What are imaginary numbers and why do we use them?

Imaginary numbers can seem very daunting when you first encounter them. The name is the most confusing part; how can a number be imaginary? Do imaginary numbers really exist? You can think of the set of imaginary (or complex) numbers as just an extension of the numbers that we're used to. The essence of imaginary numbers lies in the problem of finding the square root of a negative number. We know that 22=4, for example, and also that (-2)2=4. This means that if we take the square root of 4, 41/2, we get two possible answers, namely 2 and -2. So what happens if we take the square root of -4? By conventional means this would not be possible, as there is no 'real' number that we can square to get -4. This is where imaginary numbers come in. To solve problems such as these, we assume that there is a number called i, such that i2=-1. This allows us to say that i is equal to the square root of -1, and that gives us our basis for imaginary numbers. Now, we can build a set of numbers around this imaginary unit, and define them as a+bi, where a and b are 'real' numbers and i is the imaginary unit.

From here, we can use these numbers to solve many problems in mathematics that would conventionally be unsolvable. Consider our problem that we stated before, where we attempted to find the square root of -4. If we let x2=-4, we can then say that x=+/-(-4)1/2. We can factorise this, to get that x=+/- (4)1/2 * (-1)1/2. We know that (4)1/2=2, and since we know that (-1)1/2, or the square root of -1, is called i, we can re-write this number in a more friendly way, to say that x=+/-2i. So we have used our idea of imaginary numbers to find a solution to a problem in the real world, and this is just the tip of the iceberg. The concept of imaginary numbers has been around for hundreds of years, and it has found many uses in the likes of physics, engineering and aerodynamics, as they allow us to solve problems that don't make sense when we look at them conventionally. From finding the roots of positive quadratics to helping us learn morw about triangles, imaginary numbers are definitely useful, and are just as real as the 'real' numbers.

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