Absolute Value of A Real VS. Complex Number

Well, oh my, you've done it almost right! It is actually given as

$$|4+3i|=\sqrt{4^2+3^2}=\sqrt{25}=5$$

In general,

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

This is because the absolute value only considers $a$ and $b$ as the length of each side of the triangle. Length can't, by definition, be a complex number.

This is like considering the absolute value of a negative number:

$$|-1|=1$$

The absolute value doesn't care which side of $0$ you are at, just how far. So the same goes for complex numbers, and we end up having

$$|3i|=3$$

Now, $|3i|=|3|$ does not mean that $3i=3$, for the reason that it is simply on a different side of $0$. This kind of logic would say $1=-1$, since their absolute values are the same.

The definition (or one possible definition) of the absolute value of the complex number $a+bi$ (where $a$ and $b$ are real) is $\sqrt{a^2+b^2}$. So you ask, why is it not $\sqrt{a^2+(bi)^2}$ instead? The answer is that this is simply how we choose to define it. You could define a different quantity which is $\sqrt{a^2+(bi)^2}$, but you would have to give a different name to it because everyone else has already agreed that "absolute value" means to take $\sqrt{a^2+b^2}$ instead.

Now, a more interesting question is why everyone else decided on that definition. One reason is that you can represent complex numbers as points in the plane by letting $a+bi$ correspond to the point $(a,b)$, and then $|a+bi|$ is the distance from this point to the origin. Note that when you do this, $(a,b)$ is just a point in the ordinary Euclidean plane: it is a point that is $a$ units to the right of the origin and $b$ units above the origin. It doesn't make sense to say that the vertical distance is $bi$, since in geometry when we measure distances they are always positive real numbers. The vertical distance is $b$ because you have moved $b$ units vertically in the plane. (Actually, this is only accurate if $b$ is positive: if $b$ is negative, you have moved $-b$ units down, and so the distance is $-b$ rather than $b$. But you end up squaring this quantity when you use the Pythagorean theorem, so it doesn't matter if it's negative.)

Now ultimately this explanation is not very satisfying, because it doesn't explain why we chose to represent complex numbers in the plane this way. For instance, why don't we choose to represent them such that the complex number $i$ corresponds to a vertical distance different from $1$, or a distance in some direction other than vertical? One answer is that choosing $i$ to mean "go up one unit" happens to make distance have nice algebraic properties we would like absolute values to have. For instance, for real numbers, it is true that $|xy|=|x||y|$. Defining absolute values of complex numbers by $|a+bi|=\sqrt{a^2+b^2}$, it turns out that this is true for complex numbers as well. As a simple example, if we want $|xy|=|x||y|$ to be true for complex numbers, then we should have $|i|^2=|i^2|=|{-1}|=1$. So we should define $|i|=1$ or $-1$, and it is sensible to define it to be $1$ instead of $-1$ since absolute values are supposed to be positive.

You ask:

Wouldn't the distance from $0$ to $3i$ be $3i$, not $3$?

No, not quite, for the same reason that the distance from $3$ to $0$ is not $-3$. Distance is not simply the difference, but the absolute value of the difference.

Of course, you'll probably find this unhelpful and circular. There are a few different ways you can think about $|z|$:

• Length of the line from $0$ to $z$ (note that lengths are always positive, no matter if along the positive reals, negative reals, or any other direction of the complex plane)
• $\sqrt{z\bar z}$ (multiplying by the complex conjugate $\bar z$ takes any number to the positive reals - it's equivalent to the $\sqrt {x^2}$ formula from the real numbers)
• the radius $r$ when the number is expressed as $r(\cos\theta+i\sin\theta)$, also known as polar form
• the nonnegative real number you "land on" if you rotate the plane around the origin so your point ends up on the nonnegative part of the real axis

But all of these are restricted to being positive.

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If complex absolute value was defined as $\sqrt{a^2 + (bi)^2}$, then starting at $z=3$ and "moving" upwards along the positive imaginary direction would decrease the absolute value. If we want absolute value to measure "distance from $0$" in some sense, then moving farther away from $0$ would make $\sqrt{a^2 + (bi)^2}$ a smaller number! We want our absolute value to increase when we move away from $0$, not decrease.

We want our definition of "absolute value" to not change if we rotate the plane around the origin, and we want the absolute value of a nonnegative real number to equal itself. These make our only option for the definition of $|z|$ to be $\sqrt{a^2 + b^2}$.