Math Is Fun Forum

soccerfanatic

Guest

Complex numbers

The value

is a positive real number. What real number is it?

Find all complex numbers z such that |z-1|=|z+3|=|z-i|

Prove that if w,z are complex numbers such that |w|=|z|=1 and

, then

is a real number.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

Re: Complex numbers

hi soccerfanatic,

Welcome to the forum.

I have replaced your $ signs with maths tags

to make your questions display properly on this forum.

Q1.Work out the modulus EDIT: and argument of the complex number inside the bracket  You'll find this simplifies easily using De Moivre after that.

Q2.  I re-wrote with z = a +bi.  The a squared and b squared terms cancel out and it's then straight forward to determine a and b. 

Q3.  Once again I re-wrote with z = a + bi and w = c + di.

Then I worked out the denominator in terms of a, b, c, d and i.  If you multiply top and bottom by the complex conjugate of the denominator the denom. is then real.  So all you need to prove is that the imaginary part of the numerator is zero.  So most of the algebra can be left un-simplified.

You'll need to use a^2 + b^2 = c^2 + d^2 = 1.

Bob

Last edited by Bob (2015-10-18 19:09:08)

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