Real and Complex Mixed!

Algebra Level 2

It is easy to see that z = 1 z = 1 and z = − 1 z = -1 both satisfy ∣ z ∣ = 1 |z| = 1 in the set of complex numbers.

Must z = i z = i and z = − i z = -i both satisfy ∣ z ∣ = i |z| = i ?

Note: i = − 1 i = \sqrt{-1}

No, both are wrong. No, but only z = i z = i works. No, but only z = − i z = -i works. Yes, both work.

Michael Huang by Michael Huang , 29, USA

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2 solutions

Blan Morrison
Oct 7, 2018

The absolute value (or magnitude) of any number (including real, complex, and quaternion) is the distance between 0 (or the origin) and the point representing said number. The absolute value of any number, ∣ z ∣ |z| , is measured as a real value. So, ∣ i ∣ = ∣ − i ∣ = ∣ 1 ∣ = ∣ − 1 ∣ = 1 |i|=|-i|=|1|=|-1|=1 .

7 Helpful 0 Interesting 0 Brilliant 0 Confused

Let f : C → C f : C \to C , be a complex valued function such that f ( z ) = ∣ z ∣ f(z) =|z| . Then, by definition, if z = a + i b z=a+i b for real a a , and b b : f ( z ) = ∣ z ∣ = a 2 + b 2 f(z) =|z|=\sqrt{a^2+b^2} Which gives only real values. Therefore, the equation ∣ z ∣ = i |z|=i has no solution in the complex plane.

3 Helpful 0 Interesting 0 Brilliant 0 Confused

Be careful here. By saying that the absolute value of Z is the Pythagorean theorem, there indeed is a complex solution. For example, if both A and B are i/√(2), √(a^2+b^2)=i, and a^2+b^2=-1 are all true.

Michael Boyd - 2 years, 5 months ago

Is not absolute value of z z is just modulus, also a a and b b are just real numbers, so there is no complex solution with no real part

Hjalmar Orellana Soto - 2 years, 3 months ago

1 pending report

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