复数：大小值

Magnitude of Complex Numbers

The magnitude of a complex number is its distance from the origin in the complex plane. The concept may seem a little bit foreign right now as you build up your knowledge of complex numbers, but the process is very similar to finding the distance between two points on a coordinate plane or a number line.

Recall the distance formula for points on the Cartesian plane: $$\text{Distance }=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$ As always, be careful with the square root!

If you're finding the distance from the origin in the Cartesian plane, this formula gets simplified: $$\text{Distance }=\sqrt{x^2+y^2}$$ As $x_1$ and $y_1$ would both be $0$.

When we move onto the complex plane, we use the same process. We're still only looking at the positive square root since distance is always positive. Let's look at the complex plane to refresh our minds.

The $x$-axis is replaced with the real axis, and the $y$-axis is replaced with the imaginary axis. We can use this information to alter our distance formula from the origin to be used with the complex plane.

Let's say we have a complex number in rectangular form: $z=a+ib.$ We sometimes denote the magnitude (or distance from the origin) of $z$ as $|z|.$ (Careful: this is the same signage we use to find the absolute value).

Similar to the distance formula, the formula for the magnitude of a complex number $z=a+ib$ is as follows: $$|z|=\sqrt{a^2+b^2}$$

Where $a$ is our real component and $b$ is our imaginary component. Let's look at an example of this.

This is a graph of the point $3+2i$ on the complex plane. It is $3$ units to the right, and $2$ units above the origin, similar to the point $(3,2)$ in the Cartesian plane. To break this down even further, this complex number has a real component of $3$ and an imaginary component of $2$. Let's plug this into our magnitude formula: $$\begin{align} |z| &= \sqrt{a^2+b^2}\\ |z| &=\sqrt{(3)^2+(2)^2}\\ |z| &= \sqrt{13}\end{align}$$

Here's a problem for practice:

First, let's see this point on a graph.

We can see that it is $4$ units to the left and $3$ units below the origin. Let's plug our numbers into the magnitude formula. $$\begin{align} \text{Magnitude }&= \sqrt{a^2+b^2}\\ \text{Magnitude }&=\sqrt{(-4)^2+(-3)^2}\\ \text{Magnitude }&=\sqrt{16+9}\\ \text{Magnitude }&=\sqrt{25}\\ \text{Magnitude }&=5 \end{align}$$

The big takeaway here is remembering the formula. $$|z|=\sqrt{a^2+b^2}$$

The Magnitude of a Complex Number

Earlier, we related complex numbers to vectors on the complex plane. We saw how adding complex numbers is essentially equivalent to adding vectors.

Now, vectors have magnitudes. The magnitude of a vector is the length of the vector; the vector $\binom ab$ has magnitude $\sqrt{a^2+b^2}$. We wish to extend this to complex numbers.

However, it's not trivial how to find the "length" of a complex number. We need a segment to find the length of. So we draw a diagram with $z=a+bi$ and consider the possible cases.

The only natural segment to draw is the one from $z$ to the origin, so we draw that.

Aha! Now we have something. We can use the Pythagorean Theorem to find that the length of the segment from $z$ to the origin is $\sqrt{a^2+b^2}$ (just like in vectors)! We call this the magnitude of $z$, and we denote it by $|z|$.

Magnitude in terms of $z$, $\overline z$

Let's look at the magnitude formula again: $$|z| = |a+bi| = \sqrt{a^2+b^2}$$ We can square both sides (since they're positive reals) to get: $$|z|^2 = a^2 + b^2$$

This is the same as the expression we got when we multiplied a complex number by its conjugate! Therefore, we have: $$|z|^2 = z\overline z = a^2 + b^2$$

That's all there is to this explanation about magnitudes. Thanks for reading!

(Video) The Magnitude of a Complex Number

by James Hilburn

In this video by James Hilburn, you can see an example of how to find the magnitude of a complex number by graphing.

Summary

The complex plane looks a lot like the cartesian plane with real numbers that we are more familiar with. The horizontal access is the $x$ axis (the real axis), and the vertical axis is the $y$ axis (the imaginary axis). We know this because in a complex number, $x+yi$, the $x$ term is the real part, and the $yi$ part is the imaginary part.

Say we want to graph the complex number, $1+3i$. We know the first term is the real part, and the second term is the imaginary part. On our complex plane, we will move $1$ unit to the right and $3$ units up.

Watch a couple more of the examples to fully understand!

Graphing a Complex Number and Finding the Magnitude

In our previous lessons, we learned the basics of complex numbers and how to multiply them. In this lesson, we’re going to dive into two important concepts: graphing complex numbers and finding their magnitude.

By the end of this lesson, you'll be able to plot complex numbers on a graph and calculate their magnitudes!

You may remember that a complex number has two parts: a real part (like $2$ or $0.5$) and an imaginary part (involving the imaginary unit $i$, where $i=\sqrt{-1}$).

In general, a complex number looks like: $$z=a+bi$$

, where $a$ is the real part, $b$ is the imaginary part (with $i$), and $i$ is the square root of $-1$.

For example, $3+4i$ is a complex number, where the real part is $3$ and the imaginary part $4i$.

#####Graphing Complex Numbers

To better understand complex numbers, we can graph them on a special kind of coordinate plane called the complex plane.

• The horizontal axis represents the real part of the complex number.
• The vertical axis represents the imaginary part of the complex number.

When you graph a complex number like $3+4i$, you're plotting a point on this plane.

• The real part is $3$, so move $3$ units to the right along the real (horizontal) axis.
• The imaginary part is $4i$, so move $4$ units up along the imaginary (vertical) axis.

The point where these two lines meet is the location of the complex number on the complex plane.

Magnitude of a Complex Number

The magnitude of a complex number tells us how far the number is from the origin on the complex plane. We can think of it as the distance from the origin to the point representing the complex number.

The find the magnitude of a complex number, we can use the Pythagorean Theorem.

$$|z| = \sqrt{a^2+b^2}$$

In our previous example, to calculate the magnitude of $3+4i$:

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

So, the magnitude of $3+4i$ is $5$.

Conclusion

In this lesson, you learned:

• A complex number is written as $a+bi$, where $a$ is the real part and $b$ is the imaginary part.
• The complex plane helps us graph complex numbers, with the real part on the horizontal axis and the imaginary part on the vertical axis.
• The magnitude of a complex number is the distance from the origin and is calculated using the formula $|z| = \sqrt{a^2+b^2}$

In the next lessons, you will learn about the complex conjugate and dividing complex numbers.