In mathematics, it can be said that real numbers are often used to measure various quantities, such as time, size, or the contrast of natural positive numbers. It includes a positive and negative number, as well as fractions. The fractions consist of irrational and rational numbers. While irrational numbers do not have decimal expansions to repeat themselves, rational numbers do. Hence, many might be confused if every rational number is an integer. Before being able to recognize which rational numbers are integers, we should know the definitions of integer, irrational and rational numbers.
Definition of an Integer
An integer is referred to as a whole number that is not a fraction number. It can be positive and negative, including zero. The integers are generated from the set of counting numbers and the operation of subtraction. As such, if the numbers are subtracted from themselves, they will become zero. Also, when the small numbers subtract large numbers, the result becomes negative. Thus, it can be seen that every integer can be derived from counting numbers, which will result in a set of numbers undergoing the operation of subtraction.
Meaning of Irrational Numbers
An irrational number is a number that cannot be written as a ratio of fractions for any integers. It is the number that does not have decimal expansions to repeat itself. In other words, its decimal form does not stop and does not repeat. Most square roots are irrational numbers because they cannot be expressed as fractions. However, there are some square roots of the perfect square that are rational numbers as they are always whole numbers, like √4 and √9.
The most famous examples of irrational numbers are √2, and π. To clarify, the √2 is approximately 1.414, but the exact value cannot be determined due to the digits coming after the decimal point being infinite. The same goes for the value of π. In 2018, π had been calculated to almost 22 billion digits and no pattern has yet to be found.
Meaning of Rational Numbers
In contrast to an irrational number, a rational number can express fractions of two integers or can be the result of a regular division equation. Even though the result is not a fraction but repeats infinitely, it can still be a rational number. Below are three qualifications that are used to identify rational numbers:
- It can be in a simple form of a fraction with a numerator and a denominator.
- Both numerators and denominators must be normal integers themselves.
- The denominator cannot be zero. However, the numerators can be zero, which the outcome is zero itself.
Furthermore, it should be noted that there is more than one way to address the same rational numbers as a ratio of integers. These are examples of rational numbers:
- 1/4 and 2/8 are the same rational number.
- It can be said that all integers are rational numbers. For example, the number 7 is considered to be a rational number as it can be rewritten as 7/1.
- All mixed numbers are also rational numbers. For instance, 3 1/5 is a rational number that can be rewritten as 16/5.
- All decimals with an end or that have repeating patterns are rational numbers. For example, the numbers 0.25 and 0.333… are rational numbers as they can be written as 1/4 and 1/3 respectively.
Examples of rational numbers that are not integers
It is found that all integers are rational numbers, but not every rational number is an integer. Here are good examples of a rational number that is not an integer due to its decimal number that has reoccurring, repeating decimal numbers:
- A decimal number like 0.363636 is a rational number, but it is not an integer because it has a repeating pattern.
- 0.44444, or 0.111111 are also rational numbers that are not integers because the decimal repeats itself infinitely.
- The numbers like 8/11, 2/3, and 5/7 are rational numbers that are not integers because their decimal forms are not in a repeating pattern and they are definite.
- A number like -75/-20 is a rational number that is not integer because even if it is simplified to 15/4, it still cannot be expressed without fractions or decimal terms.
The Importance of Being Able to Identify Rational Numbers
In the future, when you start to dive into more complex mathematics, being able to identify which numbers are irrational and which ones are rational is critical. For instance, in an algebraic calculation, there will be rules that you need to follow for you to get the right results. That also depends on whether you can differentiate irrational numbers from rational numbers. Therefore, knowing the basics of mathematics, such as understanding the rules and terms, will allow you to go further with a better comprehension of these systems.
To put it in the simplest way, the rational number that is not an integer will be in a fraction form in which both terminators and numerators are integers. May this guide be of any help to you in the future, good luck!