Algebra 2 Flashcards: Using Conjugates With Complex Numbers

What this deck covers

This deck focuses on Using Conjugates With Complex Numbers, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra 2.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

Answer: $\frac{2}{5}+\frac{6}{5}i$. Multiply by conjugate $1+3i$: $\frac{4(1+3i)}{(1-3i)(1+3i)}=\frac{4+12i}{10}$

Flashcard 2: What is $\left|\frac{3+4i}{1-2i}\right|$?

Answer: $\sqrt{5}$. Use $|\frac{z}{w}|=\frac{|z|}{|w|}$: $\frac{|3+4i|}{|1-2i|}=\frac{5}{\sqrt{5}}=\sqrt{5}$

Flashcard 3: State the formula for the modulus of $z=a+bi$.

Answer: $|z|=\sqrt{a^2+b^2}$. Distance from origin in the complex plane.

Flashcard 4: State the quotient formula for $\frac{a+bi}{c+di}$ using a conjugate in the denominator.

Answer: $\frac{a+bi}{c+di}=\frac{(a+bi)(c-di)}{c^2+d^2}$. Multiply numerator and denominator by the conjugate of the denominator.

Flashcard 5: What is $\frac{2-2i}{2+2i}$ written as $a+bi$?

Answer: $-i$. Factor and simplify: $\frac{2(1-i)}{2(1+i)}=\frac{1-i}{1+i}=-i$

Flashcard 6: What is $\frac{i}{1+i}$ written as $a+bi$?

Answer: $\frac{1}{2}+\frac{1}{2}i$. Multiply by conjugate $1-i$: $\frac{i(1-i)}{(1+i)(1-i)}=\frac{i+1}{2}$

Flashcard 7: What is $|-9i|$?

Answer: $9$. For pure imaginary numbers, $|bi|=|b|$.

Flashcard 8: What is the complex conjugate of $z=7-3i$?

Answer: $7+3i$. Change the sign of the imaginary part from negative to positive.

Flashcard 9: What is $|-8+6i|$?

Answer: $10$. $\sqrt{(-8)^2+6^2}=\sqrt{100}=10$

Flashcard 10: What is $|\overline{z}|$ in terms of $|z|$?

Answer: $|\overline{z}|=|z|$. Taking the conjugate doesn't change the modulus.

Flashcard 11: What is $(2-5i)(2+5i)$?

Answer: $29$. $(a+bi)(a-bi)=a^2-b^2i^2=a^2+b^2=4+25=29$

Flashcard 12: What is $\frac{1}{2+i}$ written as $a+bi$?

Answer: $\frac{2}{5}-\frac{1}{5}i$. Multiply by conjugate $2-i$: $\frac{1(2-i)}{(2+i)(2-i)}=\frac{2-i}{5}$

Flashcard 13: What is $\frac{3-4i}{2+i}$ written as $a+bi$?

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

Flashcard 14: What is $|3+4i|$?

Answer: $5$. $\sqrt{3^2+4^2}=\sqrt{25}=5$

Flashcard 15: What is $\frac{1}{3-2i}$ written as $a+bi$?

Answer: $\frac{3}{13}+\frac{2}{13}i$. Multiply by conjugate $3+2i$: $\frac{1(3+2i)}{(3-2i)(3+2i)}=\frac{3+2i}{13}$

Flashcard 16: State the property of conjugation for products: $\overline{zw}=?$

Answer: $\overline{zw}=\overline{z}\,\overline{w}$. The conjugate distributes over multiplication.

Flashcard 17: State the modulus product rule: $|zw|=?$

Answer: $|zw|=|z||w|$. The modulus of a product equals the product of moduli.

Flashcard 18: What is $(a+bi)(a-bi)$ written in terms of $a$ and $b$?

Answer: $a^2+b^2$. Using the difference of squares pattern with $i^2=-1$.

Flashcard 19: State the value of $\frac{z}{\overline{z}}$ for $z=2+i$ written as $a+bi$.

Answer: $\frac{3}{5}+\frac{4}{5}i$. $\frac{z}{\overline{z}}=\frac{2+i}{2-i}=\frac{3+4i}{5}$ after rationalizing.

Flashcard 20: Find $|z|$ if $z\overline{z}=49$ and $|z|\ge 0$.

Answer: $7$. Since $|z|^2=z\overline{z}=49$, we have $|z|=7$.

Flashcard 21: What is the conjugate you multiply by to rationalize $\frac{1}{3-2i}$?

Answer: $3+2i$. Multiply by the conjugate to eliminate the imaginary part in the denominator.

Flashcard 22: What is $\frac{1+i}{1-i}$ written as $a+bi$?

