Rational number

Number-systems (NZQRC)

Rational Representation

Diagonal argument

Rational number refers to any number that can be expressed as the quotient or fraction \(\frac{p}{q}\) 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 denoted by the symbol \(\mathbb{Q}\), which stands for "quotient".

Definition

A rational number is defined as a number that can be written in the form \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q \neq 0\). The numerator \(p\) represents how many parts of a certain size there are, while the denominator \(q\) represents the size of each part. For example, the rational number \(\frac{3}{4}\) represents 3 parts of a size that is 1/4th of a whole.

Properties

Rational numbers have several important properties:

• They can be positive, negative, or zero.
• They are closed under addition, subtraction, multiplication, and division (except division by zero).
• They can be represented as decimals, but unlike irrational numbers, rational numbers either terminate after a finite number of digits or begin to repeat a sequence of digits infinitely.

Comparison

Rational numbers can be compared by converting them to a common denominator and then comparing the numerators. For example, to compare \(\frac{1}{2}\) and \(\frac{3}{4}\), one could convert them to \(\frac{2}{4}\) and \(\frac{3}{4}\), respectively, and then observe that \(\frac{2}{4} < \(\frac{3}{4}\).

Operations

The basic arithmetic operations on rational numbers are as follows:

• Addition: To add two rational numbers, convert them to a common denominator and then add the numerators.
• Subtraction: Similar to addition, to subtract one rational number from another, convert them to a common denominator and then subtract the numerators.
• Multiplication: To multiply two rational numbers, multiply their numerators and denominators separately.
• Division: To divide one rational number by another, multiply the first by the reciprocal of the second.

Decimal Representation

Rational numbers can be represented as either terminating or repeating decimals. A terminating decimal has a finite number of digits after the decimal point, while a repeating decimal has one or more digits that repeat infinitely. Every rational number can be converted into a decimal form by dividing its numerator by its denominator.

Rational Numbers and the Real Number System

Rational numbers are a subset of the real numbers. However, not all real numbers are rational. Numbers that cannot be expressed as a fraction of two integers are called irrational numbers. The real number system is composed of both rational and irrational numbers.

Applications

Rational numbers are used in various fields, including mathematics, engineering, science, and finance. They are essential for measuring, calculating proportions, and performing quantitative analysis.