If $|z|=\sqrt{a^2+b^2}$, then what is $z$?

Question

Perhaps I’m having some difficulty understanding the complex plane. Say you have a complex number $z=a+bi$, where $a$ is the real part and $b$ is the imaginary part. Why do you plot the real part on one axis and the imaginary part on the other axis in a complex plane? I understand that real goes on real and imaginary goes on imaginary, but I don’t understand why $a+bi=(a,bi)$. How does the complex number suddenly become an ordered pair?

I believe I need that question answered before I can understand what $z$ is. I know that $|z|$ is the distance from the origin in the ReIm plane of $P(a,bi)$, and a geometric argument creates the equation $|z|=\sqrt{a^2+b^2}$. However, seeing as I don’t really know what $z$ is at all, I don’t quite understand the relationship between $z$ and $|z|$. $|z|$ is the absolute value of $z$… which means $|z|=|a+bi|=\sqrt{a^2+b^2}$? How does that work?

Answer 1

It is a choice we've made to depict complex numbers in the manner we have. But there are reasons for the choice. Complex numbers are all linear combinations of $1$ and $i$ using two real parameters (the real and imaginary parts), meaning this is number system is "two-dimensional" and hence may be depicted as a 2D plane by specifying an origin and two directions to represent the real and imaginary axis. Since there is no reason the imaginary axis should be bent at an awkward and asymmetric angle, it makes the most sense to depict the imaginary axis as perpendicular to the real axis. And hence the choice to make the real axis the $x$-axis and the imaginary axis the $y$-axis.

Some consider expressions like $a+bi$ a bit informal. What actually is $i$ after all? In math we have formal constructions of most things. We can construct integers from naturals using ordered pairs (in order to mimic formal differences $m-n$), we can construct rationals from natural numbers using ordered pairs (to signify formal quotients $m/n$), and we can construct real numbers from so-called "Cauchy" sequences of rational numbers (e.g. 3, 3.1, 3.14, ... can represent pi). Of course in all of these representations there is a bit of redundancy (mathematicians take care of that by considering "equivalence classes.") Point is, the same is sometimes done for $\mathbb{C}$: it is sometimes defined as a collection of ordered pairs with $(a,b)(c,d)$ defined to be $(ac-bd,ad+bc)$, and this is done algebraically purely for the purpose of having a formal construction!

It is another decision we've made to define the absolute value $|z|$ of a complex number to be the distance between it and the origin as depicted in the complex plane. (This restricts to the usual interpretation on the real number line, since $|x|$ for real numbers $x$ is just the distance between $x$ and $0$.) If we write $z=a+bi$, then by the Pythagorean theorem we have $|z|^2=a^2+b^2$.

There are 2D number systems where it doesn't make sense to use the same method of depiction. For instance $\mathbb{Q}(\sqrt{2})=\{a+b\sqrt{2}:a,b\in\mathbb{Q}\}$ or the "dual ring" $\mathbb{R}[\varepsilon]/(\varepsilon^2)$, i.e. the number system with elements that look like $a+b\varepsilon$ with $a,b$ real and the property that $\varepsilon^2=0$ by definition. What makes the accepted complex plane depiction of complex numbers a good choice is that it satisfies the property $|zw|=|z||w|$ and allows us to write complex numbers using polar form $re^{i\theta}$ which is very useful. (In addition to the complex numbers, there is a number system of so-called "split complex" numbers $a+bj$ with $j^2=+1$, not to be confused with the engineers' decision to use $j$ for a square root of negative one, which is sensible to depict as a plane as well because of the connections to conic sections and hyperbolic geometry.)

Answer 2

As a set, $\mathbb{C} $ is just $\mathbb{R^2}$, only we name their elements differently. So for every $(a, b) \in \mathbb{R^2}$ there is some $a+bi \in \mathbb{C}$. Viewing both as vector spaces, the canonical base $(1,0),(0,1)$ is mapped to $1,i$ giving an isomorphism.

If you plot both planes, you'll see the points have a natural correspondence, meaning, as I've already mentioned, they're the "same" point — only with a different name.

Thus, if you understand $\mathbb{R^2} $ you should have no problem understanding how points are represented in $\mathbb{C} $. Of course, this does not mean they're the same, since the latter is also endowed with an inner product.

