Theory of Intelligence
11TH•Mathematics•Complex Numbers•Exercise 1.1

Introduction

Practice set 1.1 for Complex Numbers.

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Q1

Question 1

Find the multiplicative inverse of each complex number.

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(i)

(−4,7)(-4, 7)

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Step-by-Step Solution

Question 1 • Part (i)

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Solution

Write the complex number in a+ib form:z=−4+7iFormula for multiplicative inverse:z−1=1a+ib=a−iba2+b2Substitute a=−4 and b=7:z−1=−4−7i(−4)2+72Simplify denominator:=−4−7i16+49=−4−7i65Separate real and imaginary parts:=−465−765iWrite in ordered pair form:(−465,−765)\begin{aligned} & \boxed{\text{Write the complex number in } a+ib \text{ form:}} \\ \\ & z = -4 + 7i \\ \\ & \boxed{\text{Formula for multiplicative inverse:}} \\ \\ & z^{-1} = \frac{1}{a+ib} = \frac{a-ib}{a^2+b^2} \\ \\ & \boxed{\text{Substitute } a=-4 \text{ and } b=7:} \\ \\ & z^{-1} = \frac{-4 - 7i}{(-4)^2 + 7^2} \\ \\ & \boxed{\text{Simplify denominator:}} \\ \\ & = \frac{-4 - 7i}{16 + 49} = \frac{-4 - 7i}{65} \\ \\ & \boxed{\text{Separate real and imaginary parts:}} \\ \\ & = -\frac{4}{65} - \frac{7}{65}i \\ \\ & \boxed{\text{Write in ordered pair form:}} \\ \\ & \boxed{\left(-\frac{4}{65}, -\frac{7}{65}\right)} \end{aligned}
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