Real Numbers

The union of the irrational and rational numbers forms the real number line.¹⁰ Real numbers are of great importance in physics. They are closed under the arithmetical operations of addition, subtraction, multiplication and division, where one must exclude only division by zero. Real exponential functions such as $a^b$ or $e^x$ (where $a, b, e, x$ are all presumed to be real) will have real values, as will algebraic functions such as $(a + b)^n$ where $n$ is an integer.

However, as before we can discover arithmetical operations such as the square root operation that lead to problems with closure. For positive real arguments $x \ge 0$ , $y = \sqrt{x}$ is real, but probably irrational (irrational for most possible values of $x$ ). But what happens when we try to form the square root of negative real numbers? In fact, what happens when we try to form the square root of $-1$ ?

This is a bit of a problem. All real numbers, squared, are positive. There is no real number that can be squared to make $-1$ . All we can do is imagine such a number, and then make our system of numbers bigger to accomodate it. This process leads us to the imaginary unit $i$ such that $i^2 = -1$ , and thereby to numbers with both real and imaginary parts: Complex numbers.