পোলার এবং আয়তাকার ফর্মের মধ্যে রূপান্তর

The distance from a number to $0$ is call the absolute value, modulus, or magnitude of the number. Note that this definition is simply a generalization of the definition of absolute value of a real number. The previously described angle is called the argument of a number. For example, if $z = 1 + i$, we would write $|z| = \sqrt{2}$ and $\arg(z) = \pi/4$.

It should be relatively easy to see that, if a complex number $z$ has magnitude $r$ and argument $\theta$, then: $$z = r(\cos\theta + i\sin\theta)$$ This is called the polar form of a complex number.

Thus, if you want to convert from polar form to rectangular form, remember that $Re(z) = r\cos\theta$ and $Im(z) = r\sin\theta$.

If you want to convert a complex number $z = a+bi$ from rectangular from to polar form, then use the following conversions, which are also quite easy to show: $$r = \sqrt{a^2+b^2}$$ $$\tan\theta = \frac{b}{a}$$

There are two things of which to be cautious when calculating $\theta$:

First, if $a=0$, then $\tan\theta$ is undefined. In this case, if $b>0$, then $\theta = \pi/2$, and if $b<0$, then $\theta = 3\pi/2$. If $a=b=0$, then the complex number is simply $0$, which does not have a well-defined argument. In this case, you are free to choose any argument you want, since $r=0$ anyway.

Second, $\theta$ can lie anywhere in the range $[0,2\pi)$, but $\tan\theta$ is periodic with period $\pi$. Thus, when you compute $\tan\theta$, there will be $2$ possible values of $\theta$; in order to determine which is which, check the signs of $a$ and $b$ to determine which quadrant contains the complex number.

###Looking Back In the previous lesson, we talked about the polar form of a complex number. We saw that the horizontal axis is the real axis and the vertical axis is the imaginary axis. In this lesson, we will convert numbers in polar form to numbers in the rectangular form, and vice versa.

Converting between polar and rectangular form

Note: $a$ is the real part, and $b$ is the imaginary part of any complex number $z$. In the polar form of $z$, $r$ is the absolute value and $\theta$ is the argument. Basically, $z=a+bi=r(\cos\theta+i\sin\theta)$.

Inside our definition of the polar form, we implicitly created a conversion formula from polar back to rectangular. The real part is $r\cos\theta$, and the imaginary part is $r\sin\theta$.

$$\text{Polar }\rightarrow\text{ Rectangular}$$ $$\boxed{\begin{align}r\cos\theta &= a \\ r\sin\theta &= b\end{align}}$$

Let's use those formulas we just defined above to look at the other way around.

To get $r$ we can square $a$ and $b$ and add them; we exploit the Pythagorean Identity to eliminate $\theta$.

$$a^2+b^2=r^2\cos^2\theta+r^2\sin^2\theta=r^2$$ $$r=\sqrt{a^2+b^2}$$

Getting $\theta$ is somewhat easier: we can try to divide $b$ by $a$ to eliminate $r$:

$$\frac{b}{a}=\frac{r\sin\theta}{r\cos\theta}=\tan{\theta}$$ $$\theta=\arctan{\frac{b}{a}}.$$

So, $$\text{Rectangular }\rightarrow\text{ Polar}$$ $$\boxed{\begin{align} \sqrt{a^2+b^2} &= r\\ \arctan{\frac{b}{a}} &= \theta \end{align}}$$

(Video) Converting Complex Numbers from Rectangular to Polar Form

by Centennial Math Drop-In

In this video by Centennial Math Drop-In, you will get a quick example of how to convert a complex number from rectangular to polar form.

Convert Complex Numbers Between Polar and Rectangular Form

A complex number can be written in two different ways: rectangular form and polar form.

In rectangular form, $a$ is the real part and $b$ is the imaginary part: $$z = a + bi$$

In polar form, $r$ is the modulus (distance from the origin) and $\theta$ is the angle (direction from the positive $x$-axis: $$z=r(\cos\theta + i \sin\theta)$$

Converting from Rectangular to Polar Form

To find the modulus: $$r= \sqrt{a^2 + b^2}$$

To find the angle (argument): $$\theta = \tan ^{-1} \frac{b}{a}$$

You'll then plug these values into polar form.

Converting from Polar to Rectangular Form

Use the following formulas: $$a = r \cos \theta$$

$$b = r \sin \theta$$

Then write: $z=a+bi$