Chapter 1. Complex numbers: introduction FREE
1.1 What is a complex number?
In Grade 11 you have already seen that some equations, such as \( x^2 = -1 \), have no solution in the real numbers, because no real number's square is negative. To solve this problem, we extend the number system with a new number that we call the imaginary unit and write with the letter \( i \).
We define \( i = \sqrt{-1} \), and from that follows the most important property: \( i^2 = -1 \). With this one rule we can now write the square root of any negative number, for example \( \sqrt{-9} = \sqrt{9} \times \sqrt{-1} = 3i \).
1.2 The form \( a+bi \)
A complex number is written in the standard form \( a + bi \), where \( a \) and \( b \) are real numbers. We call \( a \) the real part and \( b \) the imaginary part. In \( 3 + 4i \) the real part is therefore \( 3 \) and the imaginary part \( 4 \). If \( b = 0 \) we get an ordinary real number, and if \( a = 0 \) we call it a purely imaginary number.
1.3 Adding and subtracting
To add or subtract complex numbers, we work only with like parts: the real parts together and the imaginary parts together.
Example 1: Calculate \( (2 + 5i) + (4 - 3i) \).
- Add the real parts: \( 2 + 4 = 6 \).
- Add the imaginary parts: \( 5i + (-3i) = 2i \).
- The answer is \( 6 + 2i \).
1.4 Multiplication
We multiply complex numbers just as we expand binomials in algebra, and then we replace \( i^2 \) with \( -1 \).
Example 2: Calculate \( (3 + 2i)(1 + 4i) \).
- Expand: \( 3 \times 1 + 3 \times 4i + 2i \times 1 + 2i \times 4i \).
- This gives \( 3 + 12i + 2i + 8i^2 \).
- Replace \( i^2 = -1 \): \( 3 + 14i + 8(-1) = 3 + 14i - 8 \).
- Simplify: \( -5 + 14i \).
1.5 Powers of \( i \)
The powers of \( i \) repeat in a pattern of four: \( i^1 = i \), \( i^2 = -1 \), \( i^3 = i^2 \times i = -i \), and \( i^4 = i^2 \times i^2 = (-1)(-1) = 1 \). After that the pattern starts again from the beginning. To work out any high power of \( i \), you divide the exponent by \( 4 \) and use only the remainder.
Example 3: Calculate \( i^{10} \). Divide \( 10 \) by \( 4 \): the remainder is \( 2 \), so \( i^{10} = i^2 = -1 \).
Complex numbers are not just theoretical; they are used in Technical Mathematics to describe alternating current, waves and vibrations in electrical and mechanical systems.