In this lecture the following are introduced:

• Real and Imaginary numbers
• Cartesian form for complex numbers
• Basic operations with complex numbers
• Complex conjugates
• Absolute value or modulus
• Polar form for complex numbers
• Exponential form for complex numbers

Real numbers

The real numbers include integers (+1, -2), fractions (+½, -¾) and irrationals (like √2). They can be ordered along a real number axis as shown.

When a number on the real number axis is multiplied by -1 it ends up rotated anticlockwise by 1800, e.g.

Imaginary numbers

When a number is squared, the number is multiplied by itself. Now what number squared will produce -1?
Call the number that does this "j". Now j² = j×j = -1 = anticlockwise rotation of 180⁰.

As an example, to change +99 to -99 needs +99×j×j, i.e. 99 times two multiples of "j" gives 180⁰.
This must mean that one multiple of j is an anticlockwise rotation of 90⁰.

The imaginary numbers are defined as numbers multiplied by j, and they lie on an axis rotated at right angles to the real numbers. The letter "j" is used here instead of the usual mathematics "i" so it will not be confused with the real electric current "i".

Cartesian form for complex numbers

Complex numbers lie in the plane formed by the real and imaginary axes and are written in Cartesian form as, z = x = j·y.

Basic Operations with complex numbers

Take two complex numbers:

Addition:

Multiplication:

Division:

The commutative, associative and distributive laws hold for addition and multiplication.

Complex Conjugates

The complex conjugate is a reflection about the real axis.

Absolute value or Modulus

The absolute value (or modulus) of a complex number is the straight-line length from the origin to the number's co-ordinate point in the complex plane.

Polar Form for complex numbers

In polar form, z = (r,θ) = r·cosθ + j·r·sinθ. Where r is the absolute value or modulus and θ is the argument i.e. the angle measured anticlockwise up from the positive real axis. A complex number in Cartesian form can be written in Polar form as:

Multiplying two complex numbers in Polar form:

The moduli are multiplied and the arguments are added.

Multiplying by j:

Multplying by j causes the expected 90⁰ anticlockwise rotation.

Dividing one complex number by another in Polar form:

One modulus is divided by the other and the arguments are subtracted.

Forming the reciprocal of complex number in Polar form (using the result above):

The modulus is reciprocal and the complex conjugate is formed with the argument.

Exponential Form for complex numbers

The Polar form may now be written as:

Multiplying and dividing:

This enables the nth roots of a complex number to be calculated and is a form of De Moivre's theorem.

Note that multiplying a complex number by a·e^jφ multiplies its modulus by a and rotates it by φ.

Dividing a complex number by a·e^jφ shrinks its modulus by 1/a and rotates it by -φ.

In the following π = the Greek letter pi and from the cosine and sine values:
e^j·(π/2) = j
e^j·π = -j
e^j·(2π) = 1, hence for n an integer e^{(z + j·2πn)} = e^z

For the complex conjugate

Summarising:

Imaginary numbers are rotated 90⁰ from the real numbers

Complex numbers are written as:

The commutative, associative and distributive laws hold for addition and multiplication.

Complex Conjugate:

Modulus:

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