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Real numbers are all the numbers that you typically use in everyday life. They include:

• Positive numbers (1, 2, 3, ...).
• Negative numbers (-1, -2, -3, ...).
• Zero (0).
• Fractions (1/2, 0.3, ...).
• Irrational numbers (like \(\backslashsqrt{2}\)).

Real numbers are represented on a number line, stretching infinitely in both directions toward positive and negative infinity. One important feature of real numbers is that any number squared will always result in a non-negative number.

Imaginary numbers come into play when we deal with the square root of negative numbers, which is impossible within the realm of real numbers. The imaginary unit, denoted as \i, is defined by: \(\backslashsqrt{-1} = i\).
This means that imaginary numbers are multiples of this imaginary unit. Some examples are:

• \(i\),
• \(2i\),
• \(3.5i\),
• \(-i\).

When working with negative square roots, we use imaginary numbers because every real number squared is non-negative.

Complex numbers combine both real numbers and imaginary numbers. They have the form:
\[ z = a + bi \]
where:

• \(a\) is the real part,
• \(bi\) is the imaginary part.

For example:

• \(3 + 4i\)
• \(-2 - 5i\)
• \(7\) (where the imaginary part is 0)

The set of complex numbers provides a way to solve equations that do not have solutions within the set of real numbers, like \(\sqrt{-64}\). In this set, \(\sqrt{-64} = 8i\).

The square root function, denoted by \(\sqrt{}\), is used to find a number which, when multiplied by itself, equals the given number. This function is straightforward for non-negative real numbers, as every positive number has two square roots (one positive and one negative) and zero has one square root (0).
However, taking the square root of a negative number is not possible in the real number system because no real number squared results in a negative value. To deal with this, we enter the realm of complex numbers, where we use the imaginary unit \(i\).
For instance, \( \sqrt{-64} \) in the complex number system equals \( 8i \), since:
\(\sqrt{-64} = \sqrt{64} \times \sqrt{-1} = 8 \times i = 8i\).