# What is NOT a real number?

Apr 9, 2015

A number which is imaginary, is NOT a real number. All real numbers can be expressed on the number line, while imaginary numbers cannot be expressed on the number line

Apr 9, 2015

I will assume that by "real number" you mean a member of the set of things called "the Real numbers"
(As opposed to meaning "what is really a number?")
A real number is a number that can be expressed in decimal form. Everything else is not a real number.

Colors, sounds, plants, $\sqrt{\textcolor{red}{\text{red}}}$, and
$15 + \times 26.78 .24 .36$ are not real numbers.

Within the realm of numbers: even roots of negative numbers (square, 4th, 6th, etc roots of negative numbers) are not real numbers. So $\sqrt{- 4}$, and $\sqrt[6]{- 64}$ are not real numbers

Some things that look like they describe numbers do not:

"The solution to $x + 3 = x + 8$" is not a number. (There is no solution to the equation.)

$\frac{0}{0}$ is not a number so it is not a real number.

In mathematics in general ${0}^{0}$ is not a number (Some branches give a definition for it.)

infinity is not a number (not a real number and not a complex number)

May 25, 2015

There are many things that are not real numbers. Perhaps the most interesting question is "what numbers are there that are not real numbers?"

(1) Complex numbers.

The simplest and most natural extension of the real numbers is to add $i = \sqrt{- 1}$ and everything else required to complete it as what is called a field - closed under addition, subtraction, multiplication and division by non-zero numbers.

In fact $\mathbb{C}$ is in some sense much more natural than $\mathbb{R}$.

Some things like Taylor's Theorem behave much better.

(2) Quaternions.

If you drop the requirement that multiplication be commutative then instead of just one pair $\pm i$ of square roots of $- 1$ you get 3 pairs called $\pm i$, $\pm j$ and $\pm k$. Some properties of these are: $i j = k$, $j i = - k$, $j k = i$, $k j = - i$, etc.

(3) Single complex infinity.

Imagine a sphere sitting on the origin of the complex plane. Given any point $z$ on the complex plane, draw a line from the top of the sphere through the point $z$. This will intersect the surface of the sphere at one point other than the top. If you use that point on the surface of the sphere to represent the number $z$ then you have defined a one-one mapping between all points of the complex plane and all points on the surface of the sphere - except the top. Call the top $\infty$ and let ${\mathbb{C}}_{\infty}$ stand for $\mathbb{C} \cup \left\{\infty\right\}$.

This is a simple example of what's called a Riemann surface. Functions like $f \left(z\right) = \frac{a z + b}{c z + d}$ can then be defined as taking the value $\infty$ when $c z + d = 0$ and $f \left(\infty\right)$ can be defined as $\frac{a}{c}$. Then the resulting $f \left(z\right)$ definition is continuous and infinitely differentiable at all points in ${\mathbb{C}}_{\infty}$. It also has the property that it maps circles to circles (including ones passing through $\infty$).

(4) Circle at infinity.

Rather than project from the top of the sphere, project from the centre. This defines a mapping between $\mathbb{C}$ and the open lower hemispherical surface. Add the equator and you have a ring of infinities with different polar angles. The ones corresponding to the real line are $+ \infty$ and $- \infty$, but there's a unique complex inifinity $\infty \left(\cos \theta + i \sin \theta\right)$ for all $\theta \in \left[0 , 2 \pi\right)$.

(5) Infinitesimals.

At the other end of the scale, what happens if you try to add infinitely small numbers. Well you can. It's generally a bit messy and does tend to break various things, but it can be useful.

(6) Finite fields.

(7) Rings.

...