This said, $|z|$ is not exactly the absolute value of $z$ but its modulus, which turns out to be the same of $(a, b)$ in $\mathbb{R^2} $ — as you pointed, its distance to $0$.

Answer 3

For your first question, to see the geometric meaning of $z$, think about $z$ as a point in the normal plane. If we choose a coordinate for this plane, namely, two axis $Ox$ and $Oy$, then we get some coordinates for $z$, say (a,b). This is the way to see $z$ as an element of $\mathbb{R}^2$. Moreover, if you want to consider $z$ as a complex number, i.e., an element of $\mathbb{C}^1:=\mathbb{R}\oplus i\mathbb{R}$, we identify $z=(a,b)$ with $a+ib$, and call $a$ the real part and $b$ the imaginary part of $z$. (If you wish, you can identify $z=(a,b)$ with $ia+b$ and then reverse the names and other things, however we should follow the original one...)

For your second question, the fact that $|z|=\sqrt{a^2+b^2}$ now comes from the geometric meaning of $z$.

Answer 4

So say you have an imaginary number $i$.

$$i^0 = 1$$

$$i^1 = \sqrt -1$$

$$i^2 = i^1i^1 = \sqrt{-1} \sqrt{-1} = -1$$

$$i^3 = i^1i^2 = \sqrt{-1} (-1) = -\sqrt{-1}$$

$$i^4 = i^0 = 1$$

As you can see, when you raise an imaginary number to every fourth power, it value if quantity. This is because on the real number axis, if you have a number on it (example: $5$). If you raise $5^n$, it would tend to positive infinity the higher the value of $n$. If you have $-5$, as soon as you square it, the number rotates 180deg around the real axis and jumps all the way to positive $25$. Then when you cube $-5$, it does another 180deg turn and jumps all the way to negative $125$. And then to 625 and so on.

For the $\sqrt{-1}$, when we square it, it obviously becomes a negative number: $-1$. With all other negative numbers on the real axis, it does not happen that way. Beginning from $i^0$ being $1$, when we raise it to the power of $1$, it cannot belong on the real axis, but it cannot make a $180^{\circ}$ jump either. So it turns anti-clockwise $90^{\circ}$ and lands on another axis: the imaginary axis. Then when we square it, it rotates 90deg again and lands on $-1$. We cube it, it rotates another $90^{\circ}$ and lands on $-\sqrt{-1}$, and finally when we raise it to the power of four, it rotates another $90^{\circ}$ around the complex plane and finishes its lap, ready to start the cycle again on $1$.

Real numbers on the real axis jump $180^{\circ}$, however imaginary numbers on the imaginary axis jump $90^{\circ}$. You can think about it that way and then it is easier to understand how numbers can have forms and dimensions with certain properties to them. They can behave like moving objects outside of space and time. This is known as Platonian Number Theory, and how numbers are "abstract objects existing outside of space and time".

$\sqrt{-1}$ may not "behave" like a real number, especially algebraically at times, but it does behave quite well geometrically. Numbers used to be denoted on a one-dimensional line, and now imaginary numbers - they are two dimensional. Numbers are two dimensional. This is why we have a "second axis" in two dimensions for the imaginary numbers.

Answer 5

Q:$\;$ If $\left| z \right| = \sqrt {a^{\,2} + b^{\,2} } $ then what is $z$ ?

R: $\; z = \left| z \right|e^{\,i\,\alpha } = \sqrt {a^{\,2} + b^{\,2} } \;e^{\,i\,\alpha } = \left| z \right|\left( {\cos \alpha + i\sin \alpha } \right)\quad \left| {\; - \pi < \alpha \le \pi } \right. $
that is, any (one of the) complex numbers represented as lying on a circle, centered at the origin, and of radius $|z|$, and which also contains the point $a+ib$.

Answer 6

z is a complex number on the circle of radius $ \sqrt{a ^2+b^2}$ centered at the origin... Any such point $z=re^{i\theta}=a+b\mathbb i $ has $ r=\sqrt {a^2+b^2} $. $\theta $, the angle with the real axis, can be anything you want. ..