Question 2 of 28beginner🔧 ApplyNumerical1 mark

Express the following complex number in the form a+iba + ib: i9+i19i^9 + i^{19}

Correct Answer

0

Exercise: EXERCISE 4.1 | Q: 2 | (Page 7)
For More Understanding

Explanation

This question tests the understanding of powers of i. The textbook Summary states that for any integer k, i^(4k) = 1, i^(4k+1) = i, i^(4k+2) = -1, i^(4k+3) = -i. Students need to express each power in the form 4k + r where r ∈ {0, 1, 2, 3} to determine the value.

Solution Steps

  1. Step 1: Express i9i^{9} in the form i^(4k+r). We have i9i^{9} = i^(4×24 \times 2+1) = i^(4k+1) where k=2. From the Summary, i^(4k+1) = i. So i9i^{9} = i.

  2. Step 2: Express i19i^{19} in the form i^(4k+r). We have i19i^{19} = i^(4×44 \times 4+3) = i^(4k+3) where k=4. From the Summary, i^(4k+3) = -i. So i19i^{19} = -i.

  3. Step 3: Add the two values: i9i^{9} + i19i^{19} = i + (-i) = i - i = 0.

  4. Step 4: Express in the form a + ib: 0 = 0 + i(0).

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