Rational Numbers – Definition with Examples
Rational numbers are commonly used in engineering, physics, and finance. However, they aren’t necessary just for niche fields and scenarios; understanding rational numbers is essential for everyday life. Thus, kids need to understand the properties and applications of rational numbers.
What Are Rational Numbers?
A rational number is any number that can be expressed as the ratio of two integers where the denominator is not zero. Simple rational numbers definition is that rational numbers are fractions in which the numerators and denominators are both integers.
The examples of rational numbers include -5/3, 4/1, and -1/2. You can use rational numbers to calculate proportions in cooking recipes, divide quantities in construction projects, and interpret statistical data in surveys and polls.
Types of Rational Numbers
There are many rational numbers, but the most important are the positive and negative ones.
Positive and negative rational numbers
Positive rational numbers can be expressed as a fraction where the numerator and denominator are positive integers. An example of rational numbers that are positive is 3/4.
Negative rational numbers can be expressed as a fraction where the numerator is a negative integer and the denominator is a positive integer, or vice versa. For example, -2/3, -7/8, and 5/-6 are all negative rational numbers.
There are also types of rational numbers that kids may not explore at the beginning of studies, and here are some of them:
- Proper fractions
- Improper fractions
- Mixed numbers
- Terminating decimals
- Non-terminating decimals
- Repeating decimals
Standard form of rational numbers
The standard form of rational numbers expresses it as a fraction in the lowest terms where the numerator and denominator have no common factors other than 1. The numerator and denominator are divided by their most significant common factor (GCF) until no further simplification is possible. For example, 6/8 can be simplified to 3/4 by dividing the numerator and denominator by their GCF of 2.
An example of the standard form of rational numbers is as follows:
Convert 0.75 to a rational number in standard form. First, you can write it as a fraction with a denominator that is a power of 10:
0.75 = 75/100
Then simplify this fraction by dividing both the numerator and denominator by their greatest common factor, which is 25:
75/100 = (75 ÷ 25)/(100 ÷ 25) = 3/4
So the rational number 0.75 in standard form is 3/4.
How to Identify Rational Numbers?
Identifying rational numbers requires you first express it as a fraction of two integers, and here are a few steps on how you can get this done:
Step 1- Check if the number is a fraction:
Any number expressed as a fraction of two integers is rational. For example, 3/4, -7/8, and 1/2 are all rational numbers.
Step 2- Convert the decimal to a fraction:
Any terminating decimal can be expressed as a fraction by putting the decimal over 1 and simplifying it. For example, you can express 0.25 as 25/100 and then simplify it to 1/4, a rational number.
Step 3- Convert the repeating decimal to a fraction:
Any repeating decimal can be expressed as a fraction by writing the repeating digits as the numerator and a number with the same number of digits as the repeating block as the denominator. For example, you can write 0.666… as 2/3, a rational number.
Step 4- Check if the number is an integer:
Any integer is a rational number since it can be expressed as a fraction with a denominator of 1. 5 is a rational number since you can write it as 5/1.
Arithmetic Operations with Rational Numbers
With rational numbers, you can perform four basic arithmetic operations: addition, subtraction, multiplication, and division. But before performing these operations, you must first express them in the same form or with a common denominator.
Here are the steps for each of these operations:
Addition and subtraction
Adding rational numbers or subtracting rational numbers starts with finding a common denominator. The common denominator is the lowest common multiple (LCM) of the denominators of the given fractions. Once you have a common denominator, you can add or subtract the numerators and write the result over the common denominator. Then, you simplify the fraction to its lowest terms.
For example, if you need to add 2/3 to 1/4, here is how it will go:
First, find the LCM of 3 and 4, which is 12.
Then, convert the fractions to have a denominator of 12. And you can get this by multiplying the numerator and denominator by 4 to get 2/3 = 8/12.
Then multiply both again, but this time by 3
1/4 = 3/12
Add the fractions.
8/12 + 3/12 = 11/12
Usually, you would have to simplify the result, but this one is already in its simplest form.
Multiplying rational numbers
When multiplying rational numbers, you need to individually multiply the numerators and denominators and then simplify the resulting fraction to its lowest terms if possible.
For example, if you need to multiply 2/3 and 3/4.
You do this: 2/3 * 3/4 = 6/12
Simplify 6/12 to 1/2.
Dividing rational numbers
When you want to divide rational numbers, you first multiply the first fraction by the reciprocal of the second fraction. You can find the reciprocal of the second fraction by flipping it upside down (interchanging the numerator and denominator) and then multiplying it with the first fraction.
For example, if you want to divide 2/3 by 3/4.
You do this:
2/3 ÷ 3/4 = 2/3 * 4/3 = 8/9
8/9 is already in its simplest form.
Multiplicative Inverse of Rational Numbers
The multiplicative inverse or reciprocal of a rational number is a value that, when multiplied by the original rational number, gives 1. For instance, the reciprocal of 3/4 is 4/3 because 3/4 multiplied by 4/3 gives 1. To get the reciprocal of a rational number, you swap the numerator and denominator.
Rational Numbers Properties
Here are some of the properties of rational numbers:
The closure property principle states that any two rational numbers’ sum, difference, product, and quotient is always a rational number.
The principle of associative property states that the sum and product of three or more rational numbers are the same regardless of how the numbers are grouped.
According to the commutative property, the sum and product of two rational numbers are the same regardless of the order in which the numbers are added or multiplied.
According to the distributive property, the product of a rational number and the sum or difference of two other rational numbers equals the sum or difference of the products of the first number and the other two numbers.
Identity property states that the sum of a rational number and zero equals the original rational number, and the product of a rational number and 1 equals the original rational number.
According to the inverse property, every non-zero rational number has a multiplicative inverse or reciprocal, which, when multiplied by the original number, results in the value of 1.
Rational Numbers and Irrational Numbers
Rational and irrational numbers are two types of real numbers that can be shown on a number line. Rational numbers are made by dividing two integers and can be positive or negative. Irrational numbers cannot be divided into two integers and cannot be written as repeating or ending decimals. The examples of irrational numbers include the square root of 2, pi, and the golden ratio.
Rational numbers can be written as decimals that end or repeat, but irrational numbers cannot. Both rational and irrational numbers together make up the set of real numbers.
Rational numbers examples include 1/2, 3/4, 2/3, -5/6, 7/8. Irrational number examples include the Euler-Mascheroni constant (γ) and the golden ratio (φ).
Ordering Rational Numbers
To order rational numbers, follow these steps:
- First, convert them to a common denominator.
- Then, arrange them based on their numerators.
- Compare denominators for ties.
For example, to order 2/3, 5/6, 1/2, and 7/12 from the least to greatest, convert them to a common denominator of 12, arrange them based on their numerators, and compare denominators for ties to get 1/2, 2/3, 7/12, 5/6.
Frequently Asked Questions on Rational Numbers
What is a rational number in math?
A rational number is any number that can be expressed as the ratio of two integers where the denominator is not zero. A rational numbers list include 5/6, 6/7, 7/8, 8/9.
How to identify a rational number?
You can identify a rational number by expressing it as a fraction of two integers.
What are terminating rational numbers?
A terminating rational number is a rational number that can be expressed as a decimal with a finite number of digits after the decimal point.
What is the difference between rational and irrational numbers?
Rational numbers are those numbers that can be expressed as the ratio of two integers, while irrational numbers cannot.
What are irrational numbers?
Irrational numbers are numbers that cannot be divided into two integers.
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