The introduction to Complex Numbers

Explore complex numbers as the sum of a real part and an imaginary part, showing how real, purely imaginary, and fully complex forms appear on the number line.

Learn how i powers cycle every four, with i^n equaling i, -i, -1, or 1, using patterns 4x+1, 4x+2, 4x+3, and multiples of four.

Explore Cartesian, polar, and exponential representations of complex numbers, detailing the real and imaginary parts, and the polar magnitude r with theta.

Learn the cartesian form of complex numbers, identify real and imaginary parts in a plus bi notation, and convert expressions like 3-4i+i^2 into a+bi.

Explore the polar form of complex numbers, z = r(cos theta + i sin theta); relate r to the modulus and theta to the argument.

Demonstrate Euler's formula e^{i x} = cos x + i sin x by expanding e^{i x}, cos x, and sin x into power series and grouping even and odd terms.

Multiplying a complex number by a real constant scales its magnitude while preserving its argument, as shown with z = 3 + 5i and 2z = 6 + 10i.

Explore the conjugate of complex numbers, observe its reflection across the real axis, and apply key relations like z+z* = 2a and z z* = r^2.

Rationalize the denominator of complex fractions by multiplying by i for purely imaginary denominators. Use the conjugate for complex denominators to produce a real denominator.

Moivre's theorem states that (cos θ + i sin θ)^n equals cos(nθ) + i sin(nθ) for integer n and real θ in radians.

Learn to prove De Moivre's theorem by applying Euler's formula and exponential laws to (cos theta + i sin theta)^n, deriving cos(n theta) + i sin(n theta) for integer n.

Apply De Moivre's theorem to compute z^n from a complex number in polar form, showing equivalence with exponential laws by cubing z and converting back to polar form.

Explore solving for unknowns in equations with complex numbers, then learn to find nth roots in Cartesian, polar, and exponential forms. Finally, examine solving polynomial equations with complex roots.

Solve for real unknowns in complex-number equations by equating the real parts and the imaginary parts; deduce a = 3 and b = -4.

Find the nth roots of a complex number by converting to exponential form and applying z = r^{1/n} e^{i(θ+2πk)/n} for k = 0,...,n−1, with θ adjusted to (−π, π).

Find the nth roots of a complex number in polar form using De Moivre's theorem, converting to exponential form, and selecting roots with arguments in [-pi, pi], yielding n solutions.

Solve the nth root of a complex number in cartesian form by equating real and imaginary parts. For higher powers, convert to exponential form.

Explore how to solve polynomial equations with complex roots, noting that real-coefficient polynomials yield conjugate pairs and a real root when needed; apply the quadratic formula and factoring.

Explore addition and subtraction of complex numbers on the Argand diagram, using vector addition and the parallelogram rule to visualize z=2+i and w=1+2i.

Learn how multiplying a complex number by a scalar scales its magnitude along the same line from the origin in the Argand diagram, illustrated with z = 2+i.

Explore complex multiplication on the Argand diagram, doubling magnitudes, rotating by the product's argument, and combining magnitude and angle effects in z and w.

Explore loci as sets of points that satisfy a condition, and understand why this section is named loci, the plural of locus, in the context of complex numbers.

The locus of z is a circle centered at a+bi with radius r, representing all points at a fixed distance r from a+bi on the Argand diagram.

Identify the locus of z as a half-line starting at a+bi, forming an angle theta with the positive real axis, excluding a+bi, as shown by arg(z-3+i)=3pi/4 and arg(z-1-i)=-pi/3.