Express the following complex numbers in the form a + ib.
- A: 2 e^(3 ipi /2)
- B: 4?(3 e^(ipi /4))
- C: e^(ipi)
- D: ((sqrt 5)/(1+sqrt2)) (e^(ipi /4))
The Correct Answer and Explanation is:
To express the given complex numbers in the form a+iba + ib, we convert the exponential form using Euler’s formula: eiθ=cos(θ)+isin(θ)e^{i\theta} = \cos(\theta) + i\sin(\theta)
A: 2e3iπ22e^{\frac{3i\pi}{2}}
e3iπ2=cos(3π2)+isin(3π2)=0−i=−ie^{\frac{3i\pi}{2}} = \cos\left(\frac{3\pi}{2}\right) + i\sin\left(\frac{3\pi}{2}\right) = 0 – i = -i ⇒2e3iπ2=2(−i)=0−2i\Rightarrow 2e^{\frac{3i\pi}{2}} = 2(-i) = \boxed{0 – 2i}
B: 43eiπ44\sqrt{3}e^{\frac{i\pi}{4}}
eiπ4=cos(π4)+isin(π4)=22+i22e^{\frac{i\pi}{4}} = \cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} ⇒43eiπ4=43(22+i22)=26+i26=26+2i6\Rightarrow 4\sqrt{3}e^{\frac{i\pi}{4}} = 4\sqrt{3}\left(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\right) = 2\sqrt{6} + i2\sqrt{6} = \boxed{2\sqrt{6} + 2i\sqrt{6}}
C: eiπe^{i\pi}
eiπ=cos(π)+isin(π)=−1+0i=−1e^{i\pi} = \cos(\pi) + i\sin(\pi) = -1 + 0i = \boxed{-1}
D: 51+2eiπ4\frac{\sqrt{5}}{1+\sqrt{2}} e^{\frac{i\pi}{4}}
We simplify: eiπ4=22+i22e^{\frac{i\pi}{4}} = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} So: 51+2(22+i22)=522(1+2)(1+i)\text{So: } \frac{\sqrt{5}}{1+\sqrt{2}} \left(\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}\right) = \frac{\sqrt{5}\sqrt{2}}{2(1+\sqrt{2})}(1 + i)
Now rationalize the denominator: 102(1+2)=10(1−2)2(12−(2)2)=10(1−2)2(1−2)=−10(1−2)2\frac{\sqrt{10}}{2(1+\sqrt{2})} = \frac{\sqrt{10}(1-\sqrt{2})}{2(1^2 – (\sqrt{2})^2)} = \frac{\sqrt{10}(1-\sqrt{2})}{2(1 – 2)} = \frac{-\sqrt{10}(1-\sqrt{2})}{2} ⇒−10(1−2)2(1+i)\Rightarrow \boxed{\frac{-\sqrt{10}(1-\sqrt{2})}{2}(1 + i)}
Explanation
To convert complex numbers from exponential (polar) form to rectangular (Cartesian) form a+iba + ib, we use Euler’s formula: eiθ=cos(θ)+isin(θ)e^{i\theta} = \cos(\theta) + i\sin(\theta)
This formula links trigonometry and exponential functions, enabling the transformation of expressions involving angles and magnitudes into real and imaginary parts.
For A, the angle is 3π2\frac{3\pi}{2}, where cosine is 0 and sine is -1. Multiplying by 2 yields 0−2i0 – 2i.
For B, we multiply 434\sqrt{3} by eiπ/4e^{i\pi/4}, whose cosine and sine values are both 22\frac{\sqrt{2}}{2}. Multiplying yields 26+2i62\sqrt{6} + 2i\sqrt{6}.
C involves the identity eiπ=−1e^{i\pi} = -1, which is a famous result from Euler’s formula and demonstrates the connection between ee, ii, and π\pi.
D combines a real coefficient with an exponential. After expressing the exponential using cosine and sine, we simplify and rationalize the denominator to express it as a complex number. Rationalization ensures the final form doesn’t contain irrational numbers in the denominator, which is standard in mathematics.
These conversions are crucial in fields like electrical engineering, quantum mechanics, and signal processing, where complex numbers are often used in polar form for convenience in multiplication and division, but rectangular form is preferred for calculations involving addition and visualization.
