Using Conjugates with Complex Numbers - Algebra 2

What is $\frac{4}{1-3i}$ written as $a+bi$ ?

$\frac{2}{5}+\frac{6}{5}i$ . Multiply by conjugate $1+3i$ : $\frac{4(1+3i)}{(1-3i)(1+3i)}=\frac{4+12i}{10}$
What is $\left|\frac{3+4i}{1-2i}\right|$ ?

$\sqrt{5}$ . Use $|\frac{z}{w}|=\frac{|z|}{|w|}$ : $\frac{|3+4i|}{|1-2i|}=\frac{5}{\sqrt{5}}=\sqrt{5}$
State the formula for the modulus of $z=a+bi$ .

$|z|=\sqrt{a^2+b^2}$ . Distance from origin in the complex plane.
State the quotient formula for $\frac{a+bi}{c+di}$ using a conjugate in the denominator.

$\frac{a+bi}{c+di}=\frac{(a+bi)(c-di)}{c^2+d^2}$ . Multiply numerator and denominator by the conjugate of the denominator.
What is $\frac{2-2i}{2+2i}$ written as $a+bi$ ?

$-i$ . Factor and simplify: $\frac{2(1-i)}{2(1+i)}=\frac{1-i}{1+i}=-i$
What is $\frac{i}{1+i}$ written as $a+bi$ ?

$\frac{1}{2}+\frac{1}{2}i$ . Multiply by conjugate $1-i$ : $\frac{i(1-i)}{(1+i)(1-i)}=\frac{i+1}{2}$
What is $|-9i|$ ?

$9$ . For pure imaginary numbers, $|bi|=|b|$ .
What is the complex conjugate of $z=7-3i$ ?

$7+3i$ . Change the sign of the imaginary part from negative to positive.
What is $|-8+6i|$ ?

$10$ . $\sqrt{(-8)^2+6^2}=\sqrt{100}=10$
What is $|\overline{z}|$ in terms of $|z|$ ?

$|\overline{z}|=|z|$ . Taking the conjugate doesn't change the modulus.
What is $(2-5i)(2+5i)$ ?

$29$ . $(a+bi)(a-bi)=a^2-b^2i^2=a^2+b^2=4+25=29$
What is $\frac{1}{2+i}$ written as $a+bi$ ?

$\frac{2}{5}-\frac{1}{5}i$ . Multiply by conjugate $2-i$ : $\frac{1(2-i)}{(2+i)(2-i)}=\frac{2-i}{5}$
What is $\frac{3-4i}{2+i}$ written as $a+bi$ ?

$\frac{2}{5}-\frac{11}{5}i$ . Multiply by conjugate $2-i$ : $\frac{(3-4i)(2-i)}{(2+i)(2-i)}=\frac{2-11i}{5}$
What is $|3+4i|$ ?

$5$ . $\sqrt{3^2+4^2}=\sqrt{25}=5$
What is $\frac{1}{3-2i}$ written as $a+bi$ ?

$\frac{3}{13}+\frac{2}{13}i$ . Multiply by conjugate $3+2i$ : $\frac{1(3+2i)}{(3-2i)(3+2i)}=\frac{3+2i}{13}$
State the property of conjugation for products: $\overline{zw}=?$

$\overline{zw}=\overline{z}\,\overline{w}$ . The conjugate distributes over multiplication.
State the modulus product rule: $|zw|=?$

$|zw|=|z||w|$ . The modulus of a product equals the product of moduli.
What is $(a+bi)(a-bi)$ written in terms of $a$ and $b$ ?

$a^2+b^2$ . Using the difference of squares pattern with $i^2=-1$ .
State the value of $\frac{z}{\overline{z}}$ for $z=2+i$ written as $a+bi$ .

$\frac{3}{5}+\frac{4}{5}i$ . $\frac{z}{\overline{z}}=\frac{2+i}{2-i}=\frac{3+4i}{5}$ after rationalizing.
Find $|z|$ if $z\overline{z}=49$ and $|z|\ge 0$ .

$7$ . Since $|z|^2=z\overline{z}=49$ , we have $|z|=7$ .
What is the conjugate you multiply by to rationalize $\frac{1}{3-2i}$ ?

$3+2i$ . Multiply by the conjugate to eliminate the imaginary part in the denominator.
What is $\frac{1+i}{1-i}$ written as $a+bi$ ?

