Problem 37 Suppose $z$ is a nonzero compl... [FREE SOLUTION]

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Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 6: Problem 37

Suppose $z$ is a nonzero complex number. Show that $\bar{z}=\frac{1}{z}$ if and only if $|z|=1$.

Short Answer

Expert verified

Given a nonzero complex number $z = a + bi$, we showed that $\bar{z} = \frac{1}{z}$ if and only if $|z| = 1$. We defined the conjugate of $z$ as $\bar{z} = a - bi$ and expressed the reciprocal of $z$ in the standard form as $\frac{1}{z} = \frac{a - bi}{a^2 + b^2}$. By showing that $|z| = 1$ implies $\bar{z} = \frac{1}{z}$ and vice versa, we proved the desired relationship between the conjugate, magnitude, and reciprocal of a complex number.

Step by step solution

Define and Express Complex Number Conjugate and Magnitude

Let $z = a + bi$ be a nonzero complex number, where $a$ and $b$ are real numbers. The conjugate of $z$ is given by $\bar{z} = a - bi$. The magnitude of $z$ is given by $|z|= \sqrt{a^2 + b^2}$.

Express the Reciprocal of z

The reciprocal of the complex number $z$ is given by $\frac{1}{z}=\frac{1}{a+bi}$. To express it in standard form, multiply both numerator and denominator by the conjugate of $z$: \[\frac{1}{z}=\frac{1}{a+bi}\cdot\frac{(a-bi)}{(a-bi)}=\frac{a-bi}{a^2+b^2}\]

Show $\bar{z}=\frac{1}{z}$ when $|z|=1$

First, we will show that if $|z|=1$, then $\bar{z}=\frac{1}{z}$. If $|z|=1$, then: \[\sqrt{a^2 + b^2} = 1\] \[a^2 + b^2 = 1\] Now, given the expression for $\frac{1}{z}$ from Step 2, we have: \[\frac{1}{z}=\frac{a-bi}{a^2+b^2}\] Since $a^2 + b^2 = 1$, we have: \[\frac{1}{z}=\frac{a-bi}{1}= a - bi\] This is the conjugate of $z$, so $\bar{z}=\frac{1}{z}$ when $|z|=1$.

Show $|z|=1$ when $\bar{z}=\frac{1}{z}$

Now, we will show that if $\bar{z}=\frac{1}{z}$, then $|z|=1$. Given that $\bar{z}=\frac{1}{z}$, we have $a-bi = \frac{a-bi}{a^2 + b^2}$. Equating the numerators, we have: \[a^2+2abi+b^2 = a^2+b^2\] Since $a$ and $b$ are real numbers, their product must be real, so $2abi = 0$. Since $z$ is nonzero and complex, either $a$ or $b$ must be non-zero. Therefore, we must have: \[a^2 + b^2 = a^2 + b^2\] As shown in Step 3, when this equation is true, $|z|=1$. Thus, we have shown that if $\bar{z}=\frac{1}{z}$, then $|z|=1$. #Conclusion# In conclusion, we have shown that for a nonzero complex number $z$, $\bar{z}=\frac{1}{z}$ if and only if $|z|=1$.

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91影视

Suppose \(z\) is a nonzero complex number. Show that \(\bar{z}=\frac{1}{z}\) if and only if \(|z|=1\).

Short Answer

Step by step solution

Define and Express Complex Number Conjugate and Magnitude

Express the Reciprocal of z

Show \(\bar{z}=\frac{1}{z}\) when \(|z|=1\)

Show \(|z|=1\) when \(\bar{z}=\frac{1}{z}\)

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