlink:href="#MJX-TEX-I-1D466"></use></g><g data-mml-node="mn" transform="translate(523,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g></g><g data-mml-node="mo" transform="translate(0,142.7)"><use data-c="221A" xlink:href="#MJX-TEX-SO-221A"></use></g><rect width="3157.6" height="60" x="1020" y="932.7"></rect></g></g></g></svg></mjx-container><p>Geometrically,$ \|z\|z\|z\| \geq 0z\|z\| = 0z = 0x = 0y = 0a + 0i\|a + 0i\| = \sqrt{a^2} = \|a\|3 + 4i-2 + 0i0 + 5i $You can't use 'macro parameter character #' in math mode.</p> <p><strong>Solution.</strong></p> <mjx-container aria-label="\|3 + 4i\| = \sqrt{9 + 16} = \sqrt{25} = 5" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.564ex;" xmlns="http://www.w3.org/2000/svg" width="29.531ex" height="2.821ex" role="img" focusable="false" viewBox="0 -997.3 13052.6 1246.8" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(278,0)"><use data-c="33" xlink:href="#MJX-TEX-N-33"></use></g><g data-mml-node="mo" transform="translate(1000.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2000.4,0)"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g><g data-mml-node="mi" transform="translate(2500.4,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(2845.4,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(3401.2,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="msqrt" transform="translate(4457,0)"><g transform="translate(853,0)"><g data-mml-node="mn"><use data-c="39" xlink:href="#MJX-TEX-N-39"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="36" xlink:href="#MJX-TEX-N-36" transform="translate(500,0)"></use></g></g><g data-mml-node="mo" transform="translate(0,107.3)"><use data-c="221A" xlink:href="#MJX-TEX-N-221A"></use></g><rect width="2722.4" height="60" x="853" y="847.3"></rect></g><g data-mml-node="mo" transform="translate(8310.2,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="msqrt" transform="translate(9366,0)"><g transform="translate(853,0)"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use><use data-c="35" xlink:href="#MJX-TEX-N-35" transform="translate(500,0)"></use></g></g><g data-mml-node="mo" transform="translate(0,137.2)"><use data-c="221A" xlink:href="#MJX-TEX-N-221A"></use></g><rect width="1000" height="60" x="853" y="877.2"></rect></g><g data-mml-node="mo" transform="translate(11496.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(12552.6,0)"><use data-c="35" xlink:href="#MJX-TEX-N-35"></use></g></g></g></svg></mjx-container><mjx-container aria-label="\|-2 + 0i\| = \sqrt{4 + 0} = 2" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.564ex;" xmlns="http://www.w3.org/2000/svg" width="23.956ex" height="2.765ex" role="img" focusable="false" viewBox="0 -972.8 10588.4 1222.2" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(500.2,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1500.4,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><g data-mml-node="mo" transform="translate(2222.7,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(3222.9,0)"><use data-c="30" xlink:href="#MJX-TEX-N-30"></use></g><g data-mml-node="mi" transform="translate(3722.9,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(4067.9,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(4623.7,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="msqrt" transform="translate(5679.4,0)"><g transform="translate(853,0)"><g data-mml-node="mn"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="30" xlink:href="#MJX-TEX-N-30"></use></g></g><g data-mml-node="mo" transform="translate(0,112.7)"><use data-c="221A" xlink:href="#MJX-TEX-N-221A"></use></g><rect width="2222.4" height="60" x="853" y="852.7"></rect></g><g data-mml-node="mo" transform="translate(9032.7,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(10088.4,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g></g></svg></mjx-container><mjx-container aria-label="\|0 + 5i\| = \sqrt{0 + 25} = 5" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.564ex;" xmlns="http://www.w3.org/2000/svg" width="22.321ex" height="2.753ex" role="img" focusable="false" viewBox="0 -967.3 9866 1216.8" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(278,0)"><use data-c="30" xlink:href="#MJX-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(1000.