Answer the following questions ( true or false)

Question

Answer the following questions (true or false)

1. Arg (0) is undefined 

2. Arg z₁/z₂= Arg z₁-Arg z_2, if z₁ ≠0, z₂≠0

3. Arg z= Arg z + 2pieK (K=0, ±1, ±2, ±3) and if z ≠0

4. Arg z₁z₂= Arg z₁+ Arg z₂, if z₁≠0, z₂≠0

5. Arg z= -Arg z, if z is not real

Answer 1

Answer the following questions (true or false)
1. Arg (0) is undefined
2. Arg z1/z2 = Arg z1-Arg z2, if z1 ≠0, z2≠0
3. Arg z = Arg z + 2pieK (K=0, ±1, ±2, ±3) and if z ≠0
4. Arg z1*z2= Arg z1+ Arg z2, if z1≠0, z2≠0
5. Arg z = -Arg z, if z is not real

 

Complex number, z = a + bi = r e^(i θ), where

r = |z| = √(a^2 + b^2) is called the "modulus" and

θ = Arg(z) is called the "argument".

 

1. z = 0 is the point at the origin with no length or direction, so 1 is true.

2. true; use exponential form to see it.

3. false; the principal value: 0 ≤ Arg(z) < 2 pi.

4. true; use exponential form to see it.

5. false.

 

Answer 2

In think the first four are true, but the last is false.

A counter example for  5)   is   z = .707 + .707 i

 

          Arg z =  pi/4     but the z which has Arg z = -  pi/4 is z = .707 - .707 i     (the complex conjugate)

 

3) is backwards in my mind,  there are an infinite set of z  all with the same Arg(z).   The differ formally by factors of   exp(2pi i) .