Welcome to this thorough explanation of rational numbers, a key idea in mathematics. Real numbers represented as p/q, where q is not equal to zero, are known as rational numbers. Since all fractions with non-zero denominators fall inside this range, we may describe precise amounts. For instance, rational numbers like 1/2, 1/5, and 3/4 are instances. Intriguingly, even the number “0”—which may be expressed in a variety of ways, including 0/1, 0/2, and 0/3—can be thought of as a rational number. However, as they produce infinite values, fractions like 1/0, 2/0, and 3/0 are illogical. It is important to investigate irrational numbers and establish their differences.
In this article, we’ll examine the concept of rational numbers, as well as look at their kinds, qualities, and comparisons to irrational numbers, as well as offer a variety of examples with solutions. We may deepen our comprehension and build a firm foundation in rational numbers by immersing ourselves in these ideas. To better understand their significance and develop reliable methods for recognizing rational numbers, we will also look at a variety of examples of rational numbers. We must first simplify and express rational numbers in decimals before we can correctly represent them on a number line.
We hope that this extensive manual will provide you the information and abilities you need to confidently and accurately traverse the world of rational numbers. So let’s get off on this mathematical adventure voyage and explore the complexities of rational numbers.
Let’s have a look at the subjects this essay will cover.
What is a Rational Number?
In mathematics, any integer that can be written as p/q where q ≠ 0 is considered a rational number. Additionally, every fraction that has an integer denominator and numerator and a denominator that is not zero falls into the category of rational numbers. The outcome of dividing a rational number, or fraction, will be a decimal number, either a terminating decimal or a repeating decimal.
How to Identify Rational Numbers?
Examine the following requirements to determine whether a number is logical or not.
- In the form of p/q, where q≠0, it is shown.
- Further simplification and representation of the ratio p/q in decimal form are possible.
The Collection of Rational Numbers
- Include positive, negative, and 0 numbers.
- Can be stated as a fraction
Examples of Rational Numbers:
Rational Number Types
A number is rational if it can be written as a fraction with both the denominator and numerator being integers and the denominator being non-zero.
The figure below explains the number sets in further detail.
Standard Form of Rational Numbers
The standard form of a rational number is defined as having no common factors other than one between the dividend and divisor and thus being positive.
12/36, for example, is a rational number. However, it may be reduced to 1/3; there is only one common component between the divisor and the dividend. As a result, rational number 13 is in standard form.
Also read: Trigonometry Table
Rational Numbers, Positive and Negative
The rational number, as we know, takes the form p/q, where p and q are integers. In addition, q must be a non-zero integer. Positive or negative rational numbers can be used. Both p and q are positive integers if the rational number is positive. If the rational number is in the form -(p/q), then either p or q is negative. That is to say
-(p/q) = (-p)/q = p/(-q).
Let’s look at some examples of positive and negative rational numbers.
Rational Number Arithmetic Operations
Arithmetic operations are the fundamental operations we perform on integers in mathematics. Let us now look at how we can do similar operations on rational numbers, namely p/q, and s/t.
Addition:
When we add p/q and s/t, we must make the denominator the same. As a result, we obtain (pt+qs)/qt.
Example: 1/2 + 3/4 = (2+3)/4 = 5/4
Subtraction:
Similarly, if we subtract p/q and s/t, we must first make the denominator equal before subtracting.
Example: 1/2 – 3/4 = (2-3)/4 = -1/4
Multiplication:
When multiplying two rational numbers, the numerator, and denominator of the rational numbers are multiplied separately. When p/q is multiplied by s/t, we obtain (p×s)/(q×t).
Example: 1/2 × 3/4 = (1×3)/(2×4) = 3/8
Division:
If p/q is divided by s/t, the result is:
(p/q)÷(s/t) = pt/qs
Example: 1/2 ÷ 3/4 = (1×4)/(2×3) = 4/6 = 2/3
Rational Numbers Multiplicative Inverse
Because the rational number is represented as a fraction in the form p/q, the multiplicative inverse of the rational number is the reciprocal of the provided fraction.