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2000.4,0)"><use data-c="35" xlink:href="#MJX-TEX-N-35"></use></g><g data-mml-node="mi" transform="translate(2500.4,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g><g data-mml-node="mo" transform="translate(2845.4,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(3401.2,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="msqrt" transform="translate(4457,0)"><g transform="translate(853,0)"><g data-mml-node="mn"><use data-c="30" xlink:href="#MJX-TEX-N-30"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use><use data-c="35" xlink:href="#MJX-TEX-N-35" transform="translate(500,0)"></use></g></g><g data-mml-node="mo" transform="translate(0,107.3)"><use data-c="221A" xlink:href="#MJX-TEX-N-221A"></use></g><rect width="2722.4" height="60" x="853" y="847.3"></rect></g><g data-mml-node="mo" transform="translate(8310.2,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(9366,0)"><use data-c="35" xlink:href="#MJX-TEX-N-35"></use></g></g></g></svg></mjx-container></div><p><span id="modulus-properties"></span></p> <h2>Properties of the Modulus</h2> <div class="highlight"><p><strong>Multiplicative properties of the modulus.</strong> For any complex numbers$ zww \neq 0 $You can't use 'macro parameter character #' in math mode):</p> <mjx-container aria-label="\|zw\| = \|z\|\,\|w\|, \qquad \left\|\frac{z}{w}\right\| = \frac{\|z\|}{\|w\|}" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -2.17ex;" xmlns="http://www.w3.org/2000/svg" width="29.056ex" height="5.471ex" role="img" focusable="false" viewBox="0 -1459 12842.8 2418" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mi" transform="translate(278,0)"><use data-c="1D467" xlink:href="#MJX-TEX-I-1D467"></use></g><g data-mml-node="mi" transform="translate(743,0)"><use data-c="1D464" xlink:href="#MJX-TEX-I-1D464"></use></g><g data-mml-node="mo" transform="translate(1459,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(2014.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mo" transform="translate(3070.6,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mi" transform="translate(3348.6,0)"><use data-c="1D467" xlink:href="#MJX-TEX-I-1D467"></use></g><g data-mml-node="mo" transform="translate(3813.6,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mstyle" transform="translate(4091.6,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mo" transform="translate(4258.6,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mi" transform="translate(4536.6,0)"><use data-c="1D464" xlink:href="#MJX-TEX-I-1D464"></use></g><g data-mml-node="mo" transform="translate(5252.6,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mo" transform="translate(5530.6,0)"><use data-c="2C" xlink:href="#MJX-TEX-N-2C"></use></g><g data-mml-node="mstyle" transform="translate(5808.6,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mrow" transform="translate(7975.2,0)"><g data-mml-node="mo"><svg width="278" height="1894" y="-697" x="27.5" viewBox="0 -234.6 278 1894"><use data-c="2223" xlink:href="#MJX-TEX-S4-2223" transform="scale(1,2.844)"></use></svg></g><g data-mml-node="mfrac" transform="translate(333,0)"><g data-mml-node="mi" transform="translate(345.5,676)"><use data-c="1D467" xlink:href="#MJX-TEX-I-1D467"></use></g><g data-mml-node="mi" transform="translate(220,-686)"><use data-c="1D464" xlink:href="#MJX-TEX-I-1D464"></use></g><rect width="916" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(1489,0)"><svg width="278" height="1894" y="-697" x="27.5" viewBox="0 -234.6 278 1894"><use data-c="2223" xlink:href="#MJX-TEX-S4-2223" transform="scale(1,2.844)"></use></svg></g></g><g data-mml-node="mo" transform="translate(10075,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(11130.8,0)"><g data-mml-node="mrow" transform="translate(345.5,709.5)"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mi" transform="translate(278,0)"><use data-c="1D467" xlink:href="#MJX-TEX-I-1D467"></use></g><g data-mml-node="mo" transform="translate(743,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g></g><g