A rational number is a number that can be expressed as a fraction where and are integers and . A rational number is said to have numerator and denominator . Numbers that are not rational are called irrational numbers. The real line consists of the union of the rational and irrational numbers. The set of rational numbers is of measure zero on the real line, so it is “small” compared to the irrationals and the continuum.
The set of all rational numbers is referred to as the “rationals,” and forms a field that is denoted . Here, the symbol derives from the German word Quotient, which can be translated as “ratio,” and first appeared in Bourbaki's Algèbre (reprinted as Bourbaki 1998, p. 671).
Any rational number is trivially also an algebraic
number.
Examples of rational numbers include , 0, 1, 1/2, 22/7, 12345/67, and so on. Farey sequences provide a way of systematically enumerating all rational numbers.
The set of rational numbers is denoted Rationals in the Wolfram Language, and a number can be tested to see if it is rational using the command Element[x, Rationals].
The elementary algebraic operations for combining rational numbers are exactly the
same as for combining fractions.
It is always possible to find another rational number between any two members of the set of rationals. Therefore, rather counterintuitively, the rational numbers are a continuous set, but at the same time countable.
For , , and any different rational numbers, then
is the square of the rational number
(Honsberger 1991).
The probability that a random rational number has an even denominator is 1/3 (Salamin and Gosper 1972).
It is conjectured that if there exists a real number for which both and are integers, then is rational. This result would follow from the four exponentials conjecture (Finch 2003).
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Rational Numbers – Definition, Types, Properties & Examples
In Maths, rational numbers are represented in p/q form where q is not equal to zero. It is one of the most important Maths topics.
Any fraction with non-zero denominators is a rational number. Hence, we can say that ‘0’ is also a rational number, as we can represent it in many forms such as 0/1, 0/2, 0/3, etc. But, 1/0, 2/0, 3/0, etc.
are not rational. Also, learn irrational numbers here.
We will learn here the properties of rational numbers along with with its types, the difference between rational and irrational numbers. Solved examples help to understand the concepts in a better way.
Also, learn the various rational numbers examples and learn how to find the rational numbers in a better way.
To represent rational numbers on a number line, we need to simplify and write in decimal form first.
Let us see what topics we are going to cover here in this article.
Table of contents:
Rational Number Definition
A rational number, in Mathematics, can be defined as any number which can be represented in the form of p/q where q is greater than 0. Also, we can say that any fraction fit under the category of rational numbers, where denominator and numerator are integers and the denominator is not equal to zero.
How to identify rational numbers?
To identify if a number is rational or not, check the below conditions.
- It is represented in the form of p/q, where q≠0.
- The ratio p/q can be further simplified and represented in decimal form.
The set of rational numerals are:
- Include positive, negative numbers, and zero
- Can be expressed as a fraction
Examples of Rational Numbers:
p | q | p/q | Rational/Irrational Number |
10 | 2 | 10/2 =5 | Rational |
1 | 1000 | 1/1000 = 0.001 | Rational |
50 | 10 | 50/10 = 5 | Rational |
7 | 7/0 | Irrational (because q=0) |
Types of Rational Numbers
A number is rational if we can write it as a fraction, where both denominator and numerator are integers.
Below diagram helps us to understand more about the number sets.
- Real numbers (R) include all the rational numbers (Q).
- Real numbers include the integers (Z).
- Integers involves the natural numbers(N).
- Every whole number is a rational number because every whole number can be expressed as a fraction.
Standard Form of Rational Numbers
The standard form of a rational number can be defined if it’s no common factors aside from one between the dividend and divisor and therefore the divisor is positive.
For Example, 12/36 is a rational number. But it can be simplified as 1/3; common factors between the divisor and dividend is only one. So we can say that rational number ⅓ is in standard form.
Positive and Negative Rational Numbers
Positive Rational NumbersNegative Rational Numbers
If both the numerator and denominator are of same signs. | If numerator and denominator are of opposite signs. |
All are greater than 0 | All are less than 0 |
Example: 12/17, 9/11 and 3/5 are positive rational numbers | Example: -2/17, 9/-11 and -1/5 are negative rational numbers |
Arithmetic Operations on Rational Numbers
In Maths, arithmetic operations are the basic operations we perform on integers. Let us discuss here how can we perform these operations on rational numbers, say p/q and s/t.
Addition: When we add p/q and s/t, we need to make the denominator same. Hence, we get (pt+qs)/qt.
- Example: 1/2 + 3/4 = (2+3)/4 = 5/4
- Subtraction: Similarly, if we subtract p/q and s/t, then also, we need to make the denominator same, first, and then do the subtraction.
