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Express the following in the form a+bi: -1, 1+i, -5-12i, 3-4i, -2(sqrt(3)-i).

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Here are the solutions to your complex numbers problems.

1. Complete the following table

We will fill in the missing values for each row.

  • Row a: Given z=−2+23iz = -2 + 2\sqrt{3}i

  1. Calculate the modulus rr.

    $$
r = \sqrt{(-2)^2 + (2\sqrt{3})^2} = \sqrt{4 + 12} = \sqrt{16} = 4
$$

  2. Calculate the argument θ\theta.

    The number is in the second quadrant. α=arctan⁡(∣23−2∣)=arctan⁡(3)=π3 \alpha = \arctan\left(\left|\frac{2\sqrt{3}}{-2}\right|\right) = \arctan(\sqrt{3}) = \frac{\pi}{3} θ=π−α=π−π3=2π3 \theta = \pi - \alpha = \pi - \frac{\pi}{3} = \frac{2\pi}{3}

  3. Write the polar and exponential forms.

    $$
\text{Polar form: } 4\left(\cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right)\right)
$$
$$
\text{Exponential form: } 4e^{i\frac{2\pi}{3}}
$$
    • Row b: Given modulus r=3r = 3 and argument θ=−5π4\theta = -\frac{5\pi}{4}

  4. Calculate the rectangular form x+yix+yi.

    $$
x = r\cos\theta = 3\cos\left(-\frac{5\pi}{4}\right) = 3\left(-\frac{\sqrt{2}}{2}\right) = -\frac{3\sqrt{2}}{2}
$$
$$
y = r\sin\theta = 3\sin\left(-\frac{5\pi}{4}\right) = 3\left(\frac{\sqrt{2}}{2}\right) = \frac{3\sqrt{2}}{2}
$$
$$
\text{Rectangular form: } -\frac{3\sqrt{2}}{2} + \frac{3\sqrt{2}}{2}i
$$

  5. Write the polar and exponential forms.

    $$
\text{Polar form: } 3\left(\cos\left(-\frac{5\pi}{4}\right) + i\sin\left(-\frac{5\pi}{4}\right)\right)
$$
$$
\text{Exponential form: } 3e^{-i\frac{5\pi}{4}}
$$
    • Row c: Given polar form 6(cos⁡(π6)+isin⁡(π6))6\left(\cos\left(\frac{\pi}{6}\right) + i\sin\left(\frac{\pi}{6}\right)\right)

  6. Identify the modulus and argument.

    $$
r = 6
$$
$$
\theta = \frac{\pi}{6}
$$

  7. Calculate the rectangular form x+yix+yi.

    $$
x = r\cos\theta = 6\cos\left(\frac{\pi}{6}\right) = 6\left(\frac{\sqrt{3}}{2}\right) = 3\sqrt{3}
$$
$$
y = r\sin\theta = 6\sin\left(\frac{\pi}{6}\right) = 6\left(\frac{1}{2}\right) = 3
$$
$$
\text{Rectangular form: } 3\sqrt{3} + 3i
$$

  8. Write the exponential form.

    $$
\text{Exponential form: } 6e^{i\frac{\pi}{6}}
$$
    • Row d: Given exponential form πeiπ2\pi e^{i\frac{\pi}{2}}

  9. Identify the modulus and argument.

    $$
r = \pi
$$
$$
\theta = \frac{\pi}{2}
$$

  10. Calculate the rectangular form x+yix+yi.

    $$
x = r\cos\theta = \pi\cos\left(\frac{\pi}{2}\right) = \pi(0) = 0
$$
$$
y = r\sin\theta = \pi\sin\left(\frac{\pi}{2}\right) = \pi(1) = \pi
$$
$$
\text{Rectangular form: } \pi i
$$

  11. Write the polar form.

    $$
\text{Polar form: } \pi\left(\cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right)\right)
$$

    Here is the completed table:

