Exponential form of a complex number

In summary: In that case θ would be negative and you would have to use the inverse of tan to get the correct answer.
  • #1
blues1
2
0

Homework Statement



if z = -2 + 2i

find r and θ

The Attempt at a Solution



our teacher told us that when we have z = a + bi

r = sqrt(a^2 + b^2)

and θ = tan^-1(b/a)

so here it's supposed to be r = sqrt(8) and θ = - pi/4

but using wolfram alpha to see if the results are matching I get that

sqrt(8)*e^(i*-pi/4) is 2 - 2i

what am I doing wrong? I guess this has to do with the θ, but since it's always tan^-1(b/a) why am I getting different results?
 
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  • #2
This is an example of why you should not use formulas without thinking.

b/a and therefore tan(b/a) are exactly the same if you change signs on both a and b. In particular,
[tex]\frac{-2}{2}= \frac{2}{-2}= -1[/tex]

Also, [itex]tan(\theta)= tan(\pi+ \theta)[/itex]. Since tan is "multi-valued", a calculator or computer (or table of trig functions) will get the principle value which is always between [itex]-\pi/2[/itex] and [itex]\pi/2[/itex]. To distinguish between the two you need to look at the signs of the individual numbers. (-2, 2) is in the second quadrant so [itex]\theta[/itex] is between [itex]\pi/2[/itex] and [itex]\pi[/itex]. In this problem, [itex]\theta= \pi- \pi/4= 3\pi/4[/itex], NOT [itex]-\pi/4[/itex].
 
  • #3
HallsofIvy said:
This is an example of why you should not use formulas without thinking.

b/a and therefore tan(b/a) are exactly the same if you change signs on both a and b. In particular,
[tex]\frac{-2}{2}= \frac{2}{-2}= -1[/tex]

Also, [itex]tan(\theta)= tan(\pi+ \theta)[/itex]. Since tan is "multi-valued", a calculator or computer (or table of trig functions) will get the principle value which is always between [itex]-\pi/2[/itex] and [itex]\pi/2[/itex]. To distinguish between the two you need to look at the signs of the individual numbers. (-2, 2) is in the second quadrant so [itex]\theta[/itex] is between [itex]\pi/2[/itex] and [itex]\pi[/itex]. In this problem, [itex]\theta= \pi- \pi/4= 3\pi/4[/itex], NOT [itex]-\pi/4[/itex].

if i remember correct everything is positive from 0 to pi/2, in pi/2 to p only the sin is positive, cos is positive from p to 3pi/2

so if I get a negative θ and z for example is from 0 to pi/2 I should just have tan^-1(b/a)

if I am from pi/2 to p, I should have p - tan^-1(b/a)

from p to 3pi/2 => 3pi/2 - tan^-1(b/a)?

is this how it goes? :/
 
  • #4
blues1 said:
if i remember correct everything is positive from 0 to pi/2, in pi/2 to p only the sin is positive, cos is positive from p to 3pi/2

so if I get a negative θ and z for example is from 0 to pi/2 I should just have tan^-1(b/a)
I don't understand what you are saying. If z = a + bi and both a and b are positive, θ = tan^-1(b/a) would just be positive.

from p to 3pi/2 => 3pi/2 - tan^-1(b/a)?

is this how it goes? :/
No. If z = a + bi and both a and b are negative, then when you find tan^-1(b/a) you will get an angle between 0 and π/2. But you want to be in Quadrant III, so you'll have to add pi.

And you're missing the case of z being in Quadrant IV of the complex coordinate plane.
 

Related to Exponential form of a complex number

1. What is the exponential form of a complex number?

The exponential form of a complex number is a representation of a complex number in the form re, where r is the magnitude or modulus of the complex number, and θ is the angle or argument of the complex number.

2. How do you convert a complex number from standard form to exponential form?

To convert a complex number from standard form a + bi to exponential form re, you can use the following formula: r = √(a2 + b2) and θ = tan-1(b/a). Plug in the values for a and b from the standard form into the formula to find the values for r and θ.

3. What is the advantage of using exponential form for complex numbers?

The advantage of using exponential form for complex numbers is that it makes calculations involving complex numbers much easier. In exponential form, multiplication and division can be done by simply adding or subtracting the exponents, and powers can be calculated by multiplying the exponents. This simplifies complex number operations significantly.

4. Can the exponential form of a complex number be negative?

Yes, the exponential form of a complex number can be negative. The negative sign would be placed in front of the e term, such as -re. This indicates that the angle or argument of the complex number is in the opposite direction from the positive form.

5. How is the exponential form of a complex number related to the polar form?

The exponential form of a complex number is equivalent to the polar form, which is another way of representing complex numbers. In polar form, the complex number is written as r(cosθ + isinθ), where r is the modulus and θ is the argument. The exponential form is derived from the polar form by using Euler's formula, e = cosθ + isinθ.

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