# Turning Complex Number z into Polar Form

• mr-feeno
So, can you provide a summary of the conversation?In summary, the problem statement is to convert the given equation, (z-1)/(z+1)=i, into polar form. The attempt at a solution involved finding the magnitude of z and the angle it makes with the horizontal axis, but there was confusion about the value of |i| and the angle it makes. The conversation also clarified that the given equation can be easily solved in polar form by finding the magnitude and angle of i.
mr-feeno

## Homework Statement

$$\frac{z-1}{z+1}=i$$
I found the cartesian form, z = i, but how do I turn it into polar form?

## The Attempt at a Solution

$$|z|=\sqrt{0^2+1^2}=1$$
$$\theta=arctan\frac{b}{a}=arctan\frac{1}{0}$$

Is the solution then that is not possible to convert it to polar form?

mr-feeno said:

## Homework Statement

$$\frac{z-1}{z+1}=i$$
I found the cartesian form, z = i, but how do I turn it into polar form?

## The Attempt at a Solution

$$|z|=\sqrt{0^2+1^2}=1$$
$$\theta=arctan\frac{b}{a}=arctan\frac{1}{0}$$

Is the solution then that is not possible to convert it to polar form?

Notice that ##\tan(\frac{\pi}{2} + k \pi)## with ## k \in \mathbb{Z}## is not defined. Draw ## i ## in the complex plane. What can you conclude?

mr-feeno said:

## Homework Statement

$$\frac{z-1}{z+1}=i$$
I found the cartesian form, z = i, but how do I turn it into polar form?
What is the exact problem statement? Are you given that ##\frac{z-1}{z+1}=i##?

Doing some work on this, it appears that if ##\frac{z-1}{z+1}=i##, then ##z = i##
It would have been helpful to me for you to say what is given, and what you needed to do.
mr-feeno said:

## The Attempt at a Solution

$$|z|=\sqrt{0^2+1^2}=1$$
$$\theta=arctan\frac{b}{a}=arctan\frac{1}{0}$$

Is the solution then that is not possible to convert it to polar form?
It's easy to convert to polar form. What is |i|?
What is the angle that i makes with the horizontal axis?

Mark44 said:
What is the exact problem statement? Are you given that ##\frac{z-1}{z+1}=i##?

Doing some work on this, it appears that if ##\frac{z-1}{z+1}=i##, then ##z = i##
It would have been helpful to me for you to say what is given, and what you needed to do.

It's easy to convert to polar form. What is |i|?
What is the angle that i makes with the horizontal axis?
$$90\circ$$? I felt it was clear
|z| is the length

mr-feeno said:
$$90\circ$$? I felt it was clear
|z| is the length

The angle is ##\pi/2##, in radians, or 90°.
The magnitude is NOT |z|. I asked what is the magnitude of i?

No, it wasn't clear.

Clear would be something like this:
If ##\frac{z - 1}{z + 1} = i##, solve for z, writing it in polar form.

Mark44 said:
The angle is ##\pi/2##, in radians, or 90°.
The magnitude is NOT |z|. I asked what is the magnitude of i?

No, it wasn't clear.

Clear would be something like this:

Ok, my bad. But thanks :)

mr-feeno said:
Ok, my bad. But thanks :)
You're welcome!

## 1. What is a complex number?

A complex number is a number that contains both a real part and an imaginary part. It is typically written in the form a + bi, where a is the real part and bi is the imaginary part (with i being the imaginary unit equal to √−1). Complex numbers are used in many fields of science and mathematics, including engineering, physics, and computer graphics.

## 2. How do you convert a complex number into polar form?

To convert a complex number z = a + bi into polar form, we can use the formula r(cosθ + isinθ), where r is the distance from the origin to the complex number and θ is the angle between the positive real axis and the line connecting the origin to the complex number. The values for r and θ can be found using the Pythagorean theorem and trigonometric functions.

## 3. What is the significance of polar form for complex numbers?

Polar form allows us to represent a complex number in terms of its distance from the origin and the angle it makes with the positive real axis. This form is useful for performing operations such as multiplication and division of complex numbers, as well as finding roots and powers of complex numbers.

## 4. Can negative numbers have a polar form?

Yes, negative numbers can also be represented in polar form. For example, the complex number -1 can be written as r(cosπ + isinπ), where r = 1 and θ = π. This shows that the distance from the origin is 1 unit and the angle is π radians, or 180 degrees.

## 5. How is polar form used in real-world applications?

Polar form is used in a variety of real-world applications, such as in electrical engineering for analyzing AC circuits, in physics for describing wave behavior, and in navigation for representing the position of an object using distance and direction. It is also used in computer graphics for displaying complex shapes and animations.

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