Complex numbers: a number with two coordinates

Optional course topic. Use square roots, the unit circle and angle-sum identities. Geometric concepts are explained here; a separate geometry course is unnecessary.

On the real line x²+1=0 has no solution. Complex numbers extend the reals: introduce i with i²=−1 and write numbers as a+bi. Both i and −i then satisfy that equation.

$i^2=-1,\qquad z=a+bi\quad(a,b\in\mathbb R).$

The real part is a; the imaginary part is the real coefficient b, not the whole term bi. Equality requires both parts to agree. Setting b=0 recovers a real number. Imaginary is a name, without judging the object’s reality.

Arithmetic with one new rule

Add coordinates; for products expand brackets and replace i² with −1:

$(a+bi)+(c+di)=(a+c)+(b+d)i,$

$(a+bi)(c+di)=(ac-bd)+(ad+bc)i.$

For example, (1+2i)+(3−i)=4+i and (1+2i)(3−i)=5+5i. Subtract coordinatewise too. Powers of i cycle: i,−1,−i,1, then i again. OpenStax

The conjugate of a+bi is a−bi. Their product a²+b² is real and positive for a nonzero number. To divide, multiply numerator and denominator by the denominator’s conjugate:

$\frac{1+2i}{1-i}=\frac{(1+2i)(1+i)}{(1-i)(1+i)}=\frac{-1+3i}{2}.$

Division by 0 remains forbidden; for c+di≠0 the denominator c²+d² is positive. Check: ((−1+3i)/2)(1−i)=1+2i.

Plane, length and angle

Represent a+bi by point (a,b). The horizontal axis is real, the vertical imaginary. The modulus is distance from origin: its square equals the sum of the squares of the perpendicular coordinate segments.

For nonzero z its direction from the positive real axis gives an argument φ in radians. Angles φ+2πk for integer k give the same direction. Zero has modulus 0 but no defined argument.

$|z|=\sqrt{a^2+b^2},\qquad z=r(\cos\varphi+i\sin\varphi),\quad r=|z|>0.$

This is trigonometric or polar form. Since a=r cos φ and b=r sin φ, its coefficients recover the original number. Modulus and polar form.

Multiplication changes scale and direction

For two nonzero numbers, sine and cosine addition identities give:

$z_1z_2=r_1r_2[\cos(\varphi_1+\varphi_2)+i\sin(\varphi_1+\varphi_2)].$

Moduli multiply; arguments add up to full turns. Thus multiplying by i, whose modulus is 1 and argument π/2, turns a point a quarter turn counterclockwise without changing distance:

$i(1+2i)=-2+i,\qquad |1+2i|=|-2+i|=\sqrt5.$

Multiplication by i

z=1+2i → iz=−2+i

Solid arrow to (1,2); dashed to (−2,1) after multiplying by i. Both moduli √5. Equal axis scales: one horizontal unit equals one vertical unit; angle π/2 radians.

Multiplying by 2i would give the same turn and double the distance. Multiplying by zero collapses everything to the origin, where direction is undefined. Complex arithmetic connects coordinate calculations with lengths and angles; the real line remains inside this plane.

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Powers and all complex roots

Repeated multiplication gives De Moivre’s formula for positive integer n:

$z^n=r^n[\cos(n\varphi)+i\sin(n\varphi)]\quad(n=1,2,3,\ldots).$

Conversely, wⁿ=z for nonzero z has n distinct complex roots:

$w_k=\sqrt[n]{r}\left[\cos\left(\frac{\varphi+2\pi k}{n}\right)+i\sin\left(\frac{\varphi+2\pi k}{n}\right)\right],\quad k=0,\ldots,n-1.$

Here n≥1 is integer, r>0, and the root of r is positive real. Full turns before dividing the angle generate different directions. For z=1 and n=3 these are 1 and −1/2±i√3/2; each cubes to 1. For z=0 only w=0 remains. The unique real cube root and all complex roots are different conventions. Powers and roots, OpenStax §10.5.