What is the properties of a real number?

Answer 1

A number is real simply because it isn't imaginary; that is, real numbers can never be factored in such a way that one of their factors is an even-root of a negative number or "i" (unless you include enough of those i's so they will cancel each other out). Another way to know that a number is real is to determine whether it's rational or irrational. Rational numbers can be expressed as ratios of integers. For example, -2 is rational because -2 = -2/1 = -4/2 and so forth. Irrational Numbers are usually universally recognized (such as pi and e) or they look like decimal numbers whose digits follow no particular pattern and are infinite in number, such as .0856239... If a number is neither rational nor irrational, it isn't real.

Answer 2

The real number system is a mathematical field.

To start with, the Real number system is a Group. This means that it is a set of elements (numbers) with a binary operation (addition) that combines any two elements in the set to form a third element which is also in the set. The Group satisfies four axioms: closure, associativity, identity and invertibility.

• Closure means that for all real numbers x and y, x+y is real.
• Associativity means that for all real number x, y and z, (x+y)+z = x+(y+z).
• Identity means that there is a real number, denoted by 0, such that for all real numbers x, x+0 = x = 0+x.
• Invertibility means that for any real number x, there exists a real number y such that x+y = 0. y is denoted by -x.

In addition, it is a Ring. A ring is an Abelian group (that is, addition is commutative) and it has a second binary operation (multiplication) that is defined on its elements. This second operation is distributive over the first.

• Abelian means for all real x and y, x+y = y+x.
• Distributivity requires that for all real x, y and z, x*(y+z) = x*y + x*z.

And finally, a Field is a Ring over which division - by non-zero numbers - is defined.

• For any real x and y, where y is not 0, x/y is a real number.

There are several mathematical terms above which have been left undefined to keep the answer to a manageable size. All these algebraic structures are more than a term's worth of studying. You can find out more about them using Wikipedia but be sure to select the hit that has "mathematical" in it!