Полярна форма запису комплексного числа

The most typical way to express a complex number would be to decompose it into its real and imaginary components, i.e. $z=a+bi$. This is called the rectangular form of a complex number, and it's easy to see why. We can imagine plotting all complex numbers on a plane parametrized by rectangular coordinates, with the real and imaginary components serving as the x- and y-coordinates, respectively.

However, if we visualize complex numbers as points on a plane, then switching from rectangular to polar coordinates gives us another way to think of complex numbers. We can express a complex number in terms of its distance from $0$, $r$, and the angle $\theta$ that the ray from $0$ to the number makes with the positive real axis. This angle is measured counterclockwise, as it is in polar coordinates.

The distance from a number to $0$ is call the absolute value 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: $$r = \sqrt{a^2+b^2}$$ $$\tan\theta = \frac{b}{a}$$

For convenience, we sometimes write $z = r\mathrm{cis}\theta$. This shorthand is used because "cis" has the first letter of cosine, the letter i, and the first letter of sine.

Clearly, adding numbers would be a disaster, unless we convert to rectangular form and then convert back, which negates the whole point of using a different form.

Polar form is incredibly useful for multiplication. Prepare yourselves for one of the most important and mind-blowing fact about complex numbers. Let's see what happens when we multiply two complex numbers, $r\mathrm{cis}\alpha$ and $s\mathrm{cis}\beta$:

$$(r\mathrm{cis}\alpha)(s\mathrm{cis}\beta) = [r(\cos\alpha + i\sin\alpha)](s(\cos\beta + i\sin\beta))$$ $$(r\mathrm{cis}\alpha)(s\mathrm{cis}\beta) = rs(\cos\alpha + i\sin\alpha)(\cos\beta + i\sin\beta)$$ $$(r\mathrm{cis}\alpha)(s\mathrm{cis}\beta) = rs[(\cos\alpha\cos\beta - \sin\alpha\sin\beta) + i(\sin\alpha\cos\beta + \cos\alpha\sin\beta)]$$

The two terms in parentheses should look very familiar.

These above results are just the angle addition identities.

Now, we conclude with our amazing result: $$(r\mathrm{cis}\alpha)(s\mathrm{cis}\beta) = rs[\cos(\alpha + \beta) + i\sin(\alpha + \beta)]$$ $$(r\mathrm{cis}\alpha)(s\mathrm{cis}\beta) = rs\mathrm{cis}(\alpha + \beta)$$

This is absolutely incredible. When we multiply two complex numbers, we multiply their magnitudes and add their arguments. This fact is not at all obvious, but it is very useful. As you will see in the future, this will often make multiplying using polar form much easier than multiplying using rectangular form. It will also allow for us to make many interesting and useful connections between complex numbers and seemingly unrelated areas of mathematics.

Graphical Explanation

As a refresher, a complex number is a number that can be written in the form $a + bi$ where $a$ and $b$ are real numbers and $i = \sqrt{-1}$.

With that out of the way, allow me to introduce to you the complex plane. Here is a picture:

This graph looks almost identical to the normal coordinate plane! The only difference is that instead of the $x$-axis we have the real axis and instead of the $y$-axis we have the imaginary axis. So, the real axis is the horizontal axis, and the imaginary axis is the vertical axis.

With this information, try doing the following question using intuition:

How coincidental! The point $(3,4)$ on the coordinate plane corresponds to the point $3+4i$ on the complex plane. In fact, all complex numbers $a+bi$ can be thought of the point $(a,b)$ when graphed on the complex plane.

If that makes enough sense to you, you can move on. Otherwise read through the rest of this section.

The real and imaginary parts of a complex number is exactly what it sounds like. For example, in the complex number $3+4i, 3$ would be the real part and $4i$ would be the imaginary part. For any complex number, $a+bi$ has real part $a$ and imaginary part $bi$.

The real axis replaces the x-axis of the coordinate plane. So the real part of a coordinate on the complex plane should correspond to the x-coordinate of the same point on the coordinate plane.

Similarly, the imaginary axis replaces the y-axis of the coordinate plane so the imaginary part of a coordinate on the complex plane should correspond to the y-coordinate of the same point on the coordinate plane.

If this still didn't make sense, staring at the graph for long enough may help you.

Calculating the magnitude of a complex number is also very similar to calculating the magnitude of a point on the coordinate plane. In particular, the complex number $a+bi$ corresponds to $(a,b)$ which has magnitude $\sqrt{a^2+b^2}$.

The magnitude is

$$\sqrt{3^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5$$.

We finally have all the tools we need to find the polar form of complex numbers! What we are trying to show is that every complex number $a+bi$ can be expressed as $r(\cos\theta + i\sin\theta)$. Take a moment to think about why this might be true. Hint: think about circles.

Let's try to express $3+4i$ in this form. First, let's draw a circle centered at the origin through $3+4i$ on the complex plane!

Notice that there is only one circle centered at the origin that goes through $3+4i$! This is because this circle must have radius $5$ as we know that the distance from the origin to $3+4i$ is $5$. Now, we somehow have to fiddle in some $\cos\theta$ and $\sin\theta$ terms in here. The best way to do this is by drawing right triangles! So let's draw a right triangle involving the origin and $3+4i$.

