In mathematics, a rational number is any number that can be expressed as a ratio of two integers, where the denominator is not zero. The term "rational" comes from the word "ratio." Examples of rational numbers include \(1 \over {2} \), \( 3 \over {4} \), \( 5 \over {6}\), and so on.
Identifying Rational Numbers
There are four types of rational numbers:
Rational numbers can be identified by looking for fractions or decimals that terminate or repeat. Terminating decimals are decimals that have a finite number of digits after the decimal point, such as 0.25, 0.75, 1.5, and so on. Repeating decimals are decimals that have a repeating pattern of digits after the decimal point, such as 0.3333..., 0.55555..., 0.121212..., and so on.
Rational numbers can be represented on a number line. The number line is a line that represents all real numbers, with positive numbers to the right of 0 and negative numbers to the left of 0. Rational numbers are marked by dots on the number line, and they can be plotted between whole numbers. For example, the rational number 1.5 or \(1 \frac{1}{2}\) can be plotted between 1 and 2.
Examples of Rational Numbers
Let's look at some examples of rational numbers.
\(3 \over 4\) - This is a fraction that can be simplified, and it represents a rational number.
0.5 - This is a decimal that terminates, so it represents a rational number.
0.6666... - This is a repeating decimal, which represents a rational number. It can be written as \(2\over 3\).
\(-2\over 3\) - This is a negative fraction that can be simplified, so it represents a rational number.
2 - This is a positive integer, that can be expressed as \(2 \over 1\), so it is a rational number
Rational numbers are an important concept in mathematics. They are numbers that can be expressed as a ratio of two integers and can be identified by looking for fractions or decimals that terminate or repeat.
irrational numbers are numbers that cannot be expressed as a ratio of two integers. Unlike rational numbers, they cannot be written as a fraction in which both the numerator and the denominator are integers. Irrational numbers are usually expressed as decimal expansions that neither terminate nor repeat.
Some examples of irrational numbers include:
One important fact about irrational numbers is that they are not countable, meaning that there is no way to list all of them in a sequence. In contrast, the rational numbers are countable because they can be listed in a sequence, for example by listing all the fractions in order of increasing magnitude. This means that there are more irrational numbers than there are rational numbers.
Irrational numbers can be plotted on the number line, but their exact value cannot be represented. They are usually approximated by decimal expansions or by using mathematical symbols such as square roots or pi.
