What are imaginary numbers?

To understand imaginary numbers, let's first go back to the number line, one of the most basic things there is in maths. It's used in primary schools for teaching kids how to add and subtract. In these primary schools, at the earliest level, these number lines would probably be from 0 onwards.

However, mathematicians also started to question a bit more. if 5+3=10, then what is the answer to 3-5? This may seem rather obvious, but this simple question led to the creation of negative numbers. If we think of the number line again, then we'll see that these negative numbers go to the left of 0, continuing on with -1, -2, -3, etc. These numbers were very useful for things such as finance, where negative numbers are used to represent debt, and even science, to represent things like very cold temperatures.

The next question that followed on from this is that if 4÷2=2, then what is the answer to 2÷4? Again, this seems to be a simple question, but to answer this, mathematicians created fractions, to represent parts of a whole. On a number line, these fractions are inserted between the different integers.

Then, we get to some even more challenging questions, such as what is the square root of 2, or what is the ratio between the circumference and the diameter of a circle? The answer to this is less obvious than the other questions that we have asked, but mathematicians decided to invent irrational numbers. These are numbers such as pi and e (also known as Euler's number) which can't be represented as fractions.

This is because there is no pattern in the sequencing of numbers they have. For example, 1/3 = 0.33333... There is a clear pattern here. However, pi is equal to 3.14159265... There is no repeating pattern of numbers that we can use to convert pi into a fraction, therefore it is called an irrational number. However, if we go back to the number line again, pi can still be labelled onto it, as we know it is between 3 and 4 but isn't a fraction.

Now, we have already done the square root of a positive number, but what about the square root of a negative number? In this case, we'll take the square root of -1.

Mathematicians can see that it is not possible to multiply the same number by itself to get -1, as if we multiply two negative numbers together, we get a positive number, and if we multiply 2 positive numbers together, we also get a positive number.

Therefore, it would seem that the square root of -1 has no answer. However, as we have seen from the examples before, when mathematicians don't have an answer to something, they'll usually create one. So, i was created to represent the square root of -1.

Just like real numbers, we can add imaginary numbers to other imaginary numbers. For example, 5i + 2i = 7i. However, a more interesting use of imaginary numbers is to add them to real numbers, so we get numbers such as 2 + 3i.

When we go back to our number line, we'll find that there is no place for imaginary numbers, as there is no real solution to the square root of -1. However, if we want to fit all the numbers onto it we'll have to change it from a number line into a complex plane.

The idea of a complex plane is that we take the 1-dimensional, real number line and turn it into a two-dimensional plane upon which every single number can be represented. This means that we can represent complex numbers as coordinates on this plane.

This may seem kind of useless, and in the past this was exactly what people believed. The term 'imaginary numbers' comes from René Descartes, who called them this in a derogatory way.

However, in this modern age, people are finding new uses for imaginary numbers. We can use complex numbers in quantum mechanics, which helps us understand the universe at the smallest of scales. Also, complex numbers are the base of the Riemann Hypothesis, a mathematical conundrum which has remained unsolved for more than a century. The solving of this would allow us to find the exact distribution of prime numbers, which would be useful in the real world as well as the mathematical one, as it would allow mathematicians to have a formula for finding prime numbers, which have been assumed to be randomly arranged on the number line. In a more poetic way, you could say it brings order to the chaos of primes.

All in all, imaginary numbers provide a valuable answer to an unanswered question, and in doing so it provides the mathematical world with a new tool to create new problems and solutions that will have profound impacts on the world. Despite being ignored in the past, they've now been recognised as just as meaningful as any other number is.