- Example: 1/2 – 3/4 = (2-3)/4 = -1/4
Multiplication: In case of multiplication, while multiplying two rational numbers, the numerator and denominators of the rational numbers are multiplied, respectively. If p/q is multiplied by s/t, then we get (p×s)/(q×t).
- Example: 1/2 × 3/4 = (1×3)/(2×4) = 3/8
- Division: If p/q is divided by s/t, then it is represented as:
(p/q)÷(s/t) = pt/qs - Example: 1/2 ÷ 3/4 = (1×4)/(2×3) = 4/6 = 2/3
Rational Numbers Properties
- The result of two rationals is always a rational number is we multiply, add or subtract them.
- A rational number remains the same is we divide or multiply both numerator and denominator with the same number.
- Sum of zero and a rational number revert the same number itself.
Learn more properties of rational numbers here.
Rational Numbers and Irrational Numbers
There is a difference between rational and Irrational Numbers. A fraction with non-zero denominators is called a rational number. The number ½ is a rational number because it is read as integer 1 divided by the integer 2. All the numbers that are not rational are called irrational.
Rationals can be either positive, negative or zero. While specifying a negative rational number, the negative sign either in front or with the numerator and that is the standard mathematical notation. For example, we denote negative of 5/2 as -5/2.
An irrational number cannot be written as a fraction but can be written as a decimal. It has endless non-repeating digits after the decimal point. Some of the irrational numbers are mentioned below.
π = 3.142857…
√2 = 1.414213…
Rational Numbers Examples
- Example 1:
- Identify each of the following as irrational or rational: ¾ , 90/12007, 12 and √5.
- Solution:
Since a rational number is the one that can be expressed as a ratio. This indicates that it can be expressed as a fraction wherein both denominator and numerator are whole numbers.
- ¾ is a rational number as it can be expressed as a fraction.
- Fraction 90/12007 is rational.
- 12, also be written as 12/1. Again a rational number.
- Value of √5 = 2.2360…. , does not end. Cannot be written as a fraction. It is an irrational number.
- Example 2:
- Identify whether mixed fraction, 1½ is a rational number.
- Solution:
- The Simplest form of 1½ is 3/2
- Numerator = 3, which is an integer
- Denominator = 2, is an integer and not equal to zero.
- So, yes, 3/2 is a rational number.
- Example 3:
- Determine whether the given numbers are rational or irrational.
(a) 1.75 (b) 0.01 (c) 0.5 (d) 0.09 (d) √ 3
Solution:
The given numbers are in decimal format. To find whether the given number is decimal or not, we have to convert it into the fraction form (i.e.,) p/q
If the denominator of the fraction is not equal to zero, then the number is rational, or else, it is irrational.
Decimal Number | Fraction | Rational Number |
1.75 | 7/4 | yes |
0.01 | 1/100 | yes |
0.5 | 1/2 | yes |
0.09 | 1/11 | yes |
√ 3 | ? | No |
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If a number that can be expressed in the form of p/q is called rational numbers.
Here p and q are integers, and q is not equal to 0. A rational number should have a numerator (p) and denominator (q). Examples: 10/2, 30/3, 100/5.
A rational number is a number that is expressed as the ratio of two integers, where the denominator value should not be equal to zero, whereas an irrational number cannot be expressed in the form of fractions. Example of the rational number is 10/2, and for an irrational number is a famous mathematical value Pi(π) which is equal to 3.14.
Yes, 0 is a rational number because it is an integer, that can be written in any form such as 0/1, 0/2, where b is a non zero integer. It can be written in the form: p/q = 0/1. Hence, we conclude that 0 is a rational number.
Integers and rational numbers (Algebra 1, Exploring real numbers) – Mathplanet
Natural numbers are all numbers 1, 2, 3, 4… They are the numbers you usually count and they will continue on into infinity.
Whole numbers are all natural numbers including 0 e.g. 0, 1, 2, 3, 4…
Integers include all whole numbers and their negative counterpart e.g. …-4, -3, -2, -1, 0,1, 2, 3, 4,…
- All integers belong to the rational numbers. A rational number is a number
- $$frac{a}{b},: b
eq 0$$ - Where a and b are both integers.
- Example
The number 4 is an integer as well as a rational number. As it can be written without a decimal component it belongs to the integers. It is a rational number because it can be written as:
- $$frac{4}{1}$$
- or
- $$frac{8}{2}$$
- or even
- $$frac{-8}{-2}$$
- Whereas
- $$frac{1}{5}=0.2$$
- is a rational number but not an integer.
