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 the complex conjugate of the real number $z=11$?

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

Flashcard 2: What is $\overline{z}\,z$ if $z=4-3i$?

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

Flashcard 3: 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 4: Identify the value of $z-\overline{z}$ for $z=a+bi$.

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

Flashcard 5: 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 6: What is $|\overline{z}|$ in terms of $|z|$?

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

Flashcard 7: State the modulus product rule: $|zw|=?$

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

Flashcard 8: 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 9: 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 10: 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 11: 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 12: 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 13: 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 14: 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 15: State the property of conjugation for sums: $\overline{z+w}=?$

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

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 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 18: 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 19: What is $z+\overline{z}$ if $z=-3+7i$?

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

Flashcard 20: What is $\overline{\overline{z}}$ in terms of $z$?

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

Flashcard 21: What is $\overline{6-\pi i}$?

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

Flashcard 22: 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 23: What is the complex conjugate of $z=a-bi$?

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

Flashcard 24: 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 25: What is the complex conjugate of $z=-4+9i$?

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

Flashcard 26: 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 27: What is $|-9i|$?

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

Flashcard 28: What is $\overline{(2+i)(3-4i)}$?

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

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

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