Using Conjugates with Complex Numbers - Algebra 2
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What is the complex conjugate of the real number $z=11$?
What is the complex conjugate of the real number $z=11$?
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$11$. Real numbers are their own conjugates.
$11$. Real numbers are their own conjugates.
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What is $\overline{z},z$ if $z=4-3i$?
What is $\overline{z},z$ if $z=4-3i$?
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$25$. $z\overline{z}=|z|^2=4^2+(-3)^2=25$
$25$. $z\overline{z}=|z|^2=4^2+(-3)^2=25$
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What is $(a+bi)(a-bi)$ written in terms of $a$ and $b$?
What is $(a+bi)(a-bi)$ written in terms of $a$ and $b$?
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$a^2+b^2$. Using the difference of squares pattern with $i^2=-1$.
$a^2+b^2$. Using the difference of squares pattern with $i^2=-1$.
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Identify the value of $z-\overline{z}$ for $z=a+bi$.
Identify the value of $z-\overline{z}$ for $z=a+bi$.
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$2bi$. Subtracting the conjugate gives twice the imaginary part.
$2bi$. Subtracting the conjugate gives twice the imaginary part.
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State the modulus quotient rule for nonzero $w$: $\left|\frac{z}{w}\right|=?$
State the modulus quotient rule for nonzero $w$: $\left|\frac{z}{w}\right|=?$
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$\left|\frac{z}{w}\right|=\frac{|z|}{|w|}$. The modulus of a quotient equals the quotient of moduli.
$\left|\frac{z}{w}\right|=\frac{|z|}{|w|}$. The modulus of a quotient equals the quotient of moduli.
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What is $|\overline{z}|$ in terms of $|z|$?
What is $|\overline{z}|$ in terms of $|z|$?
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$|\overline{z}|=|z|$. Taking the conjugate doesn't change the modulus.
$|\overline{z}|=|z|$. Taking the conjugate doesn't change the modulus.
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State the modulus product rule: $|zw|=?$
State the modulus product rule: $|zw|=?$
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$|zw|=|z||w|$. The modulus of a product equals the product of moduli.
$|zw|=|z||w|$. The modulus of a product equals the product of moduli.
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What is the conjugate you multiply by to rationalize $\frac{1}{3-2i}$?
What is the conjugate you multiply by to rationalize $\frac{1}{3-2i}$?
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$3+2i$. Multiply by the conjugate to eliminate the imaginary part in the denominator.
$3+2i$. Multiply by the conjugate to eliminate the imaginary part in the denominator.
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What is $\frac{1}{2+i}$ written as $a+bi$?
What is $\frac{1}{2+i}$ written as $a+bi$?
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$\frac{2}{5}-\frac{1}{5}i$. Multiply by conjugate $2-i$: $\frac{1(2-i)}{(2+i)(2-i)}=\frac{2-i}{5}$
$\frac{2}{5}-\frac{1}{5}i$. Multiply by conjugate $2-i$: $\frac{1(2-i)}{(2+i)(2-i)}=\frac{2-i}{5}$
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What is $\frac{1}{3-2i}$ written as $a+bi$?
What is $\frac{1}{3-2i}$ written as $a+bi$?
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$\frac{3}{13}+\frac{2}{13}i$. Multiply by conjugate $3+2i$: $\frac{1(3+2i)}{(3-2i)(3+2i)}=\frac{3+2i}{13}$
$\frac{3}{13}+\frac{2}{13}i$. Multiply by conjugate $3+2i$: $\frac{1(3+2i)}{(3-2i)(3+2i)}=\frac{3+2i}{13}$
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What is $\frac{3-4i}{2+i}$ written as $a+bi$?
What is $\frac{3-4i}{2+i}$ written as $a+bi$?
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$\frac{2}{5}-\frac{11}{5}i$. Multiply by conjugate $2-i$: $\frac{(3-4i)(2-i)}{(2+i)(2-i)}=\frac{2-11i}{5}$
$\frac{2}{5}-\frac{11}{5}i$. Multiply by conjugate $2-i$: $\frac{(3-4i)(2-i)}{(2+i)(2-i)}=\frac{2-11i}{5}$
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What is $\frac{1+i}{1-i}$ written as $a+bi$?
What is $\frac{1+i}{1-i}$ written as $a+bi$?
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$i$. Multiply by conjugate $1+i$: $\frac{(1+i)(1+i)}{(1-i)(1+i)}=\frac{2i}{2}$
$i$. Multiply by conjugate $1+i$: $\frac{(1+i)(1+i)}{(1-i)(1+i)}=\frac{2i}{2}$
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What is $\frac{2-2i}{2+2i}$ written as $a+bi$?
What is $\frac{2-2i}{2+2i}$ written as $a+bi$?
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$-i$. Factor and simplify: $\frac{2(1-i)}{2(1+i)}=\frac{1-i}{1+i}=-i$
$-i$. Factor and simplify: $\frac{2(1-i)}{2(1+i)}=\frac{1-i}{1+i}=-i$
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What is $\frac{1}{i}$ written as a real number?
What is $\frac{1}{i}$ written as a real number?
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$-i$. Multiply by $-i$: $\frac{1}{i}\cdot\frac{-i}{-i}=\frac{-i}{-i^2}=\frac{-i}{1}=-i$
$-i$. Multiply by $-i$: $\frac{1}{i}\cdot\frac{-i}{-i}=\frac{-i}{-i^2}=\frac{-i}{1}=-i$
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State the property of conjugation for sums: $\overline{z+w}=?$
State the property of conjugation for sums: $\overline{z+w}=?$
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$\overline{z+w}=\overline{z}+\overline{w}$. The conjugate distributes over addition.
