Complex Conjugates - Arithmetic Practice Problems Online

The expression $\displaystyle \frac{1}{2-11i}$ can be rewritten with a real denominator (using the complex conjugate) in the form $\displaystyle \frac{a+bi}{c},$

where $a$ , $b$ , and $c$ are positive, co-prime integers. What is the sum of $a$ , $b$ , and $c$ ?

Details and assumptions

$i$ is the the imaginary unit, defined by $i^2 = -1$ .

Consider the complex numbers $A=12+i \mbox{ and } B=11-3i.$ If the value of $2A\overline{A}+2\overline{A}B+A\overline{B}+B\overline{B}$ is $X+Yi$ , what is the value of $X-Y$ ?

Details and assumptions

$\overline{A}$ denotes the complex conjugate of the complex number $A$ .

$i$ is the imaginary number that satisfies $i^2 = -1$ .

For complex numbers $\alpha=5+7i$ and $\beta=3-i,$ what is the value of $\alpha \overline{\alpha} +\overline{\alpha} \beta +\alpha \overline{\beta} +\beta \overline{\beta}?$

Details and assumptions

$\overline{\alpha}$ denotes the complex conjugate of the complex number $\alpha$ .

$i$ is the imaginary number that satisfies $i^2 = -1$ .

Consider the complex numbers $A=8-2i \mbox{ and } B=4-i.$ What is the value of $(A+B)^2+\left(\overline{A+B}\right)^2$ ?

Details and assumptions

$\overline{A}$ denotes the complex conjugate of the complex number $A$ .

$i$ is the imaginary number that satisfies $i^2 = -1$ .

Let $\omega$ and $z$ be the complex numbers $\omega =2-i, \quad z =\frac{\omega+4}{4\omega-1}.$

If $z\overline{z}$ can be expressed as $\frac{a}{b},$ where $a$ and $b$ are coprime positive integers, what is $a+b?$

Details and assumptions

$\overline{z}$ denotes the complex conjugate of the complex number $z$ .

$i$ is the imaginary number that satisfies $i^2 = -1$ .

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Complex Conjugates - Arithmetic

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Complex Conjugates - Arithmetic