What is a Rational Number?
In Mathematics, numbers are categorized in various ways to better understand their properties and relationships. One fundamental category is that of rational numbers. Learn about rational numbers, their characteristics and their types. Understand the concept of rational numbers and their applications through mathematical examples.
Rational Number Definition – A rational number can be expressed as a ratio of two integers. They are categorized under Real Numbers.
In simpler terms -
Rational numbers are those numbers that can be expressed as a ratio between two integers. For example, the fractions 1/3 and -1111/8 are both rational numbers.
Numbers that cannot be written as a ratio of integers are called irrational.
Content
- Characteristics of Rational Numbers
- Rational Number Analogy
- Types of Rational Numbers
- Rational Number Examples
- Is Pi A Rational Number?
- How to Identify if a Number is Rational?
- Arithmetic Operations on Rational Numbers
- C++ Program For Adding Rational Numbers
Rational numbers can be represented as a/b, where a is the numerator, b is the denominator, and b ≠ 0. Suppose a = 2 and b = 4; then its representation would be –
Rational Number Analogy
Imagine you have a pizza. You can share that pizza with your friends. If you cut the pizza into equal pieces and have some of those pieces, like 1 piece or 2 pieces or even 3 pieces, those are like rational numbers.
A rational number is a fancy way of saying a number that can be made by dividing one whole number by another. The top number is called the ‘numerator’ (pieces of pizza you have), and the bottom number is called the ‘denominator’ (the number of parts the whole pizza is divided into).
So, if you have 1/2 of the pizza, that’s a rational number because you took 1 piece out of 2 equal pieces. If you have 3/4 of the pizza, that’s also rational because you took 3 out of 4 equal pieces.
But, if you have a strange number like the square root of 2 (about 1.41421356…), you can’t make it by dividing whole numbers, so it’s not rational. It’s called an ‘irrational’ number.
So, in simple words, a rational number is like a piece of pizza that you can get by dividing a pizza into equal parts, and both the top and bottom numbers are whole numbers.
In other words, rational numbers are real numbers that can be rewritten as the ratio of two integers because both the numerator and denominator are known.
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What is an Irrational Number?
An irrational number is a type of number that cannot be expressed as a ratio or fraction of two integers (a/b, where a and b are integers and b≠0). In decimal form, irrational numbers are unique because they:
- Do not terminate: Their decimal representation goes on and on.
- Do not repeat: There is no recurring pattern in their digits.
The discovery of irrational numbers traces back to ancient Greece, where mathematicians realized that certain equations could not be solved using rational numbers.
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Characteristics of Rational Numbers
Rational numbers have the following properties and characteristics:
- Infinite Set: There are infinite rational numbers. There is no limit to the number of fractions that can be generated.
- No First or Last Element: There is no smallest or largest rational number. For any rational number, another smaller or larger rational number exists.
- Density: Rational numbers form a dense set. Between any two rational numbers, many other rational numbers infinitely exist, meaning there are no immediate successors or predecessors, unlike integers.
- Not Continuous: Rational numbers are not a complete or continuous set on the number line. There are "gaps" between rational numbers occupied by irrational numbers, which cannot be expressed as fractions.
- Decimal Representation: Every rational number can be expressed in decimal form.
- Representation on the Number Line: Rational numbers can be represented as points on a number line, which demonstrates their density and order.
- Between two rational numbers, there are infinitely many rational numbers.
- Rational numbers are a superset that contains the integers, whole, and natural numbers
Types of Rational Numbers
The real numbers are divided into irrational and rational numbers. And rational numbers can be reduced to integers, whole, and natural numbers. There are two main types of rational numbers: integers and natural numbers.
Rational numbers can be categorized into the following types based on their forms:
Integers: Integers are whole numbers, which can be positive, negative, or zero.
Examples:
-
- Positive integers: 3,5,103, 5, 103,5,10
- Negative integers: −4,−2,−8-4, -2, -8−4,−2,−8
- Zero: 0
Integers are a subset of rational numbers because they can be expressed as fractions with a denominator of 1 (e.g., 3 = 3/1;−4 = −4/1 )
Fractions: Fractions are numbers expressed as a/b, where a (numerator) and b (denominator) are integers, and b≠0.
Examples:
-
- Positive fractions: 2/7, 3/4
- Negative fractions: −4/5,−9/11
Terminating Decimals: These are decimal numbers that end after a finite number of digits. They can be expressed as fractions.
Examples:
-
- 0.35=35/100
- 0.7116=7116/10000
- 0.9768=9768/10000
Non-Terminating Repeating Decimals: These are decimals that do not end but have a repeating pattern of digits after the decimal point. They can also be expressed as fractions.
