← Complex numbers
1 Reading the notation
Complex-number notation is compact and each part carries meaning. Here is every symbol used on this page.📖 Symbols used with complex numbers
8 termsCover the right column and check you can say what each one means.
- i
- The imaginary unit, defined by i² = −1. It is not an approximation. It is a definition.
- z
- The conventional letter for a complex number: z = a + bi, where a is the real part and b the imaginary part.
- z̄ (z with a bar: the conjugate)
- The conjugate: flip the sign of the imaginary part. If z = a + bi then z̄ = a − bi. z × z̄ = a² + b², which is REAL. That is why you multiply by the conjugate to divide.
- |z| (modulus)
- The distance from the origin on an Argand diagram: |z| = √(a² + b²). Always positive. The same bars mean absolute value for a real number. It is the same idea, distance from zero.
- arg(z) (argument)
- The angle from the positive real axis to the point, measured anticlockwise. A calculator’s arctan only returns −90° to 90°, so always sketch the point to get the right quadrant.
- cis θ
- Shorthand for cos θ + i sin θ. So z = r cis θ is the polar form, with r = |z|.
- r
- The modulus when written in polar form: the same thing as |z|.
- Argand diagram
- A graph with the real part on the x-axis and the imaginary part on the y-axis. Complex numbers become points.