    Complex Number x+yiModulus rArgument θPolar form r(cos⁡θ+isin⁡θ)Exponential form reiθa) −2+23i42π34(cos⁡(2π3)+isin⁡(2π3))4ei2π3b) −322+322i3−5π43(cos⁡(−5π4)+isin⁡(−5π4))3e−i5π4c) 33+3i6π66(cos⁡(π6)+isin⁡(π6))6eiπ6d) πiππ2π(cos⁡(π2)+isin⁡(π2))πeiπ2\begin{array}{|l|c|c|c|c|} \hline \textbf{Complex Number } \boldsymbol{x+yi} & \textbf{Modulus } \boldsymbol{r} & \textbf{Argument } \boldsymbol{\theta} & \textbf{Polar form } \boldsymbol{r(\cos\theta + i\sin\theta)} & \textbf{Exponential form } \boldsymbol{re^{i\theta}} \\ \hline \text{a) } -2+2\sqrt{3}i & \boxed{4} & \boxed{\frac{2\pi}{3}} & \boxed{4\left(\cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right)\right)} & \boxed{4e^{i\frac{2\pi}{3}}} \\ \hline \text{b) } \boxed{-\frac{3\sqrt{2}}{2} + \frac{3\sqrt{2}}{2}i} & 3 & -\frac{5\pi}{4} & \boxed{3\left(\cos\left(-\frac{5\pi}{4}\right) + i\sin\left(-\frac{5\pi}{4}\right)\right)} & \boxed{3e^{-i\frac{5\pi}{4}}} \\ \hline \text{c) } \boxed{3\sqrt{3} + 3i} & \boxed{6} & \boxed{\frac{\pi}{6}} & 6\left(\cos\left(\frac{\pi}{6}\right) + i\sin\left(\frac{\pi}{6}\right)\right) & \boxed{6e^{i\frac{\pi}{6}}} \\ \hline \text{d) } \boxed{\pi i} & \boxed{\pi} & \boxed{\frac{\pi}{2}} & \boxed{\pi\left(\cos\left(\frac{\pi}{2}\right) + i\sin\left(\frac{\pi}{2}\right)\right)} & \pi e^{i\frac{\pi}{2}} \\ \hline \end{array}

    2. Express the following in the form reiθre^{i\theta} with −π<θ≤π-\pi < \theta \le \pi

    • a) z=−iz = -i

  12. Identify x=0,y=−1x=0, y=-1.

    Calculate the modulus. r=02+(−1)2=1 r = \sqrt{0^2 + (-1)^2} = 1

  13. Calculate the argument.

    Since zz is on the negative imaginary axis, θ=−π2\theta = -\frac{\pi}{2}.

  14. Write in exponential form.

    $$
z = \boxed{e^{-i\frac{\pi}{2}}}
$$
    • b) z=1+iz = 1+i

  15. Identify x=1,y=1x=1, y=1.

    Calculate the modulus. r=12+12=2 r = \sqrt{1^2 + 1^2} = \sqrt{2}

  16. Calculate the argument.

    The number is in the first quadrant. θ=arctan⁡(11)=π4 \theta = \arctan\left(\frac{1}{1}\right) = \frac{\pi}{4}

  17. Write in exponential form.

    $$
z = \boxed{\sqrt{2}e^{i\frac{\pi}{4}}}
$$
    • c) z=−5−12iz = -5 - 12i

  18. Identify x=−5,y=−12x=-5, y=-12.

    Calculate the modulus. r=(−5)2+(−12)2=25+144=169=13 r = \sqrt{(-5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13

  19. Calculate the argument.

    The number is in the third quadrant. α=arctan⁡(∣−12−5∣)=arctan⁡(125) \alpha = \arctan\left(\left|\frac{-12}{-5}\right|\right) = \arctan\left(\frac{12}{5}\right) θ=−π+α=−π+arctan⁡(125) \theta = -\pi + \alpha = -\pi + \arctan\left(\frac{12}{5}\right)

  20. Write in exponential form.

    $$
z = \boxed{13e^{i\left(-\pi + \arctan\left(\frac{12}{5}\right)\right)}}
$$
    • d) z=3−4iz = 3 - 4i

  21. Identify x=3,y=−4x=3, y=-4.

    Calculate the modulus. r=32+(−4)2=9+16=25=5 r = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5

  22. Calculate the argument.

    The number is in the fourth quadrant. α=arctan⁡(∣−43∣)=arctan⁡(43) \alpha = \arctan\left(\left|\frac{-4}{3}\right|\right) = \arctan\left(\frac{4}{3}\right) θ=−α=−arctan⁡(43) \theta = -\alpha = -\arctan\left(\frac{4}{3}\right)

  23. Write in exponential form.

    $$
z = \boxed{5e^{i\left(-\arctan\left(\frac{4}{3}\right)\right)}}
$$
    • e) z=−2(3−1)z = -2(\sqrt{3} - 1)

  24. Simplify the number.

    z=−23+2z = -2\sqrt{3} + 2. This is a real number.
    Since 3≈1.732\sqrt{3} \approx 1.732, z≈−2(1.732)+2=−3.464+2=−1.464z \approx -2(1.732) + 2 = -3.464 + 2 = -1.464. This is a negative real number.

  25. Calculate the modulus.

    $$
r = |-2(\sqrt{3} - 1)| = 2(\sqrt{3} - 1)
$$

  26. Calculate the argument.

    For any negative real number, θ=π\theta = \pi.

  27. Write in exponential form.

    $$
z = \boxed{2(\sqrt{3} - 1)e^{i\pi}}
$$
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