Ooopps we meant to ask what is the definition of a (real) (number)?

Question

Please give an example of a number that isn't a (real) (number).

Answer 1

I don't have any objections to the definitions of reals others have posted here, but I think a more interesting question is, what makes something a "number", as opposed to some other kind of mathematical object? What properties must "numbers" absolutely have, so that anything which lacks one of those properties cannot be called a number?

My own personal prejudice, for example, says that numbers, among other things, measure or count something and can be compared to each other as more, or less. If we insisted on this as an indispensable property of number, then the complex "numbers" are right out. They are certainly important and useful and interesting, they are just not numbers. I don't think too many people feel that way, but it seems to be a reasonable requirement. I think the term "number" is rather vaguel used.

Answer 2

A real number can be specified by a point on the number line. A non "real number" cannot. Real numbers contain all transcendental numbers (such a pi and e), alegbraic numbers (solutions to some polynomial with integer coefficients), irrational numbers, and rational numbers, which themselves contain the integers, which contain the whole numbers. These are all real.

Complex numbers are good examples of sets that contain non-real numbers. "i" is a non real number, it's also a type of complex number referred to as fully imaginary, and it's typical definition is that i^2 = -1. Complex numbers with a non-zero imaginary part cannot be located on the number line, and are often located on the complex plane instead, which is an extension of the number line into the entire x-y plane.

I'm sure other systems exist or could be conceived that have no relation to the other real numbers. I'm aware of at least octonian and quaternions, but these are related to complex numbers in a certain sense.

--charlie

Answer 3

Just about any number is real, like -4, 1/2, pi, etc. However, there is no real number that is a solution to the equation
x^2 = -4, for example. (because any real number squared is not negative). So we use complex numbers to solve this. The solution is 2i, where i is defined as the square root of -1.

So in general numbers like i, 3i, 2-4i, etc are not real.

Answer 4

The field of all rational and irrational numbers is called the real numbers, or simply the "reals." The set of real numbers is also called the continuum. The set of reals is called Reals in Mathematica, and a number can be tested to see if it is a member of the reals using the command Element[x, Reals], and expressions that are real numbers have the Head of Real.

Answer 5

square roots of negative numbers aren't real, they're imaginary. Can you imagine that?!

Answer 6

Zero!