ACT Math Flashcards: Complex Numbers

What this deck covers

This deck focuses on Complex Numbers, giving you a quick way to review the definitions, rules, and examples that matter most for ACT Math.

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: Calculate the modulus of $-5 - 12i$.

Answer: $13$. Use $|z| = \sqrt{25 + 144} = \sqrt{169} = 13$.

Flashcard 2: Divide $4i$ by $2i$. What is the result?

Answer: $2$. Division of complex numbers: $\frac{4i}{2i} = 2$.

Flashcard 3: What is $\overline{\overline{z}}$ equal to for any complex number $z$?

Answer: $z$. Taking conjugate twice returns original number.

Flashcard 4: What is $\overline{z_1z_2}$ in terms of $\overline{z_1}$ and $\overline{z_2}$?

Answer: $\overline{z_1}\,\overline{z_2}$. Conjugate of product equals product of conjugates.

Flashcard 5: What is the sum of $3 + 4i$ and $5 + 2i$?

Answer: $8 + 6i$. Add real parts and imaginary parts separately.

Flashcard 6: What is the result of dividing $3 + 4i$ by $1 + i$?

Answer: $\frac{7}{2} + \frac{1}{2}i$. Multiply by conjugate of denominator and simplify.

Flashcard 7: What is the sum of $3 + 4i$ and $5 + 2i$?

Answer: $8 + 6i$. Add real parts and imaginary parts separately.

Flashcard 8: What is the standard form of a complex number?

Answer: $a + bi$ where $a, b$ are real numbers. This is the general form where $a$ is real part and $b$ is imaginary coefficient.

Flashcard 9: What is $(a+bi)(a-bi)$ equal to?

Answer: $a^2+b^2$. Multiplying by conjugate eliminates imaginary terms.

Flashcard 10: If $z = 7i$, what is $z^*$?

Answer: $-7i$. The conjugate of $7i$ is $-7i$.

Flashcard 11: Express $-2i$ as a complex number in standard form.

Answer: $0 - 2i$. Pure imaginary numbers have real part 0.

Flashcard 12: What is the point in the complex plane corresponding to $-2+5i$?

Answer: $(-2,5)$. Plot as $(\text{real part}, \text{imaginary part})$.

Flashcard 13: What is $\sqrt{-50}$ written in simplest form using $i$?

Answer: $5\sqrt{2}\,i$. $\sqrt{-50} = \sqrt{25 \cdot 2} \cdot i = 5\sqrt{2}i$.

Flashcard 14: What is the result of $(2 + 3i)^2$?

Answer: $-5 + 12i$. Expand: $4 + 12i + 9i^2 = 4 + 12i - 9 = -5 + 12i$.

Flashcard 15: What is the inverse of the complex number $1 + i$ in standard form?

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

Flashcard 16: Add the complex numbers $7 + 2i$ and $-4 - 3i$.

Answer: $3 - i$. Add real parts and imaginary parts separately.

Flashcard 17: Identify the real part of the complex number $7 + 3i$.

Answer:

1. The real part is the term without $i$.

Flashcard 18: What is the complex conjugate of $6 + 2i$?

Answer: $6 - 2i$. Change the sign of the imaginary part to get the conjugate.

Flashcard 19: What is the result of $(2 + 3i)^2$?

Answer: $-5 + 12i$. Expand: $4 + 12i + 9i^2 = 4 + 12i - 9 = -5 + 12i$.

Flashcard 20: What is $\overline{z_1+z_2}$ in terms of $\overline{z_1}$ and $\overline{z_2}$?

Answer: $\overline{z_1}+\overline{z_2}$. Conjugate of sum equals sum of conjugates.

Flashcard 21: What is $i^3$ equal to?

Answer: $-i$. $i^3 = i^2 \cdot i = (-1)(i) = -i$.

Flashcard 22: What is $i^2$ equal to?

Answer: $-1$. Since $i = \sqrt{-1}$, squaring gives $-1$.

Flashcard 23: Simplify the expression $(2i)^3$.

Answer: $-8i$. Evaluate: $(2i)^3 = 8i^3 = 8(-i) = -8i$.

Flashcard 24: What is the difference between $7 + 5i$ and $4 + 2i$?

Answer: $3 + 3i$. Subtract real and imaginary parts: $(7-4) + (5-2)i$.