Examples:
-
- 0.7777…=7/9 (digit "7" repeats)
- 0.6666667…=2/3 (digit "6" repeats)
- 0.141414…=14/99 (pattern "14" repeats)
Let us understand the above types of rational numbers through examples.
Rational Numbers Examples
There are infinite numbers so that we can make infinite fractions of whole numbers. However, we must know how to differentiate when a number is irrational.
Example 1
For example –
Is 8.75 a rational number?
Yes, because we can express it as a fraction:
Example 2
Example of a rational number
Is 2.718281828459047… a rational number?
No, because we cannot express it as a fraction:
Example 3
Example of an irrational number
Is 5.666666666666667 a rational number?
Yes, even though there are decimals and the series continues to infinity, it is a fraction:
Example 4
An example of a rational number
An interesting point to discuss is – Are all roots rational numbers?
The answer is that some roots are rational numbers, and some are irrational. For example, the square root of four is a rational number, but the square root of 93 is irrational.
Is Pi A Rational Number?
Pi is a mathematical expression. It has an approximate value is 3.14159365…
You can see that the value of π is expressed in decimal, which is non-terminating and non-repeating. Since the value of Pi is non-terminating, it is an irrational number.
Hence, Pi is not a rational number.
How to Identify if a Number is Rational?
Numbers in the form of numerators and denominators are rational numbers, but if the unknown is a decimal, we must check if it is rational or irrational. Check if a number or a set of numbers repeats in the decimal part of a number group. Below is the classification of these numbers.
- 0.857142857142857142857142 ⇒ Periodical = 857142 ← Periodical Pure
- 2.8 ⇒ Restricted
- 2.08333333333 ⇒ Periodical = 83 ← Periodical Mixed
Suppose you have an unknown as a decimal number, such as
- 0.243434343434…
- 123.187534883320…
- 6.01001000100001…
In such cases, we will check if the numbers repeat.
- Rational, Periodical = 34, Periodical Mixed
- Rational, Periodical = 187534, Periodical Pure
- Irrational
If there is repetition after a certain number, we can call it rational, but if there is no defined period, we can call it an irrational number.
Arithmetic Operations on Rational Numbers
Let us check how we can perform arithmetic operations on rational numbers, for example –
a/b and x/y.
Addition: To add a/b and x/y; we will use the following formula –
(ay+bx)/by.
Example: 1/4 + 4/5 ⇒ (1×5)+(4×4)/20 ⇒ 5+16/20 ⇒ 21/20
Subtraction: Similarly, when subtracting a/b and x/y; we will use the formula –
(ay-bx)/by
Example: 1/2 – 3/4 = (2-3)/4 = -1/4
3/4 – 1/3 ⇒ (3×3) – (4X1)/4X3 = 9 – 4/12 ⇒ 5/12
Multiplication: If a/b multiplies by x/y, then we get (a × x)/(b x y).
Example: 1/4 × 2/3 ⇒ (1×2)/(4×3) ⇒ 2/12 ⇒ 1/6
Division: If a/b is divided by x/y, then it is represented as:
(a/b)÷(x/y) = ay/bx
Example: 1/3 ÷ 4/5 = (1×5)/(3×4) = 5/12
C++ Program For Adding Rational Numbers
// Finding the sum of two fractions#include <iostream>using namespace std;int main() { // Write C++ code here float num_1, num_2, den_1, den_2; cout << "Enter the first fraction numerator and denominator"; cin>>num_1; cin>>den_1; cout << "Enter the second fraction numerator and denominator"; cin>>num_2; cin>>den_2;
float sum; sum = (num_1*den_2 + num_2*den_1)/(den_1*den_2); cout<<sum;
return 0;}
Output:
In a Nutshell
Rational numbers are all those that a mathematical fraction can express. We hope this article helped you understand the concept of rational numbers through examples.
FAQs
What is a rational number in math?
Any number that can be written as a ratio or fraction of two integers is rational.
What is an example of a rational number?
A rational number exists in the form of p/q where q not equal to 0. Any fraction that has non-zero denominators is a rational number. Some examples of rational numbers are 1/2, 1/4, 5/4, and so on.
Is two a rational number?
Yes, 2 is a rational number. It satisfies the condition for rational number, and we can write it in the p/q form, represented as 2/1.
How do you know if a number is rational?
A rational number can be written as a ratio or as a fraction. In such numbers, both the numerator and the denominator are whole numbers. The number 8 is rational because we can write it as a fraction, 8/1.
Which are irrational numbers?
An irrational number is a type of real number that cannot be represented as a simple fraction, i.e., if N is an irrational, then N cannot be expressed in the form of p/q where p and q are integers and q is not equal to 0. Example: pi, e (Euler number), 0.3435687697......
Is zero an irrational number?
No, zero is a rational number since, it can be expressed as 0/1 (i.e., p/q form).
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