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Thought-Process to Discover Knowledge

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mathsComplex NumbersAlgebra of Complex Numbers

### Powers of i

The value of i^n for various values of n is discussed.

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What is the value of i?

• -1
• sqrt(-1)
• sqrt(-1)

The answer is 'sqrt(-1)'

What is the value of i^2?

• -1
• -1
• sqrt(-1)

The answer is '-1'

It is defined that i=sqrt(-1). Substitute that in the following
i^2
quad quad = (sqrt(-1))^2
quad quad = (-1^(1/2))^2
quad quad = -1^(2/2)
quad quad = -1

What is the value of i^3?

• -1
• i
• -i
• -i

The answer is '-i'

i^3
quad quad = i^(2+1)
quad quad = i^2 xx i^1
quad quad = -1 xx i
quad quad = -i

What is the value of i^4?

• 1
• 1
• -1

The answer is '1'

i^4
quad quad = i^(2+2)
quad quad = (i)^2 xx i^2
quad quad = -1 xx -1
quad quad = 1

What is the value of i^0?

• 1
• 1
• -1
• i

The answer is '1'

i^0
quad quad = i^(1-1)
quad quad = i/i
quad quad = 1

What is the value of i^n, if n=4p+2q+r?

• i
• (-1)^q xx i^r
• (-1)^q xx i^r
• i^r

The answer is '(-1)^q xx i^r'

i^n
quad quad = i^(4p+2q+r)
quad quad = i^(4p) xx i^(2q) xx i^r
quad quad = (i^4)^p) xx (i^2)^q) xx i^r
quad quad = (-1)^q xx i^r

To calculate i^n, use i^4=1 and i^2=-1.

Power of i: To calculate i^n, express n in the form n=4p+2q+r, where p,q, r in NN natural numbers. Then
i^n = (-1)^q xx i^r.

Solved Exercise Problem:

What is the value of i^27?

• 1
• -i
• -i
• i

The answer is '-i'

Solved Exercise Problem:

What is i^20?

• 1
• 1
• -1
• i
• -i

The answer is '1'

Solved Exercise Problem:

What is i^7?

• 1
• -1
• i
• -i
• -i

The answer is '-i'

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