what is the difference between real numbers and rational numbers ?

Answer 1

http://en.wikipedia.org/wiki/Real_number

http://en.wikipedia.org/wiki/Rational_number

Answer 2

In mathematics, the set of real numbers, denoted R, is the set of all rational numbers and irrational numbers. A real number may be thought of as any point on an infinitely long number line.

The discovery of more rigorous definitions of the real numbers was one of the most important developments of 19th century mathematics. Definitions in use today include equivalence classes of Cauchy sequences of rational numbers; Dedekind cuts; a more sophisticated version of "decimal representation"; and an axiomatic definition of the real numbers as the unique complete Archimedean ordered field. These definitions are all described in detail below.

The term "real number" is a retronym coined in response to "imaginary number".

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

Real numbers measure continuous quantities. They may in theory be expressed by decimal representations that have an infinite sequence of digits to the right of the decimal point; these are often represented in the same form as 324.823211247…. The three dots indicate that there would still be more digits to come.

More formally, real numbers have the two basic properties of being an ordered field, and having the least upper bound property. The first says that real numbers comprise a field, with addition and multiplication as well as division by nonzero numbers, which can be totally ordered on a number line in a way compatible with addition and multiplication. The second says that if a nonempty set of real numbers has an upper bound, then it has a least upper bound. These two together define the real numbers completely, and allow its other properties to be deduced. For instance, we can prove from these properties that every polynomial of odd degree with real coefficients has a real root, and that if you add the square root of minus one to the real numbers, obtaining the complex numbers, the result is algebraically closed.

and

In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero.

Each rational number can be written in infinitely many forms, for example 3 / 6 = 2 / 4 = 1 / 2, but the simplest form is when a and b have no common divisors. Every non-zero rational number has exactly one simplest form of this type with a positive denominator. A fraction in this simplest form is said to be an irreducible fraction, or a fraction in reduced form.

The decimal expansion of a rational number is eventually periodic (in the case of a finite expansion the zeroes which implicitly follow it form the periodic part). The same is true for any other integral base above one, and is also true when rational numbers are considered to be p-adic numbers rather than real numbers. Conversely, if the expansion of a number for one base is periodic, it is periodic for all bases and the number is rational.

A real number which is not a rational number is called an irrational number.

You could get more information from the 2 links below...

Answer 3

A rational number can be expressed in the form a/b where a and b are integers. So, 1/2 and 3/1 (=3) are rational. An irrational number cannot be expressed in the form a/b where a and b are integers. So, PI and the square root of 2 are irrational. The discovery that the square root of 2 is irrational was of great concern to the Pythagoreans.

Answer 4

The rational numbers are all numbers that can be written as fractions. The real numbers (very loosely speaking) are all the points on the number line.
The rational numbers are not "complete" that is all Cauchy sequences do not converge. For example: 3, 3.1, 3.14, 3.141,3.1415, 3.14159,... The terms are getting closer and closer to each other, yet the "last" number in the sequence is pi, which is not a rational number. The reals are complete.

To further confuse you, the reals are divided into "algebraic" and "transcendental" numbers. The algebraic numbers are roots of polynomials with rational coefficients.

Answer 5

Rational numbers are a subset of real numbers.

More specifically, real numbers are the "closure" of rational numbers. Every real number is a limit of a sequence of rational numbers. We define the real numbers by using the rational numbers.

A rational number is "rational" because it is written as a "ratio." Every rational number can be written as p/q where p and q come from the integers {..., -3, -2, -1, 0, 1, 2, 3, ...} with q not equal to 0.

Thus, an irrational number is represented as an infinite sequence of rational numbers (i.e., ratios of integers). That's why numbers like pi have decimal expressions that are infinitely long. The decimal notation is a shorthand for a sum of many rational numbers, and each of those partial sums is itself a rational number. However, there is no rational number to represent pi. For every rational number there is always another rational number that is "closer" to pi. Thus, pi can only be expressed as a LIMIT of rational numbers. Every such number that can only be expressed as a limit of a rational number is known as an "irrational number" (i.e., not a ratio of integers).

Answer 6

A RATIOnal number is a number that can be expressed as a RATIO of two integers (p/q where q is not equal to zero). ALL rational numbers can be expressed as either repeating or terminating decimals.

(NOTE: An IRrational number cannot be expressed as such a ratio and therefore cannot ne expressed as a repeating or terminating decimal.

Numbers such as square root (2), e, pi, phi are examples of irrational numbers (irrational meaning NO ratio)

A real number is any number n such that n^2 is also real and must be greater than or equal to zero. This set of numbers consists of the union (combination) of the set of rationals and the set of irrationals.

NOTE Numbers such as the square root of (-1) are NOT real (After all when you square either a negative or a positive number you get a positive answer an the least value of the square of any real number is 0 as 0^2 = 0 These numbers form another set when combined with the set of real numbers called the complex numbers

Answer 7

Rational numbers are a subset of the real numbers. Real numbers are made up of rationals (numbers that can be expressed as p/q where p and q are integral) and irrationals eg the square root of two.

Answer 8

real numbers: the rational numbers and irrational numbers

rational numbers: a number which can be expressed in the form of a/b, where a and b are integers and b not equals to 0

integers = any whole number, whether in positive or negative forms

Answer 9

Rational numbers are a subset of real numbers.

Rational numbers can be expressed in terms of 2 integers p & q
as p / q.
eg, 2, 3.5, 3/5

Irrational numbers cannot be expressed as quotient of 2 integers.
ex sqrt of 3, pi = 3.14159...

Answer 10

real numbers = rational U irrational numbers.

Answer 11

the real numbers is the set consisting of all rational numbers and all irrational numbers