What are imaginary numbers, and why do we bother thinking about them if they don't exist?

When studying anything in Mathematics, it is very easy to get lost in the technical details and lose sight of what everything really means on a conceptual level - that's why these kinds of questions are so fantastic to ask (and answer)! The trick to understanding imaginary numbers is to not be distracted by their name. There is nothing new or scary about imaginary things in mathematics. In fact, most of what we study in this field is imaginary. For example, in geometry we imagine perfect circles and exactly straight lines, however in reality no such thing exists - there will always be some level of imperfection in any attempted drawing. So with this in mind, we start to see imaginary numbers as nothing more than a new sort of number which we haven't seen before. Compare this feeling to that which you felt when learning about irrational numbers such as the square root of 2 and pi!With all this now in mind, we are prepared for the big secret - what exactly is an imaginary number? The answer is easier than you might think. In the real numbers with which we are all familiar, there is no solution to the equation x²= -1, because anything squared must not be negative. To "invent" imaginary numbers, we simply say: 'imagine there is such a number which squares to minus one. Let's call this number i.' That is really all there is to it! We define the imaginary number i to simply be some number which multiplies with itself to give -1. We can get all other imaginary numbers just by scaling this up or down. 2i is imaginary. -6i is imaginary. 142.4325235i is imaginary. It really is as easy as that! What's more, we can combine imaginary numbers with real ones to form "complex numbers", such as i+1. When manipulating expressions involving our imaginary i, just follow all the usual algebraic rules along with the additional property that i²= -1.Amazingly, we can use these numbers to prove results which exist in the real number system, such as De Moivre's theorem. You will use and understand this theorem later in your course! As an aside, some authors use j instead of i to denote the exact same thing - watch out for this!