Question 11 of 28beginner🔧 ApplyNumerical1 mark

Find the multiplicative inverse of the following complex number: 4−3i4 - 3i

Correct Answer

425+325\frac{4}{25} + \frac{3}{25}i

Exercise: EXERCISE 4.1 | Q: 11 | (Page 8)
For More Understanding

Explanation

The question asks for the multiplicative inverse of the complex number 4 - 3i. The provided context (specifically Example 5 and the general formula derived in the section on conjugate and modulus) establishes that the multiplicative inverse of a complex number z is given by the formula z^-1 = zˉ\bar{z} / |z|^2. We apply this formula by finding the conjugate and the square of the modulus.

Solution Steps

  1. Step 1: Let the given complex number be z=4−3iz = 4 - 3i.

  2. Step 2: Find the conjugate of zz, denoted as zˉ\bar{z}. The conjugate is 4+3i4 + 3i.

  3. Step 3: Calculate the square of the modulus, ∣z∣2|z|^2. Using the formula a2+b2a^2 + b^2, we get ∣z∣2=42+(−3)2=16+9=25|z|^2 = 4^2 + (-3)^2 = 16 + 9 = 25.

  4. Step 4: Apply the formula for the multiplicative inverse: z−1=zˉ∣z∣2z^{-1} = \frac{\bar{z}}{|z|^2}.

  5. Step 5: Substitute the values: z−1=4+3i25=425+325iz^{-1} = \frac{4 + 3i}{25} = \frac{4}{25} + \frac{3}{25}i.

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