Adding and multiplying: just collect like terms
In the first guide you met the imaginary unit i, the single new symbol that satisfies i^2 = -1, and the resulting complex number z = x + i y. Here we do the everyday thing: we calculate with them. The good news is that you already know how. Treat x + i y exactly like the algebraic expression "x plus i times y", obey the usual rules, and only at the very end apply the one new fact, i^2 = -1, to tidy up.
Addition is the easy half: you add the real parts and the imaginary parts separately, the same way you'd combine "3 apples + 2 oranges" with "1 apple + 5 oranges". So (3 + 2i) + (1 + 5i) = 4 + 7i. Multiplication is FOIL — multiply everything out — and then you cash in i^2 = -1. For (3 + 2i)(1 + 5i) you get 3 + 15i + 2i + 10 i^2, and since 10 i^2 = -10, that collapses to -7 + 17i. This little dance is the whole of complex arithmetic.
(a + i b) + (c + i d) = (a + c) + i (b + d) (a + i b) (c + i d) = (a c - b d) + i (a d + b c)
The conjugate: flip the sign of i
Given z = x + i y, its conjugate z-bar is just x - i y: you keep the real part and flip the sign of the imaginary part. That is the entire definition of the complex conjugate. Geometrically — and we will draw this properly in the next guide — z-bar is the mirror image of z across the real axis. Conjugating twice brings you home: the conjugate of z-bar is z again.
Why bother? Because the conjugate is the tool that converts the imaginary into the purely real. Add z and z-bar and the imaginary parts cancel: z + z-bar = 2x, twice the real part. Subtract them and the real parts cancel: z - z-bar = 2 i y. Best of all, MULTIPLY z by z-bar and watch the magic — the cross terms vanish and you are left with a non-negative real number.
The modulus: an honest size for z
The product z times z-bar is exactly x^2 + y^2 — a real number, never negative. Its square root is the modulus, written |z| = sqrt(x^2 + y^2). The modulus is the complex number's distance from the origin, the honest two-dimensional Pythagoras of the pair (x, y). For 3 + 4i it is sqrt(9 + 16) = 5. So we can package the previous section as the single clean identity z times z-bar equals |z|^2.
The modulus is multiplicative — |z w| = |z| |w| — which is the algebraic shadow of a beautiful geometric fact you will meet in guide 4: multiplying complex numbers multiplies their lengths (and adds their angles). For addition, lengths only obey an inequality, the triangle inequality |z + w| <= |z| + |w|: the direct hop from 0 to z + w can never be longer than going out to z and then on to w, exactly as walking two sides of a triangle is never shorter than the third.
Division: multiply by the conjugate
Now everything pays off. How do you divide one complex number by another, say (3 + 2i) / (1 + 2i)? The denominator has an imaginary part, which is awkward — we'd like a plain real number on the bottom. The trick is to multiply top and bottom by the conjugate of the denominator. Because (1 + 2i)(1 - 2i) = |1 + 2i|^2 = 5, the bottom turns real, and the whole quotient becomes an ordinary x + i y. This is the standard recipe for complex division.
- Write the quotient as a fraction: (3 + 2i) / (1 + 2i).
- Multiply top and bottom by the conjugate of the bottom, 1 - 2i.
- The bottom becomes (1 + 2i)(1 - 2i) = 1 + 4 = 5, a real number.
- The top: (3 + 2i)(1 - 2i) = 3 - 6i + 2i - 4 i^2 = 7 - 4i.
- Divide each part by 5: the answer is 7/5 - (4/5) i.
In particular, the reciprocal of any nonzero z is 1/z = z-bar / |z|^2 — the conjugate divided by the squared length. This is exactly why every complex number except 0 has an inverse, and it is what makes the complex numbers a genuine field, a number system where you can freely divide. The lone exception is the same as on the real line: you still cannot divide by zero, because |0| = 0 leaves nothing to divide by.
Where this is heading
Notice what we have NOT used: any picture. Everything above is pure algebra, and it works on autopilot. But algebra alone hides the meaning of these moves. Why is multiplying by the conjugate a reflection? Why does multiplication scale lengths? In the next guide we put z = x + i y on a plane and watch addition become vector addition and the conjugate become a mirror flip; the geometry will make all of today's identities feel inevitable rather than lucky.
Then comes the real reward. Writing z by its length and angle leads to the polar form, multiplication turns into rotation-and-scaling, and Euler's formula ties the whole story to trigonometry. The humble |z| you computed today reappears there as the radius, and the conjugate as the angle's sign-flip. Master the four operations now, and the geometry waiting ahead will land softly.