What is a rational number?
Rational numbers can be represented as a quotient of two whole numbers. They are expressed as a fraction a / b, where a and b are integers and b is different from zero.
Most individuals find it difficult to distinguish between simple fractions and rational numbers. Whole numbers make up fractions, whereas integers make up the numerator and denominator of rational numbers.
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What is the difference between rational and irrational numbers?
What are irrational numbers?
Irrational numbers are real numbers that are not rational numbers. The following are a few examples of commonly used irrational numbers:
- The number (pi) is irrational (Π = 3⋅14159265…) because the decimal value never comes to a halt.
- √2 is an irrational number. Consider a right-angled isosceles triangle with two equal sides of length, AB and BC. The hypotenuse AC will be √2=1.414213… according to Pythagoras’ theorem.
The difference between rational and irrational numbers
Irrational numbers are infinite and non-repeating, whereas rational numbers are finite and repeating decimals.
Rational numbers examples include:
- The number 9 can be expressed as 9/1, with both 9 and 1 being integers.
- In all terminating decimal forms, 0.5 can be written as 1/2, 5/10, or 10/20.
- √81 is a rational number since it can be reduced to 9.
- 0.7777777 is a rational number with recurring decimals.
Examples of irrational numbers:
- The denominator of 5/0 is zero, making it an irrational number.
- Π is an irrational number as it is a non-repeating, never-ending number.
- Because it cannot be simplified, the square root of 2 is an irrational number.
- Because it is non-recurring and non-terminating, 0.212112111…is an irrational number.
How to identify rational numbers
A rational number can be expressed as a fraction of integers. As a result, each of these figures is a rational figure. To determine whether a given number is rational, check if it meets any of the following criteria:
- The given number can be represented as a fraction of integers.
- We can determine whether the number’s decimal expansion is terminating or non-terminating.
- All rational numbers are whole numbers.
Types of rational numbers
Different sets of rational numbers exist. We should not assume that only rational numbers are fractions with integers. Here are the different 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 number is said to be positive. In other words, if the numerator and denominator of a rational number have the same sign, it is positive. Positive rational numbers include figures like 0.2, 6 or 2/5. In this case, 0.2 can be expressed as 1/5 and 6 as 6/1.
Negative rational numbers
If the numerator and denominator have a different sign (i.e., one is a positive integer and the other is a negative integer), the rational number is said to be negative.
For example, the following rational numbers are negative: -1/7, 4/5, -25/11, 10/19, -13/23 while these rational numbers are positive: -11/-14, 2/3, -3/-4, and 1/2.
Real Numbers
Real numbers include all rational numbers. A real number is one that can be discovered in the real world. Natural numbers are used to count things, rational numbers are used to represent fractions, irrational numbers are used to calculate the square root of a number, and integers are used to measure temperature, among other things. A collection of real numbers is made up of these many types of numbers.
Whole Numbers
Because every whole number can be written as a fraction, every whole number is a rational number. A set of numbers that includes all positive integers and 0 is known as a whole number. Whole numbers are fractions, decimals, and negative values that are not included in real numbers.
Rational numbers properties
The following are some of the most important properties of rational numbers. Let’s take a closer look at these characteristics whilst also exploring a list of rational numbers.
Closure Property
The closure property of rational numbers asserts that any two rational numbers added, subtracted, multiplied, or divided will produce a rational number in all four circumstances. Let’s look at how this property affects all of the basic arithmetic operations.
When two rational numbers are combined in any way together, the outcome is another rational number.
For example:
½ + ¾ = 10/8 or 5/4
½ – ¾ = -2/8 or -1/4
½ x ¾ = 3/8
½ ÷ ¾ = 2/3
All the above answers meet the criteria for a rational number.
Commutative Property
The commutative property of rational numbers states that adding or multiplying any two rational numbers in any order produces the same result. However, if the order of the numbers is changed in subtraction and division, the result will vary as well.
- With addition: 1/3 + 1/4 is the same as 1/4 + 1/3 as both equal 7/12. Thus, a + b is the same as b + a.
- With subtraction: 1/3 – 1/4 is not the same as 1/4 – 1/3. The first equals 1/12 while the second is -1/12. Therefore, a – b does not equal b – a.
- With multiplication: 1/3 x 1/4 and 1/4 x 1/3 both equal 1/12. Therefore, a x b is the same as b x a.
- With division: 1/3 ÷ 1/4 is not the same as 1/4 ÷ 1/3. The first equals 4/3, and the second equals 3/4. Thus a ÷ b is not the same as b ÷ a.
Associative Property
The associative characteristic of rational numbers asserts that no matter how numbers are grouped, the outcome remains the same when any three rational numbers are added or multiplied. However, if the order of the numbers is changed in subtraction and division, the result will vary as well.
- With addition: (1/3 + 1/4) + 1/2 is the same as 1/4 + (1/3 + 1/2). Both equal 13/12. Thus (a + b) + c = a + (b + c).
- With subtraction: (1/3 – 1/4) – 1/2 is not the same as 1/4 – (1/3 – 1/2). The first equals 1/24, the second equals 1/12. Thus (a – b) – c does not equal a – (b – c).
- With multiplication: (1/3 x 1/4) x 1/2 is equal to 1/4 x (1/3 x 1/2). Both present the answer of 1/24. Thus (a x b) x c = a x (b x c).
- With division: (1/3 ÷ 1/4) ÷ 1/2 is not the same as 1/4 ÷ (1/3 ÷ 1/2). The first equals 8/3, the second equals 2/3. Thus (a ÷ b) ÷ c does not equal a ÷ (b ÷ c).
Distributive Property
Any equation containing three rational numbers A, B, and C, given in the form A (B + C), is resolved as A (B + C) = AB + AC or A (B – C) = AB – AC, according to the distributive property of rational numbers. This signifies that operand A is shared by the other two operands, B and C. Multiplication distributivity over addition or subtraction is another name for this feature.
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