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When we talk about numbers, we often think of the ones we can count on our fingers or see on a number line. However, there's a whole category of numbers that go beyond the everyday understanding, known as complex numbers. These numbers comprise a real part and an imaginary part. A complex number is usually written in the form of a + bi, where 'a' represents the real part, and 'bi' represents the imaginary part. The 'i' in this expression stands for the imaginary unit, which is defined as the square root of negative one.

Let's take a real-world example – in electrical engineering, complex numbers are used to represent and solve problems involving electrical circuits. Without the concept of complex numbers, we wouldn't be able to describe the behavior of alternating currents accurately.

One of the initial challenges students face with complex numbers is the concept of the square root of negative one. In everyday arithmetic, taking the square root of a negative number doesn't make sense because no real number multiplied by itself will give a negative product. This is where 'i' comes into the picture.

So what is 'i'? Briefly, 'i' is a unique number that when squared, gives us -1. This might seem strange at first, but 'i' plays a critical role in expanding the number system to include complex numbers. Without 'i', we can't solve equations like x² + 1 = 0, which have no solutions in the set of real numbers. With 'i', the solution is simply 'x = i' or 'x = -i', unveiling a new dimension of numbers to explore.

The existence of complex numbers is further validated by the fundamental theorem of algebra. This pivotal theorem states that every non-constant polynomial equation with complex coefficients has at least one complex root. This means that no matter how complicated a polynomial equation may seem, you can always find a solution within the realm of complex numbers, real numbers being a subset of complex.

For instance, an equation like x² + 1 = 0, which we mentioned earlier, appears to have no solution if we only consider the real numbers. But according to the fundamental theorem of algebra, a solution must exist, and indeed, the complex numbers provide it: x = i and x = -i. Essentially, this theorem is a guarantee that the algebraic equations we face have a place to belong within the number system, assuring us that 'i' isn't just a mathematical curiosity but an integral part of the universe of numbers.