Flashcard 25: Identify the imaginary part of the complex number $5 - 4i$.

Answer: -4. The imaginary part is the coefficient of $i$ (without the $i$).

Flashcard 26: If $z = 4 - 3i$, what is $z^*$?

Answer: $4 + 3i$. The conjugate of $4 - 3i$ is $4 + 3i$.

Flashcard 27: What is $i^4$ equal to?

Answer: $1$. $i^4 = (i^2)^2 = (-1)^2 = 1$.

Flashcard 28: What is $i^4$ equal to?

Answer: $1$. $i^4 = (i^2)^2 = (-1)^2 = 1$.

Flashcard 29: What is $i^2$ equal to?

Answer: -1. By definition, $i^2 = -1$.

Flashcard 30: Find the sum of $-3 + 5i$ and $4 - 7i$.

Answer: $1 - 2i$. Add real parts and imaginary parts separately.

Flashcard 31: What is the cycle of powers $i^n$ in terms of $n\bmod 4$?

Answer: $n\equiv^0:1,\ 1:i,\ 2:-1,\ 3:-i$. Powers of $i$ repeat every 4 terms: $1, i, -1, -i$.

Flashcard 32: What is the conjugate of $-7+9i$?

Answer: $-7-9i$. Negate the imaginary part to find conjugate.

Flashcard 33: What is the result of $(-3 + 2i)^2$?

Answer: $5 - 12i$. Expand: $9 - 12i + 4i^2 = 9 - 12i - 4 = 5 - 12i$.

Flashcard 34: What is $i^7$ in terms of $i$?

Answer: $-i$. Use $i^7 = i^4 \cdot i^3 = 1 \cdot (-i) = -i$.

Flashcard 35: What is $|3-4i|$?

Answer: $5$. $|3-4i| = \sqrt{3^2+(-4)^2} = \sqrt{25} = 5$.

Flashcard 36: Calculate the product of $i$ and $4i$.

Answer: -4. Since $i \cdot 4i = 4i^2 = 4(-1) = -4$.

Flashcard 37: Identify the real part of the complex number $7 + 3i$.

Answer:

1. The real part is the term without $i$.

Flashcard 38: Calculate $(1 - i)^2$.

Answer: $0 - 2i$. $(1-i)^2 = 1 - 2i + i^2 = 1 - 2i - 1 = -2i$.

Flashcard 39: What is the standard form of a complex number?

Answer: $a + bi$ where $a, b$ are real numbers. This is the general form where $a$ is real part and $b$ is imaginary coefficient.

Flashcard 40: What is the imaginary unit $i$ defined as?

Answer: $i=\sqrt{-1}$. The fundamental definition of the imaginary unit.

Flashcard 41: What is the complex number corresponding to the point $(4,-6)$?

Answer: $4-6i$. Convert point $(x,y)$ to $x+yi$.

Flashcard 42: Find the modulus of the complex number $4 + 3i$.

Answer:

1. Use $|a + bi| = \sqrt{a^2 + b^2} = \sqrt{16 + 9} = 5$.

Flashcard 43: What is the modulus of $6 - 8i$?

Answer: $\text{Modulus} = 10$. Use $|z| = \sqrt{a^2 + b^2} = \sqrt{36 + 64} = 10$.

Flashcard 44: Find the imaginary part of $5 - 7i$.

Answer: $-7$. The imaginary part is the coefficient of $i$.

Flashcard 45: What is the standard form of a complex number?

Answer: $a+bi$. Where $a$ is real and $b$ is the coefficient of $i$.

Flashcard 46: What is $\frac{2-i}{1+2i}$ in standard form?

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

Flashcard 47: What is the distance between $0$ and $a+bi$ in the complex plane?

Answer: $\sqrt{a^2+b^2}$. Same as the modulus formula.

Flashcard 48: What is $(5-2i)(1+3i)$ in standard form?

Answer: $11+13i$. $(5-2i)(1+3i) = 5+15i-2i-6i^2 = 5+13i+6$.

Flashcard 49: Find the real part of the complex number $9 + 4i$.

Answer: $9$. The real part is the coefficient of the non-imaginary term.

Flashcard 50: What is $(3+2i)+(4-5i)$ in standard form?

Answer: $7-3i$. Add real parts: $3+4=7$, imaginary: $2+(-5)=-3$.

Flashcard 51: What is $i^{22}$ equal to?

