Complex Numbers - ACT Math

1. What is $\overline{\overline{z}}$ equal to for any complex number $z$?

   $z$. Taking conjugate twice returns original number.

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

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

3. What is the result of dividing $3 + 4i$ by $1 + i$?

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

4. What is $(a+bi)(a-bi)$ equal to?

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

5. Express $-2i$ as a complex number in standard form.

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

6. What is the result of $(2 + 3i)^2$?

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

7. What is the inverse of the complex number $1 + i$ in standard form?

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

8. Add the complex numbers $7 + 2i$ and $-4 - 3i$.

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

9. Identify the real part of the complex number $7 + 3i$.

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

10. Simplify the expression $(2i)^3$.

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

11. What is the difference between $7 + 5i$ and $4 + 2i$?

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

12. What is $i^4$ equal to?

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

13. Find the sum of $-3 + 5i$ and $4 - 7i$.

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

14. What is $i^7$ in terms of $i$?

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

15. What is $|3-4i|$?

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

16. Calculate the product of $i$ and $4i$.

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

17. What is the standard form of a complex number?

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

18. What is the complex number corresponding to the point $(4,-6)$?

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

19. What is the modulus of $6 - 8i$?

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

20. Find the imaginary part of $5 - 7i$.

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

21. What is $\frac{2-i}{1+2i}$ in standard form?

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

22. What is the distance between $0$ and $a+bi$ in the complex plane?

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

23. What is $(3+2i)+(4-5i)$ in standard form?

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

24. What is $i^{22}$ equal to?

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

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

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

26. What is $(1+i)^2$ in standard form?

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

27. Find the magnitude of the complex number $-3 - 4i$.

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

28. What is the imaginary part of $a+bi$?

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

29. What is the condition for two complex numbers $a+bi$ and $c+di$ to be equal?

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

30. Identify the imaginary part of the complex number $5 - 4i$.

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

31. Find the real part of the complex number $9 + 4i$.

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

32. What is the square of the complex number $1 + i$?

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

33. What is $(5-2i)(1+3i)$ in standard form?

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

34. What is the rule for adding complex numbers $(a+bi)+(c+di)$?

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

35. What is the complex conjugate of $a+bi$?

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

36. Subtract $2 - 3i$ from $4 + i$.

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

37. State the complex conjugate of $7 - 5i$.

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

38. What is $\sqrt{-9}$ written using $i$?

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

39. What is the complex conjugate of $0 + 6i$?

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

40. What is the real part of $(8-3i)$?

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

41. What is the complex conjugate of $6 + 2i$?

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

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

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

43. What is the result of subtracting $6 + i$ from $2 - 3i$?

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

44. What is the point in the complex plane corresponding to $-2+5i$?

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

45. What is the result of $i^3$?

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

46. What is the cycle of powers $i^n$ in terms of $n\bmod 4$?

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

47. Express $i^5$ in terms of $i$.

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

48. What is $i^3$ in terms of $i$?

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

49. What is $(2+i)(3-4i)$ in standard form?

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

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

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

51. State the value of $i^4$.

    1. Since $i^4 = (i^2)^2 = (-1)^2 = 1$.

52. What is the result of multiplying $(3 + 2i)(3 - 2i)$?

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

53. Express $i^2$ in terms of real numbers.

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

54. What is the imaginary part of $(8-3i)$?

    $-3$. The imaginary part is the coefficient of $i$.

55. What is $i^3$ equal to?

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

56. What is $\overline{\frac{z_1}{z_2}}$ in terms of $\overline{z_1}$ and $\overline{z_2}$?

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

57. If $z = 4 - 3i$, what is $z^*$?

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

58. What property does the equation $z + z^* = 2\text{Re}(z)$ illustrate?

    Conjugate symmetry. This shows the sum of a complex number and its conjugate equals twice the real part.

59. What is $i^0$?

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

60. State the polar form of the complex number $1 + 0i$.

    $1(\cos(0) + i\sin(0))$. Real number 1 has modulus 1 and argument 0.

