Question

Solution for negative integral power of i complex number

Answer 1

Answer:

Integral power of a complex number is also a complex number. In other words any integral power of a complex number can be expressed in the form of A + iB, where A and B are real.

If z is any complex number, then positive integral powers of z are defined as z\(^{1}\) = a, z\(^{2}\) = z ∙ z, z\(^{3}\) = z\(^{2}\) ∙ z, z\(^{4}\) = z\(^{3}\) ∙ z and so on.

If z is any non-zero complex number, then negative integral powers of z are defined as:

z\(^{-1}\) = \(\frac{1}{z}\), z\(^{-2}\) = \(\frac{1}{z^{2}}\), z\(^{-3}\) = \(\frac{1}{z^{3}}\), etc.

If z ≠ 0, then z\(^{0}\) = 1.

Integral Power of:

Any integral power of i is i or, (-1) or 1.

Integral power of i are defined as:

i0 = 1, i1 = i, i2 = -1,

i3 = i2 ∙ i = (-1)i = -i,

i4 = (i2)2 = (-1)2 = 1,

i5 = i4 ∙ i = 1 ∙ i = i,

i6 = i4 ∙ i2 = 1 ∙ (-1) = -1, and so on.

i−1 = 1i = 1i × ii = i−1 = - i

Remember that 1i = - i

i−1 = 1i2 = 1−1 = -1

i−3 = 1i3 = 1i3 × ii = ii4 = i1 = i

i−4 = 1i4 = 11 = 1, and so on.

Note that i4 = 1 and i−4 = 1. It follows that for any integer k,

i4k = 1, i4k+1= i, i4k+2 = -1, i4k+3 = - i.

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