Answer: $-1$. $22 = 4 \cdot 5 + 2$, so $i^{22} = i^2 = -1$.

Flashcard 52: Calculate the modulus of $-5 - 12i$.

Answer: $13$. Use $|z| = \sqrt{25 + 144} = \sqrt{169} = 13$.

Flashcard 53: What is $(1+i)^2$ in standard form?

Answer: $2i$. $(1+i)^2 = 1+2i+i^2 = 1+2i-1 = 2i$.

Flashcard 54: What is the complex number corresponding to the point $(4,-6)$?

Answer: $4-6i$. Convert point $(x,y)$ to $x+yi$.

Flashcard 55: What is the result of multiplying $(3 + 2i)(3 - 2i)$?

Answer:

1. Multiplying conjugates gives $a^2 + b^2 = 9 + 4 = 13$.

Flashcard 56: Find the magnitude of the complex number $-3 - 4i$.

Answer:

1. Magnitude is $\sqrt{(-3)^2 + (-4)^2} = \sqrt{25} = 5$.

Flashcard 57: What is the imaginary part of $a+bi$?

Answer: $\operatorname{Im}(a+bi)=b$. The coefficient of the imaginary unit $i$.

Flashcard 58: What is the condition for two complex numbers $a+bi$ and $c+di$ to be equal?

Answer: $a=c$ and $b=d$. Real and imaginary parts must match separately.

Flashcard 59: Identify the modulus of $0 + 7i$.

Answer: $7$. For pure imaginary $bi$, modulus is $|b|$.

Flashcard 60: What is the difference between $7 + 5i$ and $4 + 2i$?

Answer: $3 + 3i$. Subtract real and imaginary parts: $(7-4) + (5-2)i$.

Flashcard 61: Identify the imaginary part of the complex number $5 - 4i$.

Answer: -4. The imaginary part is the coefficient of $i$ (without the $i$).

Flashcard 62: What is the distance between $0$ and $a+bi$ in the complex plane?

Answer: $\sqrt{a^2+b^2}$. Same as the modulus formula.

Flashcard 63: Find the real part of the complex number $9 + 4i$.

Answer: $9$. The real part is the coefficient of the non-imaginary term.

Flashcard 64: What is the square of the complex number $1 + i$?

Answer: 2i. $(1+i)^2 = 1 + 2i + i^2 = 1 + 2i - 1 = 2i$.

Flashcard 65: What is $(5-2i)(1+3i)$ in standard form?

Answer: $11+13i$. $(5-2i)(1+3i) = 5+15i-2i-6i^2 = 5+13i+6$.

Flashcard 66: What is the rule for adding complex numbers $(a+bi)+(c+di)$?

Answer: $(a+c)+(b+d)i$. Add real parts and imaginary parts separately.

Flashcard 67: Express $-2i$ as a complex number in standard form.

Answer: $0 - 2i$. Pure imaginary numbers have real part 0.

Flashcard 68: Find the imaginary part of $5 - 7i$.

Answer: $-7$. The imaginary part is the coefficient of $i$.

Flashcard 69: Find the magnitude of the complex number $-3 - 4i$.

Answer:

1. Magnitude is $\sqrt{(-3)^2 + (-4)^2} = \sqrt{25} = 5$.

Flashcard 70: What is the complex conjugate of $a+bi$?

Answer: $a-bi$. Change the sign of the imaginary part.

Flashcard 71: Subtract $2 - 3i$ from $4 + i$.

Answer: $2 + 4i$. Subtract real parts and imaginary parts separately.

Flashcard 72: State the complex conjugate of $7 - 5i$.

Answer: $7 + 5i$. Change the sign of the imaginary part.

Flashcard 73: What is $(3+2i)+(4-5i)$ in standard form?

Answer: $7-3i$. Add real parts: $3+4=7$, imaginary: $2+(-5)=-3$.

Flashcard 74: Express $i^2$ in terms of real numbers.

Answer: $i^2 = -1$. By definition, $i^2 = -1$ since $i = \sqrt{-1}$.

Flashcard 75: What is $i^3$ in terms of $i$?

Answer: $-i$. Use $i^3 = i^2 \cdot i = -1 \cdot i = -i$.

Flashcard 76: What is $\overline{\frac{z_1}{z_2}}$ in terms of $\overline{z_1}$ and $\overline{z_2}$?

