Find the modulus and amplitude of the following:
(i) `(1+i)/(1-i)`
(ii) `(1)/(1+i)`.

Answer

To find the modulus and amplitude of the given complex numbers, we will follow a systematic approach for each part.

### Part (i): Find the modulus and amplitude of \((1+i)/(1-i)\)

**Step 1: Calculate the modulus of the numerator and denominator.**
- The modulus of a complex number \(z = a + bi\) is given by \(|z| = \sqrt{a^2 + b^2}\).
- For \(1 + i\):
\[ |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \]
- For \(1 - i\):
\[ |1 - i| = \sqrt{1^2 + (-1)^2} = \sqrt{1 + 1} = \sqrt{2} \]

**Step 2: Use the property of modulus for division.**
- The modulus of the quotient is given by:
\[ \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} \]
- Thus, we have:
\[ \left|\frac{1+i}{1-i}\right| = \frac{|1+i|}{|1-i|} = \frac{\sqrt{2}}{\sqrt{2}} = 1 \]

**Step 3: Calculate the amplitude (argument) of the numerator and denominator.**
- The argument of a complex number \(z = a + bi\) is given by \(\text{arg}(z) = \tan^{-1}\left(\frac{b}{a}\right)\).
- For \(1 + i\):
\[ \text{arg}(1+i) = \tan^{-1}\left(\frac{1}{1}\right) = \tan^{-1}(1) = \frac{\pi}{4} \]
- For \(1 - i\):
\[ \text{arg}(1-i) = \tan^{-1}\left(\frac{-1}{1}\right) = \tan^{-1}(-1) = -\frac{\pi}{4} \]

**Step 4: Use the property of arguments for division.**
- The argument of the quotient is given by:
\[ \text{arg}\left(\frac{z_1}{z_2}\right) = \text{arg}(z_1) - \text{arg}(z_2) \]
- Thus, we have:
\[ \text{arg}\left(\frac{1+i}{1-i}\right) = \frac{\pi}{4} - \left(-\frac{\pi}{4}\right) = \frac{\pi}{4} + \frac{\pi}{4} = \frac{\pi}{2} \]

### Conclusion for Part (i):
- Modulus: \(1\)
- Amplitude: \(\frac{\pi}{2}\)

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### Part (ii): Find the modulus and amplitude of \(\frac{1}{1+i}\)

**Step 1: Calculate the modulus of the denominator.**
- For \(1 + i\):
\[ |1 + i| = \sqrt{1^2 + 1^2} = \sqrt{2} \]

**Step 2: Use the property of modulus for division.**
- The modulus of the quotient is:
\[ \left|\frac{1}{1+i}\right| = \frac{|1|}{|1+i|} = \frac{1}{\sqrt{2}} \]

**Step 3: Calculate the amplitude (argument) of the numerator and denominator.**
- The argument of \(1\) is:
\[ \text{arg}(1) = 0 \]
- The argument of \(1 + i\) is:
\[ \text{arg}(1+i) = \frac{\pi}{4} \]

**Step 4: Use the property of arguments for division.**
- The argument of the quotient is:
\[ \text{arg}\left(\frac{1}{1+i}\right) = \text{arg}(1) - \text{arg}(1+i) = 0 - \frac{\pi}{4} = -\frac{\pi}{4} \]

### Conclusion for Part (ii):
- Modulus: \(\frac{1}{\sqrt{2}}\)
- Amplitude: \(-\frac{\pi}{4}\)

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