Before we get to the story, we should go over the different types of numbers that we know about:
Natural Numbers - All of the integers above 0 (i.e. 1, 2, 3, 4, 5…); Note: Some people like to include 0 here. It's really up to your preference. The natural numbers are really the first numbers we become accustomed to as we grow up and they appear to be the most intuitive (hence, the name "natural"). This is because we count things and see the world in natural numbers - ex. 3 apples, 4 cars, 5 fingers, etc…
Integers - In this set, we include all of the negative integers (so it's all of the natural numbers plus all of their negative counterparts). For a long time, these numbers were also looked "down upon" because they didn't really carry the same physical intuition as the natural numbers (ex. can you really have -3 apples?). Nonetheless, the negative numbers proved to be useful, so we now use integers all of the time.
Rational Numbers - Now, we get to expand out to the rational numbers, which are basically the ratios of integers. These result in what are commonly referred to as fractions (ex. 2/7, 1/3, 9/10, etc…). These numbers didn't face as much adversity since it can be very easy to see fractions at work in real life (ex. half a block of wood, or a quarter of a pie).
Irrational Numbers - Unlike rational numbers, these are a set of numbers that were not met with much enthusiasm or acceptance from people. Irrational numbers are basically numbers that cannot be represented by a fraction. In other words, if we were to write it out, its decimals would go on forever without ever repeating any patterns. Some examples would be pi, e, the square root of 2, etc…
Now, all of the above come together to make what we call the set of Real Numbers. These are numbers that you can plot on a number line and most people are familiar with them. So, it looks like we've covered just about every single number possible, right? Well…it turns out that there is one more set of numbers that also encompasses all of the real numbers. This is called the set of Complex Numbers.
And this is where the story begins!
What are Complex Numbers?
Complex numbers are numbers of the form: a + bi. a and b are some real numbers while i is called the "imaginary number". In this set of numbers, the real superstar is i. a and b are just there to help make more numbers. So, with that in mind, we should define i. The imaginary number is defined as:
i^2 = -1
From this, it follows that (after square rooting both sides):
i = √-1
Now, this should surprise you! We've always learnt that and number squared has to be either a positive or zero, right? How could you get negative 1 after squaring i? Well, mathematicians ran into this problem a long time ago. In attempting to get solutions for cubic functions specifically (i.e. trying to find the zeros of a third degree function), they'd often run into the square roots of negatives. After getting to this point, they would be forced to throw their hands in the air, give up on their work, and leave the room.
So, how did mathematicians solve this problem? Well, it turns out that they just decided to invent a new number! And before you cry foul, this is absolutely valid as long as your new number (or rule) is consistent with the rest of the mathematics. And if the fact that the imaginary number sounds "made up" to you, I pose the question: Aren't all numbers "made up"? But that's a topic for another day!
Anyways, over the centuries it turns out that imaginary numbers (and complex numbers and, by extension, the complex plane) have been a huge help in math and, perhaps somewhat surprisingly, have many real world applications in physics and engineering! Not so imaginary, eh?
But despite their usefulness, imaginary numbers faced a great deal of adversity and opposition from the mathematics community. Net, we'll explore the reasons for this and we'll specifically focus on the imaginary number in doing so.
A Brief History
Although it's hard to pin point their exact inception, it appears that imaginary numbers first made a prominent appearance in the writings of the Italian mathematician, Gerolamo Cardano. Right off the bat, even though they seemed to give solutions to previously undoable functions, most people completely dismissed imaginary numbers as useless; they viewed solutions involving imaginary numbers as "fake" and made up.
An interesting note is that this was very similar to the reaction that negative numbers and 0 got. People also viewed them as "fake", mainly because they didn't see an exact physical representation of these numbers. A fun fact: people took their derision for imaginary numbers one step further than they did with negative numbers and zero - the name "imaginary" was given to them as an insult! It was believed to have been first done by the famous Rene Descartes, who was attempting to essentially poke fun at their uselessness.
And to be fair, there is a pretty valid reason for the initial opposition. The things is, back in that day (and to some extent, it's still true today), people demanded to see how the math they were doing translated to the real world or to visuals. For example, when doing algebra with functions, we have this process by which we obtain "solutions". These are usually just numbers. On a graph, the solution is represented by the function's x-intercepts.
With the advent of imaginary numbers, we got solutions to functions that we couldn't solve before. So, that's good, right! Well, not really…let's take an example:
Let's find the solution to the function x^2 + 1 = 0
Using real numbers, there are no solutions. However, with imaginary numbers, the solutions we obtain are +i and -i. But the, let's take a look at the graph of this function:
Clearly, there are no x-intercepts, so if we're going to look at the physical representation of the problem, there aren't any solutions. This led people to question the validity of complex and imaginary solutions. Did they really mean anything in the real world?
Imaginary numbers didn't really become accepted until the mid 18th century, thanks to the work of Euler and Gauss (Note: the famous Euler's Identity utilizes imaginary numbers - some may view the formulation of that proof as the moment that forever put the imaginary number into the mainstream).
The Sociology and Psychology of It
So, what does this history of the imaginary number reveal. One of the first things it shows us is our resistance to change. Sometimes, that is understandable; however, often, this proclivity towards sticking with what one is familiar with is doomed to restrict new ideas and halt progression. Fundamentally, humans do not like being confronted with something that is "other". This isn't just seen in math - we saw it in the women's rights and civil rights movements as well.
Moreover, looking at the story on a philosophical level, the story acts as a symbol for the nature of mathematics and its abstractions. It raises questions about just how "real" mathematics is. Sure, we can't see imaginary numbers in real life, but when you think about it, isn't that the case with all numbers? I mean, sure you can physically see or feel 5 apples…but the number 5 is more of an abstract idea that we assign to that certain rearrangement of objects. The same principle applies to negative numbers. Can you ever see or feel negative money, for instance? No, but what we do to help us understand the abstract concept of "negative-ness" is to make up the notion of "deficit". This way, negative 5 dollars makes sense to us in that we can understand what it means to have a deficit of 5 dollars. Nonetheless, it doesn't change the fact that, really, all of these numbers are just as imaginary as the imaginary number itself!
Regardless, it's impossible to provide a definitive answer for these questions. All we know is that mathematics is very useful and that the philosophy of mathematics continues to perplex humankind!
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