$i$ . Multiply by conjugate $1+i$ : $\frac{(1+i)(1+i)}{(1-i)(1+i)}=\frac{2i}{2}$
What is $\overline{(2+i)(3-4i)}$ ?

$2+5i$ . $(2+i)(3-4i)=2-5i$ , so $\overline{2-5i}=2+5i$
What is $\overline{\overline{z}}$ in terms of $z$ ?

$\overline{\overline{z}}=z$ . Taking the conjugate twice returns the original number.
State the identity relating $z\overline{z}$ and the modulus $|z|$ .

$z\overline{z}=|z|^2$ . Multiplying a complex number by its conjugate gives the square of its modulus.
What is the complex conjugate of $z=a-bi$ ?

$\overline{z}=a+bi$ . Change the sign of the imaginary part.
Identify the value of $z+\overline{z}$ for $z=a+bi$ .

$2a$ . Adding a complex number to its conjugate gives twice the real part.
State the property of conjugation for sums: $\overline{z+w}=?$

$\overline{z+w}=\overline{z}+\overline{w}$ . The conjugate distributes over addition.
State the formula for the complex conjugate of $z=a+bi$ .

$\overline{z}=a-bi$ . Change the sign of the imaginary part.
Identify $\frac{1}{a+bi}$ written using a conjugate, assuming $a$ and $b$ are real and not both $0$ .

$\frac{1}{a+bi}=\frac{a-bi}{a^2+b^2}$ . Multiply numerator and denominator by the conjugate $a-bi$ .
What is $\overline{\frac{1+2i}{3-i}}$ written as $a+bi$ ?

$\frac{1}{2}-\frac{1}{2}i$ . $\frac{1+2i}{3-i}=\frac{1}{2}+\frac{1}{2}i$ , so its conjugate is $\frac{1}{2}-\frac{1}{2}i$
Identify the value of $z-\overline{z}$ for $z=a+bi$ .

$2bi$ . Subtracting the conjugate gives twice the imaginary part.
What is $|7|$ when $z=7$ is viewed as a complex number?

$7$ . For real numbers, the modulus equals the absolute value.
What is $|5-12i|$ ?

$13$ . $\sqrt{5^2+(-12)^2}=\sqrt{169}=13$
What is the complex conjugate of the real number $z=11$ ?

$11$ . Real numbers are their own conjugates.
What is the complex conjugate of $z=-4+9i$ ?

$-4-9i$ . Change the sign of the imaginary part from positive to negative.
What is the complex conjugate of the pure imaginary number $z=-6i$ ?

$6i$ . For pure imaginary numbers, change the sign of the imaginary part.
State the property of conjugation for quotients: $\overline{\frac{z}{w}}=?$ for $w\neq 0$ .

$\overline{\frac{z}{w}}=\frac{\overline{z}}{\overline{w}}$ . The conjugate distributes over division.
What is $\overline{6-\pi i}$ ?

$6+\pi i$ . Change the sign of the imaginary part from $-\pi i$ to $+\pi i$ .
What is $\frac{1}{i}$ written as a real number?

$-i$ . Multiply by $-i$ : $\frac{1}{i}\cdot\frac{-i}{-i}=\frac{-i}{-i^2}=\frac{-i}{1}=-i$
What is the complex conjugate of $z=\frac{1}{2}-\frac{5}{3}i$ ?

$\frac{1}{2}+\frac{5}{3}i$ . Change the sign of the imaginary part from negative to positive.
What is $|1+i|^2$ ?

$2$ . $|1+i|^2=1^2+1^2=2$
State the modulus quotient rule for nonzero $w$ : $\left|\frac{z}{w}\right|=?$

$\left|\frac{z}{w}\right|=\frac{|z|}{|w|}$ . The modulus of a quotient equals the quotient of moduli.
What is $\overline{z}\,z$ if $z=4-3i$ ?

$25$ . $z\overline{z}=|z|^2=4^2+(-3)^2=25$
What is $z-\overline{z}$ if $z=-3+7i$ ?

$14i$ . $z-\overline{z}=(-3+7i)-(-3-7i)=14i$
What is $z+\overline{z}$ if $z=-3+7i$ ?

$-6$ . $z+\overline{z}=(-3+7i)+(-3-7i)=-6$
What is $|2-5i|^2$ ?

$29$ . $|z|^2=z\overline{z}=(2-5i)(2+5i)=29$

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