In this picture, $R = 5$ is the magnitude of $3+4i$ and $a$ is the angle that the line connecting $3+4i$ makes with the $x$-axis.

Our desired point $B$ is directly above the point $C$. Therefore, $B$ and $C$ must have the same real part. But since $C$ lies on the real-axis it is a real number. Therefore, the real part of $B$ is simply equal to the length

$$AC = R \cos a$$.

Now, we can think of the imaginary part of $B$ as the i times the "distance above" the real axis. For example the imaginary part of $3+4i$ is "$4$" above the real axis. But $B$ is directly above $C$ on the real axis. Thus, the distance $B$ is above the real axis is equal to the side length $BC$. Thus, the imaginary part of $B$ is simply equal to

$$BC = R \sin a$$.

Therefore, $3 + 4i = R( \cos a + i\sin a) = 5( \cos a + i \sin a)$ where $a$ is the angle $3+4i$ makes with the real-axis.

We can even solve for $a$ if we wanted to. The real part of $3+4i$ and the real part of $5(\cos a + i \sin a)$ must be the same. Thus

$$5 \cos a = 3$$ $$\cos a = \frac{3}{5}$$

$$a = \cos^{-1} (\frac{3}{5}) \approx 0.93 \hspace{2pt} \text{raidans}$$.

Now you try!

First we have to figure out $R$. This is the magnitude, which is

$$\sqrt{5^2 + (5\sqrt{3})^2} = \sqrt{25 + 75} = \sqrt{100} = 10$$.

Next, we know that $10 ( \cos \theta + i\sin \theta) = 5 + 5i\sqrt{3}$. Since the real parts and imaginary parts on both sides must match we know that

$$10 \cos \theta = 5$$ $$\cos \theta = \frac{1}{2}$$

$$\theta = \cos^{-1} ( \frac{1}{2}) = \frac{\pi}{3}$$.

This gives us our answer. To be safe, we can check that the imaginary parts match up.

$$10 \cdot i \sin \frac{\pi}{3} = 10 \cdot i \cdot \sqrt{3}{2} = 5i\sqrt{3}$$

as desired.

Algebraic Explanation

We want to express $a + bi = R(\cos \theta + i\sin \theta)$. This means the magnitude of the left hand side and right hand side must match. Therefore

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

Recall that $\cos^2 \theta + \sin^2 \theta = 1$. Therefore, $R = \sqrt{a^2+b^2}$.

Now, we use the fact that the real part and imaginary part's must match. In particular

$$R \cos \theta = a$$ $$\cos \theta = \frac{a}{\sqrt{a^2+b^2}}$$

and

$$R \sin \theta = b$$

$$\sin \theta = \frac{b}{\sqrt{a^2+b^2}}$$.

Notice that $|a| \le |\sqrt{a^2+b^2}|$ as if we square both sides

$$b^2 \ge 0$$ $$a^2 + b^2 \ge a^2$$ $$\sqrt{a^2+b^2} \ge a$$

where the last step is valid because both sides are necessarily positive. Think about why it is important that $|a| \le \sqrt{a^2+b^2}$.

If $|a| > \sqrt{a^2+b^2}$ then $|\cos \theta| > 1$ which is impossible! Also, under these equations we have

$$\cos^2 \theta + \sin^2 \theta = \frac{a^2}{a^2+b^2} + \frac{b^2}{a^2+b^2} = \frac{a^2+b^2}{a^2+b^2} = 1$$.

All these together imply that these will produce valid solutions for $\theta$.

Let's walk through a solution. We know that

$$R = \sqrt{5^2 + (5\sqrt{3})^2} = \sqrt{25+75} = \sqrt{100} = 10$$.

Furthermore we know that

$$\cos \theta = \frac{5}{10} = \frac{1}{2}$$ $$\theta = \cos^{-1} (\frac{1}{2}) = \frac{\pi}{3}$$

which gives us the conclusion.

How to Express a Complex Number in Polar Form

by David Wright

In this video by David Wright, you will learn how to write a complex number in polar form.

The Polar Form of a Complex Number

We've learned previously that a complex number has two parts: a real part and an imaginary part.

$$z = a + bi$$

, where $a$ is the real part and $b$ is the imaginary part.

But we can also write complex numbers using polar coordinates: the polar form of a complex number.

In polar form, a complex number is written as:

$$z = r(\cos \theta + i \sin \theta)$$

We can also write it more simply, using Euler's formula:

$$z = r \text{cis} \theta$$

In both equations, $r$ is the modulus (or length) of the complex number, and $\theta$ is the argument (or angle) it makes with the positive $x$-axis.

To find $r$, use the formula:

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

To find $\theta$, use the formula:

$$\theta = \tan ^{-1} (\frac{b}{a})$$

The polar form is especially helpful for multiplying and dividing complex numbers. In polar form, multiplying is straightforward: you multiply the $r$ values and add the angles. Dividing complex numbers means dividing the $r$ values and subtracting the angles. Once a complex number is written in polar form, you can apply De Moivre’s Theorem to compute powers efficiently.

The polar form provides an efficient way to find and graph roots of unity.