Maths

2 Complex Numbers

The roots of the quadratic equation ax2 + bx + c = 0 are x= −b ± b2 − 4ac 2a (2.1)

For example, with a = 1, b = −4, c = −13 we find that the roots of x2 − 4x − 13 = 0 are x=2± √ 17.

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If we try to solve

x2 + 2x + 2 = 0

in the same way (by inserting a = 1, b = 2 and c = 2 into equation (2.1)) then we arrive at the solutions

x = −1 ±

√

−1.

The number −1 is not defined if we are restricted to the usual real numbers that we use to represent physical quantities (lengths, masses, voltages etc.). In other words, the equation x2 + 2x + 2 = 0 has no solutions that are real numbers.

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√

This encourages us to extend the real numbers by introducing a new number i, called the unit imaginary number), which is defined by the condition i2 = −1. The roots of x2 + 2x + 2 = 0 are then x = −1 ± i. This is an example of a complex number. • Complex numbers do not directly describe quantities measured in the real world, but are used to simplify many complicated engineering and mathematical problems. • For example, you will seen in later maths modules that complex numbers are tremendously important for solving problems related to vibrating or oscillating systems. • Engineers often write j instead of i

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2.1 Algebra of complex numbers

An arbitrary complex number is written in the form z = x + iy, where x and y are real numbers.

x is the real part of z, denoted y is the imaginary part of z, denoted

Re(z) Im(z).

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2.1.1 Equality

Two complex numbers z1 = x1 + iy1 and z2 = x2 + iy2 are said to be equal if x1 = x2 and y1 = y2 ,

i.e. if real parts are equal and imaginary parts are equal.

2.1.2 Addition

The sum of two complex numbers is defined by z1 + z2 = (x1 + x2) + i(y1 + y2) = (sum of real parts) +i×(sum of imaginary parts) E.g. (3 + 4i) + (1 − 2i) = (3 + 1) + i(4 − 2) = 4 + 2i

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2.1.3 Subtraction

z1 − z2 = (x1 − x2) + i(y1 − y2)

2.1.4 Multiplication

Brackets can be expanded as in ordinary...