data-mml-node="mrow" transform="translate(220,-709.5)"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mi" transform="translate(278,0)"><use data-c="1D464" xlink:href="#MJX-TEX-I-1D464"></use></g><g data-mml-node="mo" transform="translate(994,0) translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g></g><rect width="1472" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container></div><details><summary>Proof of$ \|zw\| = \|z\|\|w\|z = a + biw = c + di\|zw\| = \|z\|\|w\|z = a + bi $You can't use 'macro parameter character #' in math modeis:</p> <mjx-container aria-label="\bar{z} = a - bi" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="9.783ex" height="1.756ex" role="img" focusable="false" viewBox="0 -694 4324 776" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D467" xlink:href="#MJX-TEX-I-1D467"></use></g><g data-mml-node="mo" transform="translate(288.1,3) translate(-250 0)"><use data-c="AF" xlink:href="#MJX-TEX-N-AF"></use></g></g></g><g data-mml-node="mo" transform="translate(742.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(1798.6,0)"><use data-c="1D44E" xlink:href="#MJX-TEX-I-1D44E"></use></g><g data-mml-node="mo" transform="translate(2549.8,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mi" transform="translate(3550,0)"><use data-c="1D44F" xlink:href="#MJX-TEX-I-1D44F"></use></g><g data-mml-node="mi" transform="translate(3979,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g></g></g></svg></mjx-container><p>Geometrically,$ \bar{z}zz + \bar{z} = 2a = 2\operatorname{Re}(z)z - \bar{z} = 2bi = 2i\operatorname{Im}(z)z\bar{z} = a^2 + b^2 = \|z\|^2\overline{z + w} = \bar{z} + \bar{w}\overline{zw} = \bar{z}\,\bar{w}\overline{\bar{z}} = z\|\bar{z}\| = \|z\|z = 2 - 3i\bar{z}z + \bar{z}z - \bar{z}z\bar{z}\bar{z} = 2 + 3iz + \bar{z} = (2 - 3i) + (2 + 3i) = 4z - \bar{z} = (2 - 3i) - (2 + 3i) = -6iz\bar{z} = (2 - 3i)(2 + 3i) = 4 + 9 = 13 = \|z\|^2z \bar{z} = \|z\|^2z \neq 0z $You can't use 'macro parameter character #' in math modecan be written as:</p> <mjx-container aria-label="z^{-1} = \frac{1}{z} = \frac{\bar{z}}{z\bar{z}} = \frac{\bar{z}}{\|z\|^2}" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -2.193ex;" xmlns="http://www.w3.org/2000/svg" width="21.855ex" height="5.291ex" role="img" focusable="false" viewBox="0 -1369 9659.9 2338.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msup"><g data-mml-node="mi"><use data-c="1D467" xlink:href="#MJX-TEX-I-1D467"></use></g><g data-mml-node="TeXAtom" transform="translate(498,413) scale(0.707)" data-mjx-texclass="ORD"><g data-mml-node="mo"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(778,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g></g></g><g data-mml-node="mo" transform="translate(1729.5,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(2785.2,0)"><g data-mml-node="mn" transform="translate(220,676)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mi" transform="translate(237.5,-686)"><use data-c="1D467" xlink:href="#MJX-TEX-I-1D467"></use></g><rect width="700" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(4003,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(5058.8,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(452.5,676)"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D467" xlink:href="#MJX-TEX-I-1D467"></use></g><g data-mml-node="mo" transform="translate(288.1,3) translate(-250 0)"><use data-c="AF" xlink:href="#MJX-TEX-N-AF"></use></g></g></g><g data-mml-node="mrow" transform="translate(220,-686)"><g data-mml-node="mi"><use data-c="1D467" xlink:href="#MJX-TEX-I-1D467"></use></g><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(465,0)"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D467" xlink:href="#MJX-TEX-I-1D467"></use></g><g data-mml-node="mo" transform="translate(288.1,3) translate(-250 0)"><use data-c="AF" xlink:href="#MJX-TEX-N-AF"></use></g></g></g></g><rect width="1130" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(6706.6,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(7762.3,0)"><g data-mml-node="TeXAtom" data-mjx-texclass="ORD" transform="translate(716.3,676)"><g data-mml-node="mover"><g data-mml-node="mi"><use