Complex numbers for signals

Drag the point straight left or right and watch only Across change; drag it straight up or down and watch only Up change.

A complex number is a point

Back in 3.2 you described a sinusoid with two numbers: an amplitude and a phase, a length and an angle. It turns out those two numbers pin down a single point in a flat plane, the same way “3 blocks east, 2 blocks north” pins down one spot on a map. I want to give that point its own name and its own arithmetic, because once you can add and multiply points instead of just plotting them, a spinning sinusoid becomes something you can compute with directly. That’s where we’re headed in 3.4; this page just builds the tool.

Take any ordinary $(x, y)$ point and relabel it. Call the sideways position, measured along a horizontal axis, the real part. Call the up-down position, measured along a vertical axis, the imaginary part, and mark that axis with a unit called $j$ (engineers’ name for “one step up”; you may have seen $i$ used the same way elsewhere). A point that sits $a$ across and $b$ up is written

and called a complex number. Nothing about it is mysterious or “imaginary” in the everyday sense: it’s an ordinary point, just written as a sum instead of a pair of coordinates.

Go back to the point at the top of the page. It behaves exactly like the $(x, y)$ point you already know, with new names on the two axes.

Length and angle: polar form

The same point can be described a second way: instead of “how far across, how far up,” you can give “how far from the centre, and in which direction.” That’s exactly the length-and-angle picture from 3.2’s amplitude and phase, applied to this point instead of to a wave.

The distance from the origin is called the magnitude, written $\lvert z\rvert$. Across $a$ and up $b$ form the two short sides of a right triangle whose long side is that distance, so Pythagoras gives it directly:

The direction is called the argument, written $\arg z$: the angle the line from the origin to the point makes with the positive real axis, measured in the radians from 3.2. Writing a complex number this way, as a length and an angle, is its polar form.

Drag the point around a circle centred on the origin and watch Length hold still while Angle sweeps. Then drag it straight outward from the origin and watch Angle hold still while Length grows.

Notice the two readouts, rectangular ($a+jb$) and polar (length, angle), always describe the exact same point.

Adding two complex numbers: nose to tail

Add two complex numbers by adding their real parts and their imaginary parts separately:

Picture each number as an arrow from the origin to its point. Adding them is the same as combining two displacement instructions, “3 blocks east, 2 north” then “1 block east, 4 south,” into one net displacement: you walk the first arrow, then walk the second arrow starting from where the first one left off, nose to tail, and the sum is the single arrow from the start straight to where you end up.

Move either point and watch the sum arrow slide by exactly the same amount you moved it.

Notice the sum is nothing more than the two arrows placed end to end.

Multiplying two complex numbers: rotate and scale

Multiplying is less obvious from the numbers alone, but it has a clean geometric meaning: it multiplies the two magnitudes together, and it adds the two angles together.

So multiplying a point by $z$ rotates it by $z$‘s own angle and stretches it by $z$‘s own length. A clean special case: $j$ itself sits at length 1 and angle $90°$, so multiplying anything by $j$ spins it a quarter turn with no change in size at all. Recall $z=3+4j$ from the last two instruments: its length is $\sqrt{3^2+4^2}=5$ and its angle is about $53.13°$ (the angle whose rise over run is $4/3$; a calculator’s $\arctan$ button finds it) . Multiplying it by $j$ gives $j(3+4j) = 3j+4j^2 = -4+3j$ (using $j^2=-1$, which the next section explains), a point still of length $5$ but at angle $53.13°+90°=143.13°$, exactly a quarter turn further round.

Move either point and watch the readouts.

The product’s length is always the two lengths multiplied together, and its angle is always the two angles added together. Notice that moving Point 2 onto the unit circle, so Lengths multiply reads ”… × 1.00 = …”, only ever rotates the product; it never stretches it.

Why $j \times j = -1$

You’ve now seen $j$ used twice: as the label on the vertical axis, and as a number you can multiply by. Both uses agree, because $j$ is defined as the point at length 1, angle $90°$, and multiplying by it is exactly the rotate-and-scale rule above: rotate by $90°$, don’t stretch at all. Multiply by $j$ twice in a row and you rotate by $90°$ twice, which is $180°$ total, a half-turn, landing you on the point exactly opposite where you started, at the same distance from the origin. Starting from the point $1$ (length 1, angle $0°$), two quarter-turns land you on $-1$. That’s the whole content of the famous rule

not a new kind of arithmetic, just two quarter-turns making a half-turn, flipping a point to the opposite side of the origin.

The mirror image

Flip the sign of a complex number’s imaginary part only, leaving the real part alone, and you get its conjugate, written $\bar z$:

Geometrically, that’s a mirror image of the point across the real axis: same distance across, same distance from the axis, just on the opposite side of it. Add a number to its own conjugate and the two imaginary parts are exact opposites, so they cancel completely, leaving only twice the real part:

always a point sitting on the real axis. For $z=3+4j$, $\bar z = 3-4j$ and $z+\bar z = 6 = 2(3)$ .

Drag the point anywhere and watch its mirror image follow.

The mirror image always sits the same distance below the real axis as the point sits above it (or vice versa). Notice Point + mirror always lands on the real axis, at exactly twice Across, no matter where you drag the point.

Equal steps around the circle

One last picture, because it comes back in Chapter 13, when we break signals into frequencies. Ask: which points, raised to the $N$-th power, land exactly back on $1$? Raising a point to a power means multiplying it by itself over and over, and by the rotate-and-scale rule each multiplication adds another copy of the point’s own angle. So a point at angle $\theta$ lands back at angle $0°$ (up to a full lap) after $N$ multiplications only if $N\theta$ is a whole number of full $360°$ turns. That gives $N$ possible angles, evenly spaced:

all sitting on the unit circle (length 1), called the $N$-th roots of unity. For $N=4$ they are $1,\ j,\ -1,\ -j$, at angles $0°, 90°, 180°, 270°$; you already checked $j^2=-1$ above, and $j^4=(j^2)^2=(-1)^2=1$ confirms $j$ is one of them.

Drag the N slider and watch the points redistribute themselves evenly around the circle.

Notice the readout for Angle between neighbours always reads exactly 360° divided by N, and Next point after 1, raised to the Nth power always lands back on $1$, whatever N you pick.

Worked example

Take $z = 3+4j$, the point used as the default above.

1. Rectangular to polar. $\lvert z\rvert = \sqrt{3^2+4^2} = \sqrt{25} = 5$ (a 3-4-5 triangle), and $\arg z = \arctan(4/3) = 53.13°$ (0.9273 rad).
2. Multiplying by $j$. $j(3+4j) = 3j+4j^2 = -4+3j$. Check the length is unchanged: $\lvert -4+3j\rvert = \sqrt{16+9} = \sqrt{25} = 5$; the new angle is $53.13°+90° = 143.13°$, exactly a quarter turn further round.
3. Conjugate sum. $\bar z = 3-4j$, so $z+\bar z = 6 = 2(3)$, real, exactly on the real axis.
4. 4th roots of unity. $1,\ j,\ -1,\ -j$, at angles $0°, 90°, 180°, 270°$; check $j^4 = (j^2)^2 = (-1)^2 = 1$.

Where you’ll meet this

3.4 puts the rotate-only case front and centre: a point of length 1 whose angle grows steadily with time is a phasor, and it traces out exactly the spinning-point picture that built the cosine and sine back in 3.2. Chapter 27 uses the real-plus-imaginary pair again for the two-part signals inside every radio, and Chapter 13 measures a signal’s frequencies at points that sit, quite literally, at the roots of unity you just met.

Reference card