This article covers the definition of a rational number, the different types of rational numbers, and the distinction between rational and irrational numbers. It also provides various examples with solutions. Prior to putting rational numbers on a number line, we must first simplify and write them in decimal form.
What is a rational number?
The term “rational” is derived from the word “ratio”. A rational number is a type of number that can be expressed as the fraction (p/q) or quotient of two integers:
- A numerator (p)
- A non-zero denominator (q)
After integers, rational numbers are among the most often studied categories of numbers in mathematics. These numbers have the form p/q, where p and q are both integers but q is not allowed to be zero. Most people struggle to discern between simple fractions and rational numbers. While integers constitute the numerator and denominator of rational numbers, whole numbers make up fractions.
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What are the properties of rational numbers?
The Closure Property
According to the closure property of rational numbers, any two rational numbers added together, subtracted from, multiplied by, or split into will result in a rational number. Any combination of two rational numbers results in the creation of another rational number. For instance:
½ + ¾ = 10/8 or 5/4
½ – ¾ = -2/8 or -1/4
½ x ¾ = 6/8
½ ÷ ¾ = ⅔
The Commutative Property
According to this property, any two rational numbers can be added together or multiplied by one another in any order. However, subtraction and division results will differ if the order of the numbers is altered:
- When adding, 1/3 plus 1/4 is the same as 1/4 plus 1/3 because they both equal 7/12. A + B is equivalent to B + A.
- In terms of subtraction, 1/3 – 1/4 is distinct from 1/4 – 1/3. The first equals 1/12 while the second equals – 1/12. Therefore, a – b is not equal to b – a.
- When multiplied, the results of 1/3 x 1/4 and 1/4 x 1/3 are both 1/12. A x B is therefore equivalent to B x A.
- In terms of division, 1/4÷1/3 is distinct from 1/3÷1/4. The first equals 4/3, whereas the second equals 3/4. As a result, a/b and b/a are not the same.
The Associative Property
According to this property, any three rational numbers can be added or multiplied regardless of how they are grouped. However, subtraction and division results will differ if the order of the numbers is altered:
- With addition, 1/4 + (1/3 + 1/4) + 1/2 equals 1/3 + 1/2. Both add up to 13/12. Consequently, (a + b) + c = a + (b + c)
- In the case of subtraction, 1/4 – (1/3 – 1/2) is not the same as 1/3 – 1/2. First is equivalent to 1/24; second is equal to 1/12. As a result, (a – b) – c does not equal (a – (b – c).
- In terms of multiplication, 1/4 is equivalent to (1/3 x 1/4) x 1/2. Both equal 1/24. As a result, (a x b) x (a x) (b x c).
- For division, 1/4 (1/3 1/2) is different from 1/3 (1/4 (1/3 1/2). The first one is equivalent to 8/3 and the second to 2/3. As a result, (a b) c does not equal (a b c).
The Distributive Property
The distributive property of rational numbers states that any equation comprising three rational numbers A, B, and C, written in the form A (B + C), is resolved as A (B + C) = AB + AC or A (B – C) = AB – AC. This shows that the other two operands, B and C, share operand A. Another name for this property is multiplicative distributivity over addition or subtraction.
Types of Rational Numbers
Positive Rational Numbers
If the numerator and denominator of a rational number are both positive integers or both negative integers, the rational number is positive. In other terms, a rational number is positive if its numerator and denominator both have the same sign. Examples of positive rational numbers include 0.2, 6, and 2/5.
Negative Rational Numbers
A rational number is negative if the numerator and denominator have opposite signs (i.e., one is a positive integer and the other is a negative integer). Examples of negative rational numbers include -1/7, 4/5, -25/11, 10/19, and -13/23 are all negative.
Integer Form of Rational Numbers
All the integers are considered rational numbers because they can be written in the form of. As we know that all integers are rational numbers because we can write them in the form of (p/q). For example, 6 can be written as 6/1.
Decimal Form of Rational Numbers
Non-terminating as well as terminating decimal numbers are considered rational numbers. For example, 0.3 is a terminating decimal number that can be written as 3/10 and 0.3333…. Is a non-terminating decimal number that can be written as 1/3.
What are terminating and repeating decimals?
How to identify rational numbers
To determine if a number is a rational number check if it meets the following requirements:
- A percentage of integers can be used to represent the given number.
- The decimal growth of the number can be classified as terminating or non-terminating.
- Whole numbers are the only rational numbers.
Is the number 0.923076923076923076 rational? The integer given has a series of decimals (923076) that never ends so it is not a rational number.
Below are some rational number examples:
- The number 9 can be expressed as 9/1, with both 9 and 1 being integers.
- 0.5 can be written as 1/2, 5/10, or 10/20 as it is a terminating decimal.
- √81 is a rational number since it can be reduced to 9.
- 0.7777777 is a rational number with recurring infinite numbers after the decimal.
Standard form of rational numbers
In general, it is possible to claim that x/y is a rational number in its standard form. However, the numerator and denominator cannot share any factors other than 1. In addition, the denominator, which in this case is y must always be positive.
Positive and negative rational numbers
When the numerator and denominator of a number are both positive or both negative, the number is said to be a positive rational number. For example, 3/8, 9/10 and -34/-40 are considered positive rational numbers whereas -4/15, 5/-6 and -17/19 are negative rational numbers.
Arithmetic operations on rational numbers
Mathematical operations on two or more rational numbers are referred to as operations on rational numbers. A rational number has the form p/q, where p and q are both integers and q is greater than zero. Rational numbers include 1/2, 3/4, 0.3 (or 3/10), 0.7 (or 7/10), etc.
With rational numbers, there are four fundamental arithmetic operations:
- Addition
- Subtraction
- Multiplication
- Division
Difference between rational and irrational numbers
Irrational numbers are real numbers that cannot be classified as rational numbers. In contrast to rational numbers, irrational numbers cannot be represented precisely in fraction form. In other words, rational numbers can be represented as exact fractions, while irrational numbers cannot.
Below are some characteristics of irrational numbers:
- Irrational numbers comprise non-terminal and non-recurring decimals.
- Two irrational numbers may not have a least common multiple (LCM).
- The result of the addition, subtraction, and multiplication or irrational numbers can sometimes be a rational number.
What are all the irrational numbers?
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