InstructorPriyanka Mishra
TypeOnline Course
DateSep 2, 2016
PriceFree

In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number. The set of all rational numbers, often referred to as “the rationals“, is usually denoted by a boldface Q  for “quotient”.

The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number.

A real number that is not rational is called irrational. Irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating.

Zero divided by any other integer equals zero; therefore, zero is a rational number (but division by zero is undefined).

### Embedding of integers

Any integer n can be expressed as the rational number n/1.

### Equality

if and only if a d = b c .

### Ordering

Where both denominators are positive:

if and only if

If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations:

and

Two fractions are added as follows:

### Multiplication

The rule for multiplication is:

### Division

Where c ≠ 0:

Note that division is equivalent to multiplying by the reciprocal of the divisor fraction:

### Inverse

Additive and multiplicative inverses exist in the rational numbers:

### Exponentiation to integer power

If n is a non-negative integer, then

and (if a ≠ 0):

## Continued fraction representation

Main article: Continued fraction

A finite continued fraction is an expression such as

where an are integers. Every rational number a/b has two closely related expressions as a finite continued fraction, whose coefficients an can be determined by applying the Euclidean algorithm to (a,b).

Section 1Introduction to Number SystemFree Preview
Section 2Solution of Exercises