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mathsComplex NumbersProperties of Complex Number Arithmetic

### Multiplicative Inverse

Complex numbers have multiplicative inverse.

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What does 'inverse' mean?

• same; identical; similar
• opposite; converse
• opposite; converse

Consider two complex numbers z_1 = a+ i b and z_2 = 1 /(a+ib). What will be z_1 xx z_2?

• 1
• 1+0i
• multiplicative identity
• All the above
• All the above

The answer is 'All the above'

Consider two complex numbers z_1 = a+ i b and z_2 = 1 /(a+ib).

z_2 = 1/(a+ib)
quad quad quad = 1/(a+ib) xx (a-ib)/(a-ib)
quad quad quad = (a-ib)/(a^2+b^2)

1/z or (a-ib)/(a^2+b^2) is the multiplicative inverse.

•  For any complex number z there exists 1/z : multiplicative inverse

Multiplicative Inverse: For any complex number z = a+ib in CC, there exists 1/z = (a - i b)/(a^2+b^2) in CC such that
z xx 1/z = 1

Solved Exercise Problem:

What is the multiplicative inverse of 2-i?

• 2+i
• (2+i)/5
• (2+i)/5
• (2+i)/sqrt(5)

The answer is '(2+i)/5'

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