It may seem that this post is about very abstract mathematics. But the ideas explained here are very useful for explaining oscillations and waves (post 18.11), especially when combined with the concept of the number e (post 18.15).
Counting is like moving along the line in the picture above – from left to right. When we haven’t moved, we are at position 0. We can move to position 1. If we move the same distance again, we get to 2, then to 3, 4, 5 and so on. These are the positive whole numbers or positive integers. There are also numbers like 0.3, 1.27, 3.85. Together with the integers, these numbers may up the set of positive rational numbers. There are other numbers, like π (post 17.11) and e (post 18.15), whose positions we can’t fix exactly – but we can calculate them with the precision that we need – these are the irrational numbers. If we move to 4 twice, we arrive at 4 x 2 = 8 (4 multiplied by 2). Notice that 2 x 2 = 4, 3 x 3 = 9, 4 x 4 =16; we say that 4 = 16½ (post 18.2) is the square root of 16, and that 3 = 9½ is the square root of 9. Numbers like 4, 9, 16, 25, 36, 49…. have square roots that are integers. Others, like 2, 3, 5,6, 8… have squares roots that are not integers – they can be rational or irrational numbers.
If we move from 0 towards the left, we meet the negative numbers, like -1, -2, -3, -0.25, -3.5, -1/3… that can be rational or irrational. Moving -4 twice we get to 2 x (-4) = -8; we have multiplied -4 by 2. Multiplying -4 by -2 moves us in the opposite (positive) direction, so that (-2) x (-4) = 8. Multiplying a positive number by a negative number gives us a negative answer: multiplying a negative number by another negative number gives us a positive number.
The ideas explained in the previous paragraph mean that the number 1 has two square roots, 1 and -1. If you think about it, you will see that all positive numbers have two square roots – one positive: the other negative. But we can’t think of any number that, when multiplied by itself gives -1, or any other negative number, because -1 x -1 = 1; to get -1 we need to multiply -1 by +1 and the two numbers are not the same. So, we might conclude that negative numbers don’t have square roots. This is what I was taught before I went to university.
But this is a very unsatisfactory conclusion; it means that numbers behave completely differently depending on whether we move to the right or to the left from 0, in the picture above. As a result, the equation x2 = 1 has two solutions (1 and -1) and x2 = -1 has no solutions. To overcome this problem, we invent the number i that is defined by i2 = -1. Now -1 has two square roots; i and –i (because –i x –i =1). And all the other negative numbers have two square roots – the square roots of -16 are 4i and -4i. We call the numbers that have i as a factor (3i, -27i, 2.236i), imaginary numbers; numbers that don’t have i as a factor (5, 1.32, π) are called real numbers. Now all numbers have two square roots. Some people (mostly engineers) represent the square root of -1 by j instead of i; it doesn’t make any difference to the ideas expressed here.
Real numbers belong on the line at the beginning of this post: imaginary numbers don’t.
There are also numbers, like 2 + 3i that have a real part (2) and an imaginary part (3i); they are called complex numbers. Adding to the real part (by adding real numbers) doesn’t affect the value of the imaginary part and adding to the imaginary part (by adding further imaginary numbers) doesn’t affect the value of the real part. The real and imaginary parts are independent of each other; they are orthogonal (appendix 2 of post 16.50). We can’t represent complex numbers along a single line but we can represent them on two lines that are perpendicular to each other – like the axes of an orthogonal Cartesian coordinate system (appendix 2 of post 16.50), as shown in the picture below.
So, the values of real numbers lie along a line; the values of complex numbers lie in a plane – the Argand plane.
The picture above shows the position of the complex number z = a + ib on the Argand plane. We define │z│ (the modulus of z) as the length of z in the Argand plane. According to Pythagoras’ theorem (appendix 1 of post 16.50) z2 = a2 + b2. We define z* (the complex conjugate of z) by z* = a – ib. Now zz* = (a + ib) x (a – ib) = a2 +a(-ib) + (ib)(a) + (ib)(-ib) = a2 –abi +abi –i2b2 = a2 + b2 = │z│.
Notice (also in the picture) above, that z is associated with an angle, θ, called its argument. You can see that a = zcosθ and b = zsinθ (appendix 3 of post 16.50), so that a/b = tanθ (appendix 5 of post 16.50).
If you read post 18.11, you will see that │z│ behaves just like the amplitude of a wave or oscillator and θ behaves just like ωt, where ω is an angular frequency and t represents time. So, complex numbers are often used to represent the behaviour of oscillators and waves. To pursue this further, we need to think about some more mathematical ideas. You may also have noticed that complex numbers could be considered as an alternative way of representing two-dimensional vectors (post 17.2). But be careful because i or j here have completely different meanings to the way they are used in vector algebra (see post 17.14).
18.15 More about exponential growth
18.2 Powers of numbers