Question 23 of 28intermediate🔧 ApplyNumerical2 marks

Find the modulus of 1+i1−i−1−i1+i\frac{1+i}{1-i} - \frac{1-i}{1+i}.

Correct Answer

2

Exercise: Miscellaneous Exercise on Chapter 4 | Q: 9 | (Page 10)
For More Understanding

Explanation

To find the modulus, we first simplify the given expression by rationalizing each fraction. Using the method shown in Example 6, we multiply numerator and denominator by the conjugate of the denominator. After simplification, the expression reduces to 2i, whose modulus is calculated using the definition |z| = a2+b2\sqrt{a² + b²} from Section 4.4.

Solution Steps

  1. Step 1: Simplify (1+i)/(1-i) by multiplying numerator and denominator by (1+i): (1+i)/(1-i) × (1+i)/(1+i) = (1+i)²/[(1-i)(1+i)] = (1 + 2i + i²)/(1 - i²) = (1 + 2i - 1)/(1 + 1) = 2i/2 = i

  2. Step 2: Simplify (1-i)/(1+i) by multiplying numerator and denominator by (1-i): (1-i)/(1+i) × (1-i)/(1-i) = (1-i)²/[(1+i)(1-i)] = (1 - 2i + i²)/(1 - i²) = (1 - 2i - 1)/(1 + 1) = -2i/2 = -i

  3. Step 3: Subtract the two results: i - (-i) = i + i = 2i

  4. Step 4: Find the modulus of 2i using |z| = a2+b2\sqrt{a² + b²}: |2i| = 02+22\sqrt{0² + 2²} = 4\sqrt{4} = 2

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