Number System

In mathematics, a number system is a set of numbers, (in the broadest sense of the word), together with one or more operations, such as addition or multiplication.

Examples of number systems include: natural numbers, integers, rational numbers, algebraic numbers, real numbers, complex numbers, p-adic numbers, surreal numbers, and hyperreal numbers.

Real Numbers

In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339.... The real numbers include both rational numbers, such as 42 and -23/129, and irrational numbers, such as p and the square root of 2, and can be represented as points along an infinitely long number line.

The name real numbers arose to distinguish them from what were then called imaginary numbers (and now complex numbers).

A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero.

Properties of Real Numbers

Let R denote the set of all real numbers. Then:

The set R is a field, meaning that addition and multiplication are defined and have the usual properties.
The field R is ordered, meaning that there is a total order = such that, for all real numbers x, y and z:
if x = y then x + z = y + z;
if x = 0 and y = 0 then xy = 0.

Rational Numbers

Rational numbers sound like they should be very sensible numbers. In fact, they are! Rational numbers are simply numbers that can be written as fractions or ratios

Integer

An integer can be written as a fraction simply by giving it a denominator of one, so any integer is a rational number.

Terminating Decimal

A terminating decimal can be written as a fraction simply by writing it the way you say it: 3.75 = three and seventy-five hundredths=, then adding if needed to produce a fraction: . So, any terminating decimal is a rational number.

A repeating decimal can be written as a fraction using algebraic methods, so any repeating decimal is a rational number.

Whole Numbers

Whole Numbers: Zero and the positive integers are the whole numbers.

Natural Numbers

Natural Numbers: Also called the counting numbers, this set includes all of the whole numbers except zero (1, 2, 3, ....)

Irrational Numbers

Irrational Numbers: Any real number that cannot be written in fraction form is an irrational number. These numbers include the non-terminating, non-repeating decimals (pi, 0.45445544455544445555...,2, etc.). Any square root that is not a perfect root is an irrational number. For example, 1 and 4 are rational because 1 = 1 and 4 = 2, but 2 and 3 are irrational-there are no perfect squares between 1 and 4. All four of these numbers do name points on the number line, but they cannot be written as fractions. When a decimal or fractional approximation for an irrational number is used to compute (as in finding the area of a circle), the answer is always approximate and should clearly indicate this.

Radical Expression

Definition of Radical Expression
A radical expression is an expression containing a square root.

Examples of Radical Expression
are examples of radical expression.

More about Radical Expression
Radical: The symbol that is used to denote square root or nth roots.
Radicand: Radicand is a number or expression inside the radical symbol.