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THE NUMBER THAT DOESN'T EXIST!! yet it does.

Let's start with this simple quadratic equation x² + 2x + 5 = 0


If you've learned about quadratics in high school, you know about the discriminant Δ and how it's sign is used to determine the number of quadratic's solutions. 
In this case the Δ = -16

Since it's negative there should be no solutions, and there indeed are no solutions. REAL solutions. However we can EXTEND real numbers to make this equation solvable. That's where "i" comes into the game. 
The number "i" called imaginary unit, is defined as one of the solutions to x² + 1 = 0 (the other one is -i). It is NOT a real number, but it can be used to extend them to COMPLEX NUMBERS, which are numbers of the form a + ib, where a and b are real numbers. 
Concept of COMPLEX NUMBERS is very important in mathematics as they show up in almost every field and they are simply useful. But the question is, DO THEY EVEN EXIST?!

Before answering this question, let's look into the history of complex numbers and where they first came from. 
One surprising fact about their origin is that they showed up during the study of CUBIC EQUATIONS instead of quadratic! GEROLAMO CARDANO, a 16th century Italian matemathician published first solutions to the particular cases of cubic equations in his algebra book ARS MAGNA. 
He aviods few cases because square roots of negative numbers showed up (!), but later in the book he mentions COMPLEX NUMBERS, even though had misgivings about it. 
In Chapter 37 of Ars Magna the following problem is posed: "To divide 10 in two parts, the product of which is 40".

Cardano wrote,
"It is clear that this case is impossible. Nevertheless, we shall work thus: We divide 10 into two equal parts, making each 5. 
These we square, making 25. Subtract 40, if you will, from the 25 thus produced, as I showed you in the chapter on operations in the sixth book leaving a remainder of -15, the square root of which added to or subtracted from 5 gives parts the product of which is 40. 
These will be 5+√-15 and 5-√-15.

Putting aside the mental tortures involved, multiply 5+√-15 and 5-√-15 making 25 - (-15) which is +15. Hence this product is 40."

Even though Cardano uses complex numbers, he refuses to believe that they exist. 
Maybe he was right all along and they TRULY DON'T EXIST?!

To answer this question let's get back to times, when whole mathematics was just an art of manipulating geometric shapes. Take a look at such problem "To find length of a segment that when stretched by 1 becomes a point". 
From perspective of these times such segment doesn't exist, because there is no positive number such that x+1=0
and segments have positive lengths. But as we all know there exists a set, called integers, 
which includes negatives of natural numbers and in this set such number exists: it is -1. Negative naturals extend natural numbers to integers.

Similarily asking whether there exists a number that when squared becomes -1 doesn't have one clear answer. 
Everything depends on what you are calling numbers".

If by numbers you mean natural numbers, then there is no number x such that x+1=0, but if by numbers you mean integers, then the solution turns out to be -1. 
Similarily if by numbers you mean real numbers, then there is no number x such that x²+1=0, but if by numbers you mean complex numbers, then the solution turns out to be i.

Going even further, one can generalize and extend complex numbers to what is called QUATERNIONS. 
Also there are algebraic structures that include very non-intuitive objects which are also called numbers, for example surreal numbers or SURCOMPLEX NUMBERS. And we indeed call them numbers, even though there is no real number that possess their properties. 
One might think that real numbers are real, because they are the only ones that have applications to the real world, but that's also not true. Complex numbers are useful too, for example in Signal Processing or Quantum Mechanics. Complex numbers are as "real" as real numbers. 
Sources

- Eric Platt's answer on
quora.com/What-numbers-d…

- "A Short History of Complex Numbers", Orlando Merino, 2006

- "How Imaginary Numbers Were Invented", Veritasium on YouTube
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