3.9 Complex numbers

Write z=a+bi, where a,b are real and i²=−1. The conjugate is a−bi, modulus is |z|=√(a²+b²), and z z̄=|z|².

Collect real and imaginary parts separately, rationalise denominators with the conjugate, and use modulus as a distance from the origin.

(2+3i)(2−3i)=13, so the modulus of 2+3i is √13.

i is not a variable with arbitrary square root; i² is fixed as −1 and real/imaginary parts must match independently.

Add complex numbers componentwise, multiply by expanding and replace i² with −1, and divide by multiplying numerator and denominator by the denominator’s conjugate.

After simplifying, write the answer in a+bi form. Equality of complex numbers requires equality of both components.

(1+2i)/(3−i) =[(1+2i)(3+i)]/10=(1+7i)/10.

A complex fraction is not simplified by dividing real parts and imaginary parts separately.

A polynomial of degree n with complex coefficients has exactly n complex roots counted with multiplicity. Real roots are included among these; non-real roots occur in conjugate pairs for real-coefficient polynomials.

Use a known root or factor to reduce degree, then use coefficient relationships or conjugacy to find remaining roots. Check multiplicities rather than counting distinct values only.

If a real cubic has root 2 and root 1+i, it must also have 1−i; the three roots account for the degree.

“Three roots” can include a repeated root; distinct-root counting can undercount the degree.

z=a+bi is rectangular form. In polar form z=r(cosθ+i sinθ)=r cisθ, where r=|z| and θ is an argument measured from the positive real axis.

Use rectangular form for addition and polar form for multiplication, division and powers. Choose an argument branch consistently and identify the quadrant.

1+i=√2 cis(π/4); its square is 2 cis(π/2)=2i.

The argument is not determined by tan⁻¹(b/a) alone; the quadrant may require adding π.

Multiplying by r cisθ scales a modulus by r and adds θ to the argument. Dividing subtracts arguments and divides moduli; powers use De Moivre’s theorem.

Reduce angles modulo 2π, keep all roots when solving zⁿ=w, and convert back to rectangular coordinates only at the end if needed.

The cube roots of 8 are 2 cis(2kπ/3) for k=0,1,2, giving three equally spaced points on a circle.

Taking only the principal root misses the other n−1 roots; complex roots are distributed by equal angular steps.

Represent z=a+bi by the point (a,b). Addition is vector addition, modulus is distance from the origin and conjugation reflects the point in the real axis.

Use geometry to check algebraic results: conjugates have the same modulus, and z+w is the diagonal endpoint of the parallelogram formed by z and w.

The points 1+i and 1−i are reflections with modulus √2; their sum is the real point 2.

The Argand-plane coordinates are not a graph of y=f(x); the imaginary axis is a second coordinate, not an output scale.

|z−a|=r is a circle centred at a with radius r; arg(z−a)=θ is a ray from a. Inequalities give interiors, exteriors or angular regions.

Sketch the reference point a first, then apply the distance or angle condition. Intersections of loci must satisfy every condition simultaneously.

|z−(2+i)|=3 is a circle centred at (2,1); |z|<2 is its interior.

An argument condition describes a direction, not a distance, and arg is defined modulo 2π unless a principal range is stated.

[r(cosθ+i sinθ)]^n=r^n(cos nθ+i sin nθ). To solve z^n=w, take the nth root of the modulus and add arguments (arg w+2kπ)/n for k=0,…,n−1.

List all distinct roots, then convert to rectangular form only if needed. Their Argand points lie equally spaced on a circle.

z³=8 has roots 2 cis(2kπ/3), k=0,1,2.

Taking only the principal argument yields one root and misses the remaining equally spaced solutions.