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M01.9 · Complex numbers

A free, illustrated module of 4 lessons and one interactive in M01 Math Foundations: The number i and the complex plane; Adding and multiplying: moving, turning and scaling; The polar form and Euler’s formula; Quadratics with a negative discriminant.

M01.9 L0 · Foundation 4 lessons · about 100 min Live · in review

Part of M01 Math Foundations in the Mathematics track.

No number on the number line squares to -1, so these four lessons make one, i, and find that the new numbers a + bi are the points of a plane. Adding them slides the plane, multiplying turns and stretches it, and that one picture gives the polar form, Euler’s formula and a proof of the angle-addition formulas for every angle. The module ends where the quadratic formula of M01.3.3 stopped: with a negative discriminant, a quadratic still has two roots, a mirror-image pair.

What you need first. The algebra of brackets (M01.1.2), the coordinate plane and the distance formula (M01.2), the quadratic formula (M01.3.3), the number e (M01.4.2), and the unit circle, polar coordinates and the angle-addition formulas (M01.5). The code uses loops, functions, tuples and importing a module, from P01.2 to P01.6.

Lessons

1. The number i and the complex plane M01.9.1 25 min

   No number on the number line squares to −1, so we make one, i, and keep every rule of algebra. A complex number a + bi is then a point of the plane, its modulus is its distance from 0, and Python and NumPy calculate with it directly.

2. Adding and multiplying: moving, turning and scaling M01.9.2 25 min

   Adding a complex number slides the whole plane, multiplying by a real number stretches it, and multiplying by i turns it a quarter turn. Together they show what every product does: it multiplies the lengths and adds the angles.

3. The polar form and Euler’s formula M01.9.3 25 min

   Name a complex number by its length and its angle and multiplying becomes easy: lengths multiply and angles add. For two points of the unit circle that proves the angle-addition formulas for every angle, and Euler’s formula gives each point of the circle a short name.

4. Quadratics with a negative discriminant M01.9.4 25 min

   When the discriminant is negative, the quadratic formula needs the square root of a negative number, and i supplies it: the roots come as a mirror-image pair p + qi and p − qi. The same mirror image, the conjugate, is how complex numbers are divided.

The interactive

M01.9-X1 · Turn-and-scale explorer. Drag two complex numbers z and u, or set their lengths and angles with sliders. Their sum is drawn tip to tail and their product is z turned by the angle of u and stretched by its length. Multiply again and again: with u on the unit circle the walk goes round a circle about 0, and with u inside or outside the unit circle it spirals in or out. It lives in Adding and multiplying: moving, turning and scaling.