61. What is the argument of the complex number $1 + i$?

    $\frac{\text{pi}}{4}$. Use $\theta = \arctan(\frac{1}{1}) = \frac{\pi}{4}$.

62. What is $i^4$ in terms of real numbers?

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

63. Determine the modulus of $3 + 4i$.

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

64. What is $(2-3i)^2$ in standard form?

    $-5-12i$. $(2-3i)^2 = 4-12i+9i^2 = 4-12i-9$.

65. What is the imaginary unit $i$ defined as?

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

66. What is the product of the conjugates $3 + 4i$ and $3 - 4i$?

    $25$. Product of conjugates equals $a^2 + b^2 = 9 + 16 = 25$.

67. Find the real component of $(2 + 3i)(4 + 5i)$.

    -7. Expand: $8 + 10i + 12i + 15i^2 = 8 + 22i - 15 = -7 + 22i$.

68. What is $z-\overline{z}$ for $z=a+bi$?

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

69. If $z = 7i$, what is $z^*$?

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

70. What is $i^2$ equal to?

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

71. Multiply the complex numbers $1 + i$ and $1 - i$.

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

72. What is the conjugate of $-7+9i$?

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

73. What is $i^{17}$ equal to?

    $i$. $17 = 4 \cdot 4 + 1$, so $i^{17} = i^1 = i$.

74. What is $\sqrt{-50}$ written in simplest form using $i$?

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

75. What is the sum of $3 + 4i$ and $1 + 2i$?

    $4 + 6i$. Add real parts and imaginary parts separately: $(3+1) + (4+2)i$.

76. What is $|-1+\sqrt{3}i|$?

    $2$. $|-1+\sqrt{3}i| = \sqrt{(-1)^2+(\sqrt{3})^2} = \sqrt{4} = 2$.

77. What is the result of $(2 + 3i) + (5 - i)$?

    $7 + 2i$. Add corresponding parts: $(2+5) + (3+(-1))i$.

78. Identify the modulus of $0 + 7i$.

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

79. What is the distance between $z_1=a+bi$ and $z_2=c+di$ in the complex plane?

    $\sqrt{(a-c)^2+(b-d)^2}$. Distance formula between two points in complex plane.

80. What is the absolute value (modulus) of $a+bi$?

    $|a+bi|=\sqrt{a^2+b^2}$. The distance from origin in the complex plane.

81. What is $\overline{z_1+z_2}$ in terms of $\overline{z_1}$ and $\overline{z_2}$?

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

82. What does the imaginary unit $i$ represent?

    The square root of -1. The imaginary unit $i$ is defined as $\sqrt{-1}$.

83. What is the rule for subtracting complex numbers $(a+bi)-(c+di)$?

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

84. Simplify $(-i)^2$.

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

85. What is $z+\overline{z}$ for $z=a+bi$?

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

86. Calculate $(1 - i)^2$.

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

87. What is the real part of $a+bi$?

    $\operatorname{Re}(a+bi)=a$. The coefficient of the non-imaginary term.

88. What is $\frac{3+4i}{3-4i}$ in standard form?

    $\frac{-7}{25}+\frac{24}{25}i$. Multiply by conjugate: $\frac{(3+4i)(3+4i)}{(3-4i)(3+4i)} = \frac{-7+24i}{25}$.

89. What is $\overline{z_1z_2}$ in terms of $\overline{z_1}$ and $\overline{z_2}$?

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

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

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

91. What is the product $(a+bi)(c+di)$ in standard form?

    $(ac-bd)+(ad+bc)i$. Use FOIL: $(a+bi)(c+di) = ac + adi + bci + bdi^2$.

92. What is $(6-7i)-(2+3i)$ in standard form?

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

93. Find the modulus of the complex number $4 + 3i$.

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

94. What is the result of $(-3 + 2i)^2$?

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

Study these flashcards