Answer: $i$. Multiply by conjugate $1+i$: $\frac{(1+i)(1+i)}{(1-i)(1+i)}=\frac{2i}{2}$

Flashcard 23: What is $\overline{(2+i)(3-4i)}$?

Answer: $2+5i$. $(2+i)(3-4i)=2-5i$, so $\overline{2-5i}=2+5i$

Flashcard 24: What is $\overline{\overline{z}}$ in terms of $z$?

Answer: $\overline{\overline{z}}=z$. Taking the conjugate twice returns the original number.

Flashcard 25: State the identity relating $z\overline{z}$ and the modulus $|z|$.

Answer: $z\overline{z}=|z|^2$. Multiplying a complex number by its conjugate gives the square of its modulus.

Flashcard 26: What is the complex conjugate of $z=a-bi$?

Answer: $\overline{z}=a+bi$. Change the sign of the imaginary part.

Flashcard 27: Identify the value of $z+\overline{z}$ for $z=a+bi$.

Answer: $2a$. Adding a complex number to its conjugate gives twice the real part.

Flashcard 28: State the property of conjugation for sums: $\overline{z+w}=?$

Answer: $\overline{z+w}=\overline{z}+\overline{w}$. The conjugate distributes over addition.

Flashcard 29: State the formula for the complex conjugate of $z=a+bi$.

Answer: $\overline{z}=a-bi$. Change the sign of the imaginary part.

Flashcard 30: Identify $\frac{1}{a+bi}$ written using a conjugate, assuming $a$ and $b$ are real and not both $0$.

Answer: $\frac{1}{a+bi}=\frac{a-bi}{a^2+b^2}$. Multiply numerator and denominator by the conjugate $a-bi$.

Flashcard 31: What is $\overline{\frac{1+2i}{3-i}}$ written as $a+bi$?

Answer: $\frac{1}{2}-\frac{1}{2}i$. $\frac{1+2i}{3-i}=\frac{1}{2}+\frac{1}{2}i$, so its conjugate is $\frac{1}{2}-\frac{1}{2}i$

Flashcard 32: Identify the value of $z-\overline{z}$ for $z=a+bi$.

Answer: $2bi$. Subtracting the conjugate gives twice the imaginary part.

Flashcard 33: What is $|7|$ when $z=7$ is viewed as a complex number?

Answer: $7$. For real numbers, the modulus equals the absolute value.

Flashcard 34: What is $|5-12i|$?

Answer: $13$. $\sqrt{5^2+(-12)^2}=\sqrt{169}=13$

Flashcard 35: What is the complex conjugate of the real number $z=11$?

Answer: $11$. Real numbers are their own conjugates.

Flashcard 36: What is the complex conjugate of $z=-4+9i$?

Answer: $-4-9i$. Change the sign of the imaginary part from positive to negative.

Flashcard 37: What is the complex conjugate of the pure imaginary number $z=-6i$?

Answer: $6i$. For pure imaginary numbers, change the sign of the imaginary part.

Flashcard 38: State the property of conjugation for quotients: $\overline{\frac{z}{w}}=?$ for $w\neq 0$.

Answer: $\overline{\frac{z}{w}}=\frac{\overline{z}}{\overline{w}}$. The conjugate distributes over division.

Flashcard 39: What is $\overline{6-\pi i}$?

Answer: $6+\pi i$. Change the sign of the imaginary part from $-\pi i$ to $+\pi i$.

Flashcard 40: What is $\frac{1}{i}$ written as a real number?

Answer: $-i$. Multiply by $-i$: $\frac{1}{i}\cdot\frac{-i}{-i}=\frac{-i}{-i^2}=\frac{-i}{1}=-i$

Flashcard 41: What is the complex conjugate of $z=\frac{1}{2}-\frac{5}{3}i$?

Answer: $\frac{1}{2}+\frac{5}{3}i$. Change the sign of the imaginary part from negative to positive.

Flashcard 42: What is $|1+i|^2$?

Answer: $2$. $|1+i|^2=1^2+1^2=2$

Flashcard 43: State the modulus quotient rule for nonzero $w$: $\left|\frac{z}{w}\right|=?$

Answer: $\left|\frac{z}{w}\right|=\frac{|z|}{|w|}$. The modulus of a quotient equals the quotient of moduli.

Flashcard 44: What is $\overline{z}\,z$ if $z=4-3i$?

Answer: $25$. $z\overline{z}=|z|^2=4^2+(-3)^2=25$

Flashcard 45: What is $z-\overline{z}$ if $z=-3+7i$?

Answer: $14i$. $z-\overline{z}=(-3+7i)-(-3-7i)=14i$

Flashcard 46: What is $z+\overline{z}$ if $z=-3+7i$?

Answer: $-6$. $z+\overline{z}=(-3+7i)+(-3-7i)=-6$

Flashcard 47: What is $|2-5i|^2$?

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