- A rational number written in a decimal form can either be terminating as in:
- $$frac{1}{5}=0.2$$
- Or repeating as in
$$frac{5}{6}=0.83333…$$
All rational numbers belong to the real numbers.
If you look at a numeral line
You notice that all integers, as well as all rational numbers, are at a specific distance from 0. This distance between a number x and 0 is called a number's absolute value. It is shown with the symbol
$$left | x
ight |$$
If two numbers are at the same distance from 0 as in the case of 10 and -10 they are called opposites. Opposites have the same absolute value since they are both at the same distance from 0.
$$left | 10
ight |=10=left | -10
ight |$$
Video lesson
Which of these rational numbers are integers?
What is rational number? – Definition from WhatIs.com
A rational number is a number determined by the ratio of some integer p to some nonzero natural number q. The set…
of rational numbers is denoted Q, and represents the set of all possible integer-to-natural-number ratios p/q.
In mathematical expressions, unknown or unspecified rational numbers are represented by lowercase, italicized letters from the late middle or end of the alphabet, especially r, s, and t, and occasionally u through z.
Rational numbers are primarily of interest to theoreticians.Theoretical mathematics has potentially far-reaching applications in communications and computer science, especially in data encryption and security.
If r and t are rational numbers such that r < t, then there exists a rational number s such that r < s < t. This is true no matter how small the difference between r and t, as long as the two are not equal.In this sense, the set Q is “dense.”Nevertheless, Q is a denumerable set.
Denumerability refers to the fact that, even though a set might contain an infinite number of elements, and even though those elements might be “densely packed,” the elements can be defined by a list that assigns them each a unique number in a sequence corresponding to the set of natural numbers N = {1, 2, 3, …}..
For the set of natural numbers N and the set of integers Z, neither of which are “dense,” denumeration lists are straightforward.For Q, it is less obvious how such a list might be constructed.An example appears below.The matrix includes all possible numbers of the form p/q, where p is an integer and q is a nonzero natural number.
Every possible rational number is represented in the array.Following the pink line, think of 0 as the “first stop,” 1/1 as the “second stop,” -1/1 as the “third stop,” 1/2 as the “fourth stop,” and so on.This defines a sequential (although redundant) list of the rational numbers.
There is a one-to-one correspondence between the elements of the array and the set of natural numbers N.
To demonstrate a true one-to-one correspondence between Q and N, a modification must be added to the algorithm shown in the illustration.Some of the elements in the matrix are repetitions of previous numerical values.For example, 2/4 = 3/6 = 4/8 = 5/10, and so on.
These redundancies can be eliminated by imposing the constraint, “If a number represents a value previously encountered, skip over it.”In this manner, it can be rigorously proven that the set Q has exactly the same number of elements as the set N.
Some people find this hard to believe, but the logic is sound.
In contrast to the natural numbers, integers, and rational numbers, the sets of irrational numbers, real numbers, imaginary numbers, and complex numbers are non-denumerable. They have cardinality greater than that of the set N.This leads to the conclusion that some “infinities” are larger than others!
What Are Rational Numbers? Magoosh Math: Learn About Types of Numbers
What are rational numbers? Rational numbers are numbers that can be expressed as a fraction. In this fraction, both the numerator and denominator must be integers and the denominator cannot be 0. Let’s look at some examples of rational numbers to help you better classify them.
Natural Numbers
Natural numbers are numbers used for counting and ordering. They are positive integers, beginning with 1. Therefore, some examples include: 1, 2, 3, 4, 5 . . .
Whole Numbers
Whole numbers include all natural numbers with one big difference: it also includes 0. So, when you’re listing whole numbers, you could say 0, 1, 2, 3, 4 . . .
Integers
Integers include all natural and whole numbers. The thing that sets integers apart from these first two groups is that they also include negative numbers. Since integers are rational numbers, you can be assured that the negative counterparts of natural and whole numbers are considered rational, too. Examples of integers include . . . -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 . . .
Proper Fractions
Proper fractions are numbers that are less than 1. Examples of fractions include
You can think of these numbers as parts of a whole, such as a slice of pie. And, naturally, these numbers are expressed as fractions, so they easily fit the bill as rational numbers.
Improper Fractions
Proper fractions are numbers less than 1, and improper fractions are numbers that are greater than 1. Maybe you have an entire pie plus an addition half of a pie available. In this case, you could write the fraction as: Other examples of improper fractions include:
Mixed Fractions
You can convert improper fractions into mixed fractions. Mixed fractions include both proper fractions and whole numbers. If we look at the examples from the improper fractions, we can turn these into mixed numbers:
Decimal Numbers
Decimal numbers also express values less than 1. Maybe the baby weighed 6.5 pounds at birth. The decimal tells us that the baby weighed an extra half of a pound.
