Part 1: What Are Imaginary Numbers? Why Do We Need Them?
- This is Part 1 of the series ‘Nuts and Bolts of Fourier Transform for Time Series Forecasting — Python’.
In order to understand and appreciate the need for imaginary numbers, you should start from the very basics of math,“ the numbers”. Even though this is a high school level topic, we will revisit it as it plays an important role in Fourier transform.
Long back, we humans used the numbers for counting and ordering. If you are asked to tell the number of apples in an apple tree, you would start counting from one and go on till the last apple. In this scenario, you are not bothered about the non-existence of apple, i.e you don’t have to explicitly say that there are no apples as you are bothered only if you have it. The numbers used for this purpose starts from one and travels in the positive direction called Natural numbers denoted by N.
ℕ = {1,2,3,…}
If you add two natural numbers, you will end up in a natural number
eg: 2 + 3 = 5
this is satisfied for all possible natural numbers. This property is called closure property and is defined as ‘a set that is closed under an operation (or) collection of operations is said to satisfy a closure property’. Natural numbers satisfy closure property for addition and multiplication operations as multiplication is recursive addition. Natural numbers do not satisfy closure property for subtraction operation i.e. there exist numbers where if you subtract two natural numbers, you will not end up in a natural number.
eg: 2 – 3 = -1
Negative numbers are not a part of Natural numbers. In order to solve this problem, Integers were introduced which spans from negative infinity to positive infinity.
ℤ = {…-3,-2,-1,0,1,2,3,…}
Integers satisfy closure property for the addition, multiplication and subtraction operations. Integers do not satisfy closure property for division operation i.e. there exist numbers where if you divide two Integers you will not end up in an Integer.
eg: 1 ÷ 2 = 0.5
Ratios are not a part of Integers which lead to the introduction of rational numbers.
ℚ = {x | x=a/b, a,b∈ ℤ, b≠ 0}
Rational numbers represent the decimal numbers in exact ratios. It could also not satisfy closure property for division operation as there are numbers which cannot be represented as an exact fraction.
eg: π is approximately equal to 22/7, not exactly equal.
This lead to irrational numbers, details of which is out of the scope of this article. Among irrational numbers are the ratio π of a circle’s circumference to its diameter, Euler’s number e which are discussed in later parts.
Combining all the number systems described above leads to real numbers.
ℝ = {x | -∞ < x <∞}
Real numbers satisfy closure property for addition, subtraction, multiplication and division but not square root. As √-2 cannot be represented in terms of any numbers from the real number set.
To address this problem, new numbers were introduced called imaginary numbers i, defined by
i =√-1
√-2 is simplified as √-2 = √2 * √-1 = √2i
This resulted in a new world of opportunities for mathematicians and physicists to explore more. Imaginary numbers and real numbers put together are called complex numbers which encompasses all possible numbers in the number system.
ℂ = {z | z=a+bi, -∞<a<∞, -∞<b<∞}
Complex numbers can be represented as given below: the picture illustrates real numbers in the horizontal axis and the imaginary numbers in the vertical axis.
From the above picture, it is evident that performing multiplication by a positive number results in a 360-degree rotation (Remember that rotation is something that is represented in terms of origin).An example: Multiplying 2 by 1 results in 2, which is a 360-degree rotation( A to A).
If multiplied by a negative number, the result is a 180-degree rotation. An example: Multiplying 2 by -1 results in -2, which is a 180-degree rotation(B to A).
In order to have a 90-degree rotation, the imaginary axis was introduced, the picture below illustrates multiplying by i results in a 90-degree rotation.
The above representation helps us in understanding the Euler number e which will be explained in future parts. Stay tuned :)
