Question 13 of 28beginner🔧 ApplyNumerical1 mark

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

Correct Answer

i

Exercise: EXERCISE 4.1 | Q: 13 | (Page 8)
For More Understanding

Explanation

To find the multiplicative inverse of the complex number z=−iz = -i, we use the standard formula provided in the context: z−1=a−iba2+b2z^{-1} = \frac{a - ib}{a^2 + b^2}. The complex number −i-i can be written in the form a+iba + ib as 0+(−1)i0 + (-1)i, where a=0a = 0 and b=−1b = -1. Substituting these values into the formula yields the inverse as ii. This is also consistent with the rationalization method shown in the context for powers of ii.

Solution Steps

  1. Step 1: Write the complex number in the standard form a+iba + ib. For z=−iz = -i, we have a=0a = 0 and b=−1b = -1.

  2. Step 2: Use the formula for the multiplicative inverse: z−1=a−iba2+b2z^{-1} = \frac{a - ib}{a^2 + b^2}.

  3. Step 3: Substitute the values of aa and bb into the formula: Numerator =0−i(−1)=i= 0 - i(-1) = i, and Denominator =02+(−1)2=1= 0^2 + (-1)^2 = 1.

  4. Step 4: Divide the numerator by the denominator to get the final answer: z−1=i1=iz^{-1} = \frac{i}{1} = i.

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