In between the numbers is a decimal point, which is similar to a period. This tells us that all the numbers after the decimal point are a percentage of the whole.
And, of course, this is an integer that could be written as a fraction: which means this would still be considered a rational number.
What is a Rational Number?: Definition, Properties, Videos and Examples
What is a Rational Number? When someone asks you about your age, you may say you are 15 years old. The date, the number of pages in a book, the fingers on your hand. What numbers are these? These numbers are something known as rational numbers. Let us study in detail about rational numbers and their properties.
Suggested Videos
What is a Rational Number?
We already know about some types of numbers. The numbers that we are familiar with are natural numbers, whole numbers, and integers. Natural numbers are the ones that begin with 1 and goes on endlessly up to plus infinity. If we include 0 in these sets of numbers, then these numbers become whole numbers.
Now in these sets, if we also include the negative numbers, then we call it as integers. So all the numbers that we see collectively on the number line are called integers. But what is a rational number?
A rational number is a number that can be written in the form of a numerator upon a denominator. Here the denominator should not be equal to 0. The numerator and the denominator will be integers. A rational number is of the form
( frac{p}{q} )
p = numerator, q= denominator, where p and q are integers and q ≠0
Examples: ( frac{3}{5} ), ( frac{-3}{10} ), ( frac{11}{-15} ). Here we can see that all the numerators and denominators are integers and even the denominators should be non-zero.
Browse more Topics under Number Systems
Positive and Negative Rational Numbers
Rational Number
A rational number is any number that can be expressed as a ratio of two integers (hence the name “rational”). It can be written as a fraction in which the the top number (numerator) is divided by the bottom number (denominator).
All integers are rational numbers since they can be divided by 1, which produces a ratio of two integers. Many floating point numbers are also rational numbers since they can be expressed as fractions. For example, 1.5 is rational since it can be written as 3/2, 6/4, 9/6 or another fraction or two integers. Pi (π) is irrational since it cannot be written as a fraction.
A floating point number is rational if it meets one of the following criteria:
- it has a limited number of digits after the decimal point (e.g., 5.4321)
- it has an infinitely repeating number after the decimal point (e.g., 2.333333…)
- it has an infinitely repeating pattern of numbers after the decimal point (e.g. 3.151515…)
If the numbers after the decimal point repeat infinitely with no pattern, the number is not rational or “irrational.” Below are examples of rational and irrational numbers.
- 1 – rational
- 0.5 – rational
- 2.0 – rational
- √2 – irrational
- 3.14 – rational
- π (3.14159265359…) – irrational
- √4 – rational
- √5 – irrational
- 16/9 – rational
- 1,000,000.0000001 – rational
In computer science, it is significant if a number is rational or irrational. A rational number can be stored as an exact numeric value, while an irrational number must be estimated.
NOTE: The number zero (0) is a rational number because it can be written as 0/1, which equals 0.
Updated: June 5, 2018
Cite this definition:
https://techterms.com/definition/rational_number
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Rational Numbers
A Rational Number can be made by dividing two integers.
(An integer is a number with no fractional part.)
1.5 is a rational number because 1.5 = 3/2 (3 and 2 are both integers)
Most numbers we use in everyday life are Rational Numbers.
Here are some more examples:
Number As a Fraction Rational?5 | 5/1 | Yes |
1.75 | 7/4 | Yes |
.001 | 1/1000 | Yes |
−0.1 | −1/10 | Yes |
0.111… | 1/9 | Yes |
√2 (square root of 2) |
? | NO ! |
Oops! The square root of 2 cannot be written as a simple fraction! And there are many more such numbers, and because they are not rational they are called Irrational.
Another famous irrational number is Pi (π):
Formal Definition of Rational Number
More formally we say:
A rational number is a number that can be in the form p/q
where p and q are integers and q is not equal to zero.
So, a rational number can be:
Where q is not zero
Examples:
p q p / q =1 | 1 | 1/1 | 1 |
1 | 2 | 1/2 | 0.5 |
55 | 100 | 55/100 | 0.55 |
1 | 1000 | 1/1000 | 0.001 |
253 | 10 | 253/10 | 25.3 |
7 | 7/0 | No! “q” can't be zero! |
Just remember: q can't be zero
Using Rational Numbers
The ancient greek mathematician Pythagoras believed that all numbers were rational, but one of his students Hippasus proved (using geometry, it is thought) that you could not write the square root of 2 as a fraction, and so it was irrational.
But followers of Pythagoras could not accept the existence of irrational numbers, and it is said that Hippasus was drowned at sea as a punishment from the gods!
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