$\overline{z+w}=\overline{z}+\overline{w}$. The conjugate distributes over addition.
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State the property of conjugation for products: $\overline{zw}=?$
State the property of conjugation for products: $\overline{zw}=?$
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$\overline{zw}=\overline{z},\overline{w}$. The conjugate distributes over multiplication.
$\overline{zw}=\overline{z},\overline{w}$. The conjugate distributes over multiplication.
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State the property of conjugation for quotients: $\overline{\frac{z}{w}}=?$ for $w\neq 0$.
State the property of conjugation for quotients: $\overline{\frac{z}{w}}=?$ for $w\neq 0$.
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$\overline{\frac{z}{w}}=\frac{\overline{z}}{\overline{w}}$. The conjugate distributes over division.
$\overline{\frac{z}{w}}=\frac{\overline{z}}{\overline{w}}$. The conjugate distributes over division.
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What is $\overline{\frac{1+2i}{3-i}}$ written as $a+bi$?
What is $\overline{\frac{1+2i}{3-i}}$ written as $a+bi$?
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$\frac{1}{2}-\frac{1}{2}i$. $\frac{1+2i}{3-i}=\frac{1}{2}+\frac{1}{2}i$, so its conjugate is $\frac{1}{2}-\frac{1}{2}i$
$\frac{1}{2}-\frac{1}{2}i$. $\frac{1+2i}{3-i}=\frac{1}{2}+\frac{1}{2}i$, so its conjugate is $\frac{1}{2}-\frac{1}{2}i$
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What is $z+\overline{z}$ if $z=-3+7i$?
What is $z+\overline{z}$ if $z=-3+7i$?
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$-6$. $z+\overline{z}=(-3+7i)+(-3-7i)=-6$
$-6$. $z+\overline{z}=(-3+7i)+(-3-7i)=-6$
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What is $\overline{\overline{z}}$ in terms of $z$?
What is $\overline{\overline{z}}$ in terms of $z$?
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$\overline{\overline{z}}=z$. Taking the conjugate twice returns the original number.
$\overline{\overline{z}}=z$. Taking the conjugate twice returns the original number.
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What is $\overline{6-\pi i}$?
What is $\overline{6-\pi i}$?
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$6+\pi i$. Change the sign of the imaginary part from $-\pi i$ to $+\pi i$.
$6+\pi i$. Change the sign of the imaginary part from $-\pi i$ to $+\pi i$.
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What is $\left|\frac{3+4i}{1-2i}\right|$?
What is $\left|\frac{3+4i}{1-2i}\right|$?
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$\sqrt{5}$. Use $|\frac{z}{w}|=\frac{|z|}{|w|}$: $\frac{|3+4i|}{|1-2i|}=\frac{5}{\sqrt{5}}=\sqrt{5}$
$\sqrt{5}$. Use $|\frac{z}{w}|=\frac{|z|}{|w|}$: $\frac{|3+4i|}{|1-2i|}=\frac{5}{\sqrt{5}}=\sqrt{5}$
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What is the complex conjugate of $z=a-bi$?
What is the complex conjugate of $z=a-bi$?
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$\overline{z}=a+bi$. Change the sign of the imaginary part.
$\overline{z}=a+bi$. Change the sign of the imaginary part.
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Find $|z|$ if $z\overline{z}=49$ and $|z|\ge 0$.
Find $|z|$ if $z\overline{z}=49$ and $|z|\ge 0$.
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$7$. Since $|z|^2=z\overline{z}=49$, we have $|z|=7$.
$7$. Since $|z|^2=z\overline{z}=49$, we have $|z|=7$.
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What is the complex conjugate of $z=-4+9i$?
What is the complex conjugate of $z=-4+9i$?
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$-4-9i$. Change the sign of the imaginary part from positive to negative.
$-4-9i$. Change the sign of the imaginary part from positive to negative.
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What is $\frac{4}{1-3i}$ written as $a+bi$?
What is $\frac{4}{1-3i}$ written as $a+bi$?
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$\frac{2}{5}+\frac{6}{5}i$. Multiply by conjugate $1+3i$: $\frac{4(1+3i)}{(1-3i)(1+3i)}=\frac{4+12i}{10}$
$\frac{2}{5}+\frac{6}{5}i$. Multiply by conjugate $1+3i$: $\frac{4(1+3i)}{(1-3i)(1+3i)}=\frac{4+12i}{10}$
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What is $|-9i|$?
What is $|-9i|$?
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$9$. For pure imaginary numbers, $|bi|=|b|$.
$9$. For pure imaginary numbers, $|bi|=|b|$.
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What is $\overline{(2+i)(3-4i)}$?
What is $\overline{(2+i)(3-4i)}$?
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$2+5i$. $(2+i)(3-4i)=2-5i$, so $\overline{2-5i}=2+5i$
$2+5i$. $(2+i)(3-4i)=2-5i$, so $\overline{2-5i}=2+5i$
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State the formula for the complex conjugate of $z=a+bi$.
State the formula for the complex conjugate of $z=a+bi$.
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$\overline{z}=a-bi$. Change the sign of the imaginary part.
$\overline{z}=a-bi$. Change the sign of the imaginary part.
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State the formula for the modulus of $z=a+bi$.
State the formula for the modulus of $z=a+bi$.
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$|z|=\sqrt{a^2+b^2}$. Distance from origin in the complex plane.
$|z|=\sqrt{a^2+b^2}$. Distance from origin in the complex plane.
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