data-c="1D467" xlink:href="#MJX-TEX-I-1D467"></use></g><g data-mml-node="mo" transform="translate(288.1,3) translate(-250 0)"><use data-c="AF" xlink:href="#MJX-TEX-N-AF"></use></g></g></g><g data-mml-node="mrow" transform="translate(220,-719.9)"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mi" transform="translate(278,0)"><use data-c="1D467" xlink:href="#MJX-TEX-I-1D467"></use></g><g data-mml-node="msup" transform="translate(743,0)"><g data-mml-node="mo" transform="translate(0 -0.5)"><use data-c="7C" xlink:href="#MJX-TEX-N-7C"></use></g><g data-mml-node="mn" transform="translate(311,289) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g></g><rect width="1657.6" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container><p>This confirms the formula from Section 4.3 and shows clearly that the conjugate is the key tool for inverting a complex number.</p> <div class="example"><p><strong>Example 3.</strong> Find the reciprocal of$ 1 + 2i $You can't use 'macro parameter character #' in math mode.</p> <p><strong>Solution.</strong></p> <mjx-container aria-label="\frac{1}{1 + 2i} = \frac{\overline{1+2i}}{\|1+2i\|^2} = \frac{1 - 2i}{1 + 4} = \frac{1}{5} - \frac{2}{5}\,i" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -2.193ex;" xmlns="http://www.w3.org/2000/svg" width="39.887ex" height="5.947ex" role="img" focusable="false" viewBox="0 -1659 17630 2628.4" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mfrac"><g data-mml-node="mn" transform="translate(1253.7,676)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mrow" transform="translate(220,-686)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="32" 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width="3760" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(8618.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(9674.6,0)"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><g data-mml-node="mi" transform="translate(2222.4,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g></g><g data-mml-node="mrow" transform="translate(392.5,-686)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(722.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(1722.4,0)"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use></g></g><rect width="2767.4" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(12959.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(14015.6,0)"><g data-mml-node="mn" transform="translate(220,676)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mn" transform="translate(220,-686)"><use data-c="35" xlink:href="#MJX-TEX-N-35"></use></g><rect width="700" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(15177.8,0)"><use data-c="2212" xlink:href="#MJX-TEX-N-2212"></use></g><g data-mml-node="mfrac" transform="translate(16178,0)"><g data-mml-node="mn" transform="translate(220,676)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><g data-mml-node="mn" transform="translate(220,-686)"><use data-c="35" xlink:href="#MJX-TEX-N-35"></use></g><rect width="700" height="60" x="120" y="220"></rect></g><g data-mml-node="mstyle" transform="translate(17118,0)"><g data-mml-node="mspace"></g></g><g data-mml-node="mi" transform="translate(17285,0)"><use data-c="1D456" xlink:href="#MJX-TEX-I-1D456"></use></g></g></g></svg></mjx-container></div><h2>Frequently Asked Questions</h2> <details><summary>What is the difference between the absolute value and the modulus?</summary> <p>They are the same thing for complex numbers. The term <em>absolute value</em> extends from the real number concept$ \|a\| = \sqrt{a^2}a\|a + 0i\| = \sqrt{a^2} = \|a\|z\bar{z}z = a + biz\bar{z} = (a+bi)(a-bi) = a^2 + b^2a = b = 0z = 0\bar{z}zz = a + bi(a, b)\bar{z} = a - bi(a, -b)z_1z_2\|z_1 - z_2\|\|z\|z\|zw\| = \|z\|\|w\|w\|w\|$. This is fundamental to understanding multiplication geometrically (rotation and scaling), which is developed further in polar form and Euler's formula in more advanced courses. The Geometric Interpretation of Complex Numbers Functions Learn Most Efficiently Through Adaptive Books Quick Links Become an Author Teach a Book Privacy Policy Terms and Conditions Contact Us Forum Blog Social Media Copyright © 2026 Avidemia. All right reserved.