De Moivre's theorem — common mistake

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

A frequent error is expanding high powers by brute force. Keep it right: (cis theta)^n = cis(n theta).

Memory trick: (cis theta)^n = cis(n theta).

Roots of unity

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

The n nth-roots of unity sit equally spaced on the unit circle and sum to zero.

Memory trick: n-th roots of unity form a regular polygon.

Roots of unity — common mistake

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

A frequent error is forgetting the roots sum to zero. Keep it right: N-th roots of unity form a regular polygon.

Memory trick: n-th roots of unity form a regular polygon.

Conjugate and its uses

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

The conjugate reflects across the real axis; z times its conjugate is the modulus squared.

Memory trick: z * conj(z) = |z|^2, a real number.

Conjugate and its uses — common mistake

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

A frequent error is thinking the conjugate changes the modulus. Keep it right: Z * conj(z) = |z|^2, a real number.

Memory trick: z * conj(z) = |z|^2, a real number.

Geometry of complex numbers

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

Adding, subtracting and rotating complex numbers correspond to vector and rotation operations.

Memory trick: multiplying by i rotates 90 degrees.

Geometry of complex numbers — common mistake

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

A frequent error is ignoring the rotational meaning of multiplying by i. Keep it right: Multiplying by i rotates 90 degrees.

Memory trick: multiplying by i rotates 90 degrees.

Euler's form

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

E^(i theta) = cos theta + i sin theta unifies exponentials with trigonometry.

Memory trick: Euler links e, i and trig in one line.

Euler's form — common mistake

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

A frequent error is separating exponential and trig identities. Keep it right: Euler links e, i and trig in one line.

Memory trick: Euler links e, i and trig in one line.

|z| = sqrt(a^2 + b^2)

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

Modulus of a complex number. Use it when z = a + bi.

|z| = sqrt(a^2 + b^2)

(cos t + i sin t)^n = cos nt + i sin nt

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

De Moivre's theorem. Use it when integer n.

(cos t + i sin t)^n = cos nt + i sin nt

z * conj(z) = |z|^2

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

Product with conjugate. Use it when any complex z.

z * conj(z) = |z|^2

n-th roots of unity: e^(2 pi i k/n)

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

The n roots of z^n = 1. Use it when k = 0..n-1.

n-th roots of unity: e^(2 pi i k/n)

e^(i theta) = cos theta + i sin theta

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

Euler's formula. Use it when polar/exponential form.

e^(i theta) = cos theta + i sin theta

|z1 z2| = |z1| |z2|

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

Modulus is multiplicative. Use it when any complex numbers.

|z1 z2| = |z1| |z2|

Cartesian form vs polar form

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

Cartesian form a + bi is best for adding and subtracting; polar form r*cis(theta) is best for multiplying, dividing and taking powers.

modulus vs argument

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

The modulus is the distance from the origin (size); the argument is the angle from the positive real axis (direction).

z vs its conjugate

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

A complex number and its conjugate share the same modulus but have opposite-signed imaginary parts (arguments).

Watch out: Multiplying complex numbers adds their moduli

“'Imaginary' numbers run the very real electronics in your phone and power grid.”

It MULTIPLIES the moduli and ADDS the arguments.

Arithmetic progression

“Compound interest is a geometric progression quietly growing your money — or your loan.”

An AP has a constant common difference; its terms grow linearly.

Memory trick: AP adds a fixed d each step.

Arithmetic progression — common mistake

“Compound interest is a geometric progression quietly growing your money — or your loan.”

A frequent error is confusing the common difference with the common ratio. Keep it right: AP adds a fixed d each step.

Memory trick: AP adds a fixed d each step.

Geometric progression

“Compound interest is a geometric progression quietly growing your money — or your loan.”

A GP multiplies by a constant ratio; an infinite GP converges only if |r| < 1.

Memory trick: infinite GP sums only when |r|<1.

Geometric progression — common mistake

“Compound interest is a geometric progression quietly growing your money — or your loan.”

A frequent error is summing an infinite GP with |r| >= 1. Keep it right: Infinite GP sums only when |r|<1.

Memory trick: infinite GP sums only when |r|<1.

Arithmetic-geometric series

“Compound interest is a geometric progression quietly growing your money — or your loan.”

AGP terms mix arithmetic and geometric parts and need a special summation trick.

Memory trick: multiply an AGP by r and subtract to telescope.