Answer: $\frac{\overline{z_1}}{\overline{z_2}}$. Conjugate of quotient equals quotient of conjugates.

Flashcard 77: What is $\sqrt{-9}$ written using $i$?

Answer: $3i$. $\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i$.

Flashcard 78: What is $\sqrt{-9}$ written using $i$?

Answer: $3i$. $\sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i$.

Flashcard 79: Multiply the complex numbers $1 + i$ and $1 - i$.

Answer: $2$. Use $(1+i)(1-i) = 1 - i^2 = 1 - (-1) = 2$.

Flashcard 80: What is the rule for subtracting complex numbers $(a+bi)-(c+di)$?

Answer: $(a-c)+(b-d)i$. Subtract real parts and imaginary parts separately.

Flashcard 81: What is $|3-4i|$?

Answer: $5$. $|3-4i| = \sqrt{3^2+(-4)^2} = \sqrt{25} = 5$.

Flashcard 82: Determine the modulus of $3 + 4i$.

Answer: $5$. Use $|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = 5$.

Flashcard 83: What is the complex conjugate of $0 + 6i$?

Answer: $0 - 6i$. Change the sign of the imaginary part.

Flashcard 84: What is the real part of $(8-3i)$?

Answer: $8$. The real part is the coefficient without $i$.

Flashcard 85: What is the complex conjugate of $6 + 2i$?

Answer: $6 - 2i$. Change the sign of the imaginary part to get the conjugate.

Flashcard 86: If $z = 2 + 3i$, what is $\text{Re}(\bar{z})$?

Answer: $2$. The real part of $\bar{z} = 2 - 3i$ is 2.

Flashcard 87: What is the result of subtracting $6 + i$ from $2 - 3i$?

Answer: $-4 - 4i$. Subtract: $(2 - 3i) - (6 + i) = -4 - 4i$.

Flashcard 88: What is the point in the complex plane corresponding to $-2+5i$?

Answer: $(-2,5)$. Plot as $(\text{real part}, \text{imaginary part})$.

Flashcard 89: What is the result of subtracting $6 + i$ from $2 - 3i$?

Answer: $-4 - 4i$. Subtract: $(2 - 3i) - (6 + i) = -4 - 4i$.

Flashcard 90: What is the result of $i^3$?

Answer: -i. Since $i^3 = i^2 \cdot i = (-1) \cdot i = -i$.

Flashcard 91: What is $(6-7i)-(2+3i)$ in standard form?

Answer: $4-10i$. Subtract real: $6-2=4$, imaginary: $-7-3=-10$.

Flashcard 92: What is the cycle of powers $i^n$ in terms of $n\bmod 4$?

Answer: $n\equiv^0:1,\ 1:i,\ 2:-1,\ 3:-i$. Powers of $i$ repeat every 4 terms: $1, i, -1, -i$.

Flashcard 93: What is $i^0$?

Answer: $1$. Any number to the power of 0 equals 1.

Flashcard 94: Express $i^5$ in terms of $i$.

Answer: $i$. Use $i^5 = i^4 \cdot i = 1 \cdot i = i$.

Flashcard 95: What is $z-\overline{z}$ for $z=a+bi$?

Answer: $2bi$. $(a+bi)-(a-bi) = 2bi$.

Flashcard 96: What is $i^4$ in terms of real numbers?

Answer: $1$. Powers of $i$ cycle every 4: $i^4 = (i^2)^2 = (-1)^2 = 1$.

Flashcard 97: What is $i^3$ in terms of $i$?

Answer: $-i$. Use $i^3 = i^2 \cdot i = -1 \cdot i = -i$.

Flashcard 98: What is $(2+i)(3-4i)$ in standard form?

Answer: $10-5i$. $(2+i)(3-4i) = 6-8i+3i-4i^2 = 6-5i+4 = 10-5i$.

Flashcard 99: What is the rule for adding complex numbers $(a+bi)+(c+di)$?

Answer: $(a+c)+(b+d)i$. Add real parts and imaginary parts separately.

Flashcard 100: What is $\frac{1}{a+bi}$ in standard form (with $a,b\in\mathbb{R}$)?

Answer: $\frac{a-bi}{a^2+b^2}$. Multiply by conjugate: $\frac{1}{a+bi} \cdot \frac{a-bi}{a-bi}$.