Understanding Complex Numbers - Algebra 2

1. What is the real part of $-6+11i$?

   $-6$. The real part is the constant term.

2. What is $i^2$?

   $-1$. By definition of the imaginary unit.

3. What is the coefficient of $i$ in the complex number $12-3i$?

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

4. What condition on $a$ makes $a+bi$ a pure imaginary number?

   $a=0$. When real part is zero, number is pure imaginary.

5. What is $\sqrt{-49}$ written using $i$?

   $7i$. $\sqrt{-49} = \sqrt{49} \cdot \sqrt{-1} = 7i$

6. What is $\sqrt{-100}$ written using $i$?

   $10i$. $\sqrt{-100} = \sqrt{100} \cdot \sqrt{-1} = 10i$

7. Write $i$ in the form $a+bi$.

   $0+1i$. Positive imaginary unit in standard form.

8. What is $\sqrt{-1}$ equal to in terms of $i$?

   $\sqrt{-1}=i$. Definition of the imaginary unit.

9. What is the defining property of the imaginary unit $i$?

   $i^2=-1$. This defines the imaginary unit's fundamental property.

10. What is $i^6$ simplified?

    $-1$. $i^6 = i^4 \cdot i^2 = 1 \cdot (-1) = -1$

11. What is $\frac{1}{i}$ simplified in terms of $i$?

    $-i$. Multiply by $\frac{-i}{-i}$ to get $\frac{-i}{1} = -i$

12. What is $\sqrt{-25}$ written using $i$?

    $5i$. $\sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i$

13. What is the imaginary part of $-6+11i$?

    $11$. The imaginary part is the coefficient of $i$.

14. Write $-i$ in the form $a+bi$.

    $0-1i$. Negative imaginary unit in standard form.

15. What is $i^8$ simplified?

    $1$. $i^8 = (i^4)^2 = 1^2 = 1$

16. What is $(-i)^3$ simplified?

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

17. What is $\sqrt{-36}$ written using $i$?

    $6i$. $\sqrt{-36} = \sqrt{36} \cdot \sqrt{-1} = 6i$

18. What is $\sqrt{-121}$ written using $i$?

    $11i$. $\sqrt{-121} = \sqrt{121} \cdot \sqrt{-1} = 11i$

19. What is the standard form of a complex number using real numbers $a$ and $b$?

    $a+bi$ with $a,b\in\mathbb{R}$. Every complex number has this form with real coefficients.

20. What condition on $b$ makes $a+bi$ a real number?

    $b=0$. When imaginary part is zero, number is real.

21. What is $\frac{-1}{i}$ simplified in terms of $i$?

    $i$. Multiply by $\frac{-i}{-i}$ to get $\frac{i}{1} = i$

22. What is $\sqrt{-16}$ written using $i$?

    $4i$. $\sqrt{-16} = \sqrt{16} \cdot \sqrt{-1} = 4i$

23. What is $i^4$ simplified?

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

24. Identify whether $0-7i$ is real, imaginary, or complex with nonzero parts.

    Pure imaginary number. Has zero real part, so it's pure imaginary.

25. Identify whether $3+0i$ is real, imaginary, or complex with nonzero parts.

    Real number. Has zero imaginary part, so it's real.

26. What is $\sqrt{-64}$ written using $i$?

    $8i$. $\sqrt{-64} = \sqrt{64} \cdot \sqrt{-1} = 8i$

27. What is the real number $5$ written in complex form $a+bi$?

    $5=5+0i$. Real numbers have zero imaginary part.

28. Find and correct the error: "$\operatorname{Im}(3-4i)=-4i$".

    Correct: $\operatorname{Im}(3-4i)=-4$. Imaginary part is the coefficient, not including $i$.

29. What is the repeating cycle of powers of $i$ for $i^1,i^2,i^3,i^4$?

    $i,-1,-i,1$. Powers of $i$ repeat every 4 terms.

30. What is $\frac{7i}{i}$ simplified?

    $7$. The $i$ terms cancel out, leaving 7.

31. What is the real part of the complex number $a+bi$?

    $\operatorname{Re}(a+bi)=a$. The real part is the coefficient of the constant term.

32. What are the two real-number components in the complex form $a+bi$?

    $a$ is real part; $b$ is imaginary coefficient. Standard complex form separates real and imaginary components.

33. Identify $a$ and $b$ for the complex number $7-2i$ in the form $a+bi$.

    $a=7,\ b=-2$. Real part is 7, imaginary coefficient is -2.

34. What is $i^5$ simplified?

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

35. What is $i^7$ simplified?

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

36. Identify whether $-2+5i$ is real, imaginary, or complex with nonzero parts.

    Complex with nonzero real and imaginary parts. Both real and imaginary parts are nonzero.

37. Which statement is true about $a$ and $b$ in $a+bi$?

    $a$ and $b$ are real numbers. Both coefficients must be real for complex form.

38. What is $i^3$ simplified using $i^2=-1$?

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

39. What is the imaginary part of the complex number $a+bi$?

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

40. What is $\frac{5i^2}{i}$ simplified?

    $-5i$. $\frac{5i^2}{i} = \frac{5(-1)}{i} = \frac{-5}{i} = -5i$

41. Find and correct the error: "$i^2=1$".

    Correct: $i^2=-1$. By definition, $i^2 = -1$, not $1$.

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

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

43. Identify $a$ and $b$ for the complex number $-4+9i$ in the form $a+bi$.

    $a=-4,\ b=9$. Real part is -4, imaginary coefficient is 9.

44. What is $(-i)^2$ simplified?

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

45. Write $0$ in the form $a+bi$.

    $0+0i$. Zero has both real and imaginary parts equal to zero.

46. What is $\sqrt{-4}$ written using $i$?

    $2i$. $\sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i$

47. What is the imaginary unit written as a complex number in $a+bi$ form?

    $i=0+1i$. Pure imaginary unit with zero real part.

48. What is the coefficient of $i$ in the complex number $-8+i$?

    $1$. When written as $+i$, the coefficient is 1.

49. What is $\frac{3}{i}$ simplified in terms of $i$?

    $-3i$. Multiply by $\frac{-i}{-i}$ to get $\frac{-3i}{1} = -3i$

50. Find and correct the error: "$i=\sqrt{-1}=-1$".

    Correct: $i=\sqrt{-1}$, not $-1$. $i$ is defined as $\sqrt{-1}$, not equal to $-1$.

51. What is $\sqrt{-81}$ written using $i$?

    $9i$. $\sqrt{-81} = \sqrt{81} \cdot \sqrt{-1} = 9i$

52. What is the complex number $-3i$ written in $a+bi$ form?

    $-3i=0-3i$. Pure imaginary numbers have zero real part.

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