Introduction
The definition of a rational number is a fraction of two numbers in the form \(\frac{p}{q}\), where p and q can both be integers but q cannot be equal to 0. Rational numbers include whole numbers, integers, and numbers with terminating decimals. Although rational numbers need not necessarily be fractions, any fractions can be rational numbers. The area of mathematics that deals with symbols and variables are called algebra. Natural numbers, Integers, 0, and other types of numbers are all included in rational numbers. Positive and negative numbers are both part of integers. Therefore, we can divide rational numbers into positive rational numbers and negative rational numbers. Positive rational numbers include, for instance, \(1,\frac{3}{16},\frac{25}{2}\), etc. These are examples of negative rational numbers: \( – 3, – \frac{1}{2}, – \frac{5}{3}\), etc.
Rational Numbers
The definition of a rational number is a fraction of two numbers in the form \(\frac{p}{q}\), where p and q can both be integers but q cannot be equal to 0. Rational numbers include whole numbers, integers, and numbers with terminating decimals. Although rational numbers need not necessarily be fractions they can be converted into one, all fractions are rational numbers.
Positive and Negative Rational Numbers
Those rational numbers that have both positive or negative numerators and denominators are known as positive rational numbers. Positive numbers that follow logic are always bigger than zero. For instance, when we divide \(\frac{8}{9}\), we obtain 0.88, which is more than 0, indicating that \(\frac{8}{9}\) is a positive rational number.
Those rational numbers that are negative because their numerators and denominators have opposite signs are known as negative rational numbers. Positive irrational numbers are never greater than zero. For instance, \( – \frac{{12}}{{13}}\) yields -0.92, which is both lower than 0 and a negative rational integer.
Positive Numbers
The number line can also be used to represent positive numbers. According to the illustration below, the numbers that are on the right side of the number line are thought of as positive numbers. Positive numbers are those whose value is consistently higher than zero.
Note: When a number is left unsigned, it is considered to be positive. For instance, the positive integers 45 and +45 are identical and both 45.
Negative Numbers
Similar to the positive numbers, the number line can also be used to represent negative numbers. According to the illustration below, the numbers that are on the left side of the number line are thought of as negative numbers. Negative numbers are those whose value is consistently higher than zero.
Positive, 0 and negative rational numbers are the three subcategories of rational numbers.
A rational number is positive if both the numerator and the denominator have the same sign, such as both being positive or both being negative, and it is negative if the numerator is negative and the denominator is positive or vice versa.
Take an example of \(\frac{3}{5}\) and \(\frac{{ – 3}}{{ – 4}}\) they are both positive since the sign of both numerator and denominator are the same in the respective numbers, whereas \(\frac{{ – 3}}{4}\) and \(\frac{4}{{ – 5}}\) are both negative since the signs in numerator and denominator are opposite.
Algebra of Rational numbers
Rational Numbers like all the other number categories under the real number have four basic binary operations, i.e, addition, multiplication and the inverse operations subtraction and division.
Addition and Multiplication of Rational numbers have the following properties,
Closure Property
Addition and Multiplication of rational numbers is closed, i.e., when two rational numbers are operated with these operations the result is always rational number.
Associative Property
Addition and Multiplication of rational numbers is associative, i.e., the order of operations does not change the result when the same operation is repeated between 3 or more numbers. Mathematically, for three rational numbers a, b and c
\[a + (b + c) = (a + b) + c{\text{ and }}a \cdot (b \cdot c) = (a \cdot b) \cdot c\]
Existence of Identity Property
The Identity element is known as the element which when operated with any other element, has no effect on it. For Addition of rational numbers, 0 is the identity element, and for multiplication it is 1.
Mathematically,
\[a + 0 = 0 + a = a{\text{ and }}a \cdot 1 = 1 \cdot a = a\]
Existence of Inverse Property
The Inverse of an element is known as the element which when operated with the first element, results in the identity. Rational numbers have additive inverse for all the numbers, and they are their negative counterparts, such as for 3 it is -3, for \(-\frac{5}{6}\) it is \(\frac{5}{6}\). For multiplication however, not all rational numbers have inverse, 0 is the rational number which does not have a multiplicative inverse, because by definition multiplicative inverse of a rational number \(a\) would be \(\frac{1}{a}\), but by the definition of rational numbers, 10 is not a rational number.
Commutative Property
Addition and Multiplication of rational numbers is associative, i.e., the order of element does not change the result. Mathematically, for two rational numbers a and b,
\[a + b = b + a{\text{ and }}a \cdot b = b \cdot a\]
Summary
In this article we learned about Rational numbers, positive and negative rational numbers. A rational number is defined as the number which can be represented by the form, \(\frac{p}{q}\) where p and q are coprime integers and \(q\ne 0\). The rational numbers, just like integers, can be divided into 3 categories i.e., positive, negative and 0. Positive rational numbers are those that are greater than 0, and negative are the ones that are smaller. The positive and negative rational numbers are represented on right and left sides of the number line respectively.
FAQs
What are Rational numbers? How do you identify rational numbers in decimal form?
Ans. Rational numbers are defined as the numbers which can be represented by the form, \(\frac{p}{q}\) where p and q are coprime integers and \(q \ne 0\). In decimal form, the numbers that have either terminating decimal expansion or if non-terminating then, repeating decimal expansion are rational numbers.
What is Rule for identifying positive and negative rational numbers from their fractional form?
Ans. There is one simple rule to identifying positive and negative rational numbers from fractional form
- If the numerator and denominator have the same sign, both either positive or negative, then the number as a whole is positive.
- If the numerator and denominator have different signs, numerator positive and denominator negative or vice versa, then the number as a whole is negative.
What are the numbers that are not rational called? What is the general identification of those numbers in the decimal form?
Ans. The real numbers that are not rationals, i.e., they cannot be represented in the form, \(\frac{p}{q}\) where p and q are coprime integers and \(q \ne 0\), are known as Irrational numbers. These are the numbers whose decimal expansion has infinite digits after the decimal, and they never repeat the same pattern.
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