# What is NOT a real number?

##### 3 Answers

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

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,

Within the realm of numbers: even roots of negative numbers (square, 4th, 6th, etc roots of negative numbers) are not real numbers. So

Some things that look like they describe numbers do not:

"The solution to

In mathematics in general

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

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 *field* - closed under addition, subtraction, multiplication and division by non-zero numbers.

In fact

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

(3) Single complex infinity.

Imagine a sphere sitting on the origin of the complex plane. Given any point

This is a simple example of what's called a Riemann surface. Functions like

(4) Circle at infinity.

Rather than project from the top of the sphere, project from the centre. This defines a mapping between

(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.

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