Absolute Value of a Complex Number and Complex Conjugates

The Geometric Interpretation of Complex NumbersFunctions

Absolute Value of a Complex Number and Complex Conjugates

The absolute value (or modulus) of a complex number z = x + y i is its distance from the origin in the complex plane:

| z | = x 2 + y 2

The complex conjugate of z = a + b i is z ¯ = a − b i , obtained by reflecting z across the real axis.

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Concept Formula Example
Modulus < / t d >< t d > z < / t d >< / t r >< t r >< t d > C o n j u g a t e < / t d >< t d > \bar{z} = a - bi < / t d >< t d > \overline{3 + 4i} = 3 - 4i < / t d >< / t r >< t r >< t d > z \bar{z} < / t d >< t d > z
Product of moduli < / t d >< t d > z w < / t d >< / t r >< t r >< t d > Q u o t i e n t o f m o d u l i < / t d >< t d > z/w
Modulus of conjugate < / t d >< t d > z ¯ < / t d >< / t r >< / t b o d y >< / t a b l e >< p >< s p a n i d = " m o d u l u s − d e f i n i t i o n " >< / s p a n >< / p >< h 2 > T h e M o d u l u s ( A b s o l u t e V a l u e ) < / h 2 >< d i v c l a s s = " h i g h l i g h t " >< p >< s t r o n g > D e f i n i t i o n . < / s t r o n g > T h e < s t r o n g > m o d u l u s < / s t r o n g > ( o r < s t r o n g > a b s o l u t e v a l u e < / s t r o n g > ) o f t h e c o m p l e x n u m b e r z = x + yi is:</p> ___MATH_BLOCK_1___<p>Geometrically, |z|$ is the distance from the point $z t o t h e o r i g i n i n t h e c o m p l e x p l a n e . < / p >< / d i v >< p > K e y f a c t s a b o u t t h e m o d u l u s :< / p >< u l >< l i > |z| \geq 0$ for all complex numbers $z . < / l i >< l i > |z| = 0$ if and only if $z = 0 ( i . e . , x = 0 a n d y = 0 ) . < / l i >< l i > F o r a r e a l n u m b e r a + 0i , |a + 0i| = \sqrt{a^2} = |a| , c o n s i s t e n t w i t h t h e u s u a l a b s o l u t e v a l u e . < / l i >< / u l >< d i v c l a s s = " e x a m p l e " >< p >< s t r o n g > E x a m p l e 1. < / s t r o n g > F i n d t h e m o d u l u s o f 3 + 4i , -2 + 0i , a n d 0 + 5i .</p> <p><strong>Solution.</strong></p> ___MATH_BLOCK_2______MATH_BLOCK_3______MATH_BLOCK_4___</div><p><span id="modulus-properties"></span></p> <h2>Properties of the Modulus</h2> <div class="highlight"><p><strong>Multiplicative properties of the modulus.</strong> For any complex numbers z a n d w$ (with $w \neq 0 ):</p> ___MATH_BLOCK_5___</div><details><summary>Proof of |zw| = |z||w| ( c l i c k t o e x p a n d ) < / s u m m a r y >< p > L e t z = a + bi a n d w = c + di . Then:</p> ___MATH_BLOCK_6______MATH_BLOCK_7___<p>Expanding:</p> ___MATH_BLOCK_8______MATH_BLOCK_9___<p>Taking square roots (both sides are non-negative): |zw| = |z||w| . < / p >< / d e t a i l s >< p >< s p a n i d = " c o n j u g a t e − p r o p e r t i e s " >< / s p a n >< / p >< h 2 > T h e C o m p l e x C o n j u g a t e < / h 2 >< d i v c l a s s = " h i g h l i g h t " >< p >< s t r o n g > D e f i n i t i o n . < / s t r o n g > T h e < s t r o n g > c o m p l e x c o n j u g a t e < / s t r o n g > o f z = a + bi is:</p> ___MATH_BLOCK_10___<p>Geometrically, \bar{z}$ is the reflection of $z a c r o s s t h e r e a l a x i s i n t h e c o m p l e x p l a n e . < / p >< / d i v >< p > K e y p r o p e r t i e s o f c o n j u g a t e s :< / p >< u l >< l i > z + \bar{z} = 2a = 2\operatorname{Re}(z) , w h i c h i s a r e a l n u m b e r . < / l i >< l i > z - \bar{z} = 2bi = 2i\operatorname{Im}(z) , w h i c h i s p u r e l y i m a g i n a r y . < / l i >< l i > z\bar{z} = a^2 + b^2 = |z|^2 , w h i c h i s a n o n − n e g a t i v e r e a l n u m b e r . < / l i >< l i > \overline{z + w} = \bar{z} + \bar{w} < / l i >< l i > \overline{zw} = \bar{z}\,\bar{w} < / l i >< l i > \overline{\bar{z}} = z ( t h e c o n j u g a t e o f t h e c o n j u g a t e i s t h e o r i g i n a l n u m b e r ) < / l i >< l i > |\bar{z}| = |z| < / l i >< / u l >< d i v c l a s s = " e x a m p l e " >< p >< s t r o n g > E x a m p l e 2. < / s t r o n g > L e t z = 2 - 3i$. Find $\bar{z} , z + \bar{z} , z - \bar{z} , a n d z\bar{z} . < / p >< p >< s t r o n g > S o l u t i o n . < / s t r o n g >< / p >< p > \bar{z} = 2 + 3i < / p >< p > z + \bar{z} = (2 - 3i) + (2 + 3i) = 4 ( r e a l ) < / p >< p > z - \bar{z} = (2 - 3i) - (2 + 3i) = -6i ( p u r e l y i m a g i n a r y ) < / p >< p > z\bar{z} = (2 - 3i)(2 + 3i) = 4 + 9 = 13 = |z|^2 < / p >< / d i v >< p >< s p a n i d = " c o n j u g a t e − a n d − r e c i p r o c a l " >< / s p a n >< / p >< h 2 > U s i n g t h e C o n j u g a t e t o F i n d t h e R e c i p r o c a l < / h 2 >< p > S i n c e z \bar{z} = |z|^2$ is a positive real number (for $z \neq 0$), the reciprocal of $z can be written as:</p> ___MATH_BLOCK_11___<p>This confirms the formula from Section 4.3 and shows clearly that the conjugate is the key tool for inverting a complex number.</p> <div class="example"><p><strong>Example 3.</strong> Find the reciprocal of 1 + 2i .</p> <p><strong>Solution.</strong></p> ___MATH_BLOCK_12___</div><h2>Frequently Asked Questions</h2> <details><summary>What is the difference between the absolute value and the modulus?</summary> <p>They are the same thing for complex numbers. The term <em>absolute value</em> extends from the real number concept |a| = \sqrt{a^2} , w h i l e < e m > m o d u l u s < / e m > i s t h e s t a n d a r d t e r m f o r c o m p l e x n u m b e r s . F o r a r e a l n u m b e r a$, the modulus $|a + 0i| = \sqrt{a^2} = |a| m a t c h e s t h e u s u a l a b s o l u t e v a l u e . < / p >< / d e t a i l s >< d e t a i l s >< s u m m a r y > W h y i s z\bar{z} a l w a y s a n o n − n e g a t i v e r e a l n u m b e r ? < / s u m m a r y > W r i t i n g z = a + bi , w e g e t z\bar{z} = (a+bi)(a-bi) = a^2 + b^2$. This is a sum of squares of real numbers, so it is always non-negative. It equals zero only when $a = b = 0 , i . e . , z = 0 . < / d e t a i l s >< d e t a i l s >< s u m m a r y > W h a t d o e s t h e c o n j u g a t e l o o k l i k e g e o m e t r i c a l l y ? < / s u m m a r y > I n t h e c o m p l e x p l a n e , t h e c o n j u g a t e \bar{z}$ is the reflection of $z$ across the real (horizontal) axis. If $z = a + bi$ is plotted at $(a, b)$, then $\bar{z} = a - bi$ is plotted at $(a, -b) . < / d e t a i l s >< d e t a i l s >< s u m m a r y > H o w d o e s t h e m o d u l u s r e l a t e t o d i s t a n c e b e t w e e n t w o c o m p l e x n u m b e r s ? < / s u m m a r y > T h e d i s t a n c e b e t w e e n t w o c o m p l e x n u m b e r s z_1 a n d z_2$ in the complex plane equals $|z_1 - z_2|$. This generalizes the idea that $|z|$ is the distance from $z t o t h e o r i g i n . < / d e t a i l s >< d e t a i l s >< s u m m a r y > W h y i s t h e p r o p e r t y |zw| = |z||w| u s e f u l ? < / s u m m a r y > I t m e a n s t h a t m u l t i p l i c a t i o n b y a c o m p l e x n u m b e r w$ scales distances by the factor $|w|$. 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 NumbersFunctions
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