For example, if 4/7 is a rational integer, then the multiplicative inverse of 4/7 is 7/4, such that (4/7)x(7/4) = 1
Properties of Rational Numbers
Because a rational number is a subset of the real number, it will obey all of the real number system’s characteristics. The following are some of the most important features of rational numbers:
- If we multiply, add, or subtract any two rational numbers, the outcome is always a rational number.
- If we divide or multiply both the numerator and denominator with the same factor, a rational number remains the same.
- When we add zero to a rational number, we receive the same number.
- Addition, subtraction, and multiplication are all close to rational numbers.
Rational Numbers and Irrational Numbers
There is a distinction to be made between rational and irrational numbers. A rational number is a fraction having non-zero denominators. Because it is interpreted as integer 1 divided by integer 2, the number 12 is a rational number. Irrational numbers are any numbers that are not rational. Check out the chart below to tell the difference between reasonable and irrational behavior.
Rationals might be positive, negative, or nil. The negative sign is either in front of or with the numerator of a negative rational integer, as is the conventional mathematical notation. For example, we write the inverse of 5/2 as -5/2.
Irrational numbers cannot be stated as simple fractions, although they can be represented as decimals. It has an infinite number of non-repeating digits following the decimal point. Some examples of irrational numbers include:
Pi (π) = 3.142857…
Euler’s Number (e) = 2.7182818284590452…….
√2 = 1.414213…
Also Read: Table 2 to 20
How to Find the Rational Numbers between Two Rational Numbers?
Between two rational numbers, there exists an unlimited amount of rational numbers. Two distinct strategies may be used to easily get the rational numbers between two rational numbers. Let us now examine the two distinct techniques.
Method One:
Determine the equivalent fraction for the given rational numbers and the rational numbers in between. Those figures should be the requisite reasonable figures.
Method number two:
Determine the mean value of the two rational numbers given. The needed rational number should be the mean value. Repeat the method using the old and newly obtained rational numbers to find more rational numbers.
Set of rational numbers examples
Exemplification 1:
Determine if the following are unreasonable or rational: ¾, 90/12007, 12, and √5.
Solution:
Because a rational number is one that can be stated in terms of a ratio. This means that it may be written as a fraction with both the denominator and the numerator being full integers.
- Because it may be written as a fraction, ¾ is a rational number. 3/4 = 0.75
- Fraction 90/12007 is rational.
- 12 is alternatively written as 12/1. Once again, a logical number.
- The √5 = 2.2360679775……..Because it is a non-terminating value, it cannot be expressed as a fraction. It is an illogical number.
Exemplification 2:
Determine whether 11/2, a mixed fraction, is a rational number.
Solution:
The Simplest form of 11/2 is 3/2
The denominator is 3, which is an integer.
Denominator = 2 is an integer that is not zero.
So, yes, 3/2 is a rational number.
Exemplification 3:
Determine if the numbers provided are reasonable or irrational.
(a) 1.75 (b) 0.01 (c) 0.5 (d) 0.09 (d) √3
Solution:
The numbers supplied are in decimal format. To determine if a given integer is decimal or not, we must convert it to fraction form (i.e., p/q).
If the fraction’s denominator is not equal to 0, the number is rational; otherwise, it is irrational.
Frequently Asked Questions on Rational Numbers
What exactly are rational numbers? Provide examples.
A rational number is one that has the form p/q, where p and q are integers and q is greater than zero. Some rational number examples are 1/3, 2/4, 1/5, 9/3, and so on.
What exactly is the distinction between rational and irrational numbers?
A rational number is one that can be stated as the ratio of two integers with the denominator not equal to zero, whereas an irrational number cannot be expressed as a fraction. Irrational numbers are non-terminating and non-recurring, whereas rational numbers are. A rational number is 10/2, whereas an irrational number is the renowned mathematical value Pi(), which is 3.141592653589…….
Is the number 0 a rational number?
Yes, because it is an integer that may be represented in any form, such as 0/1, 0/2, where b is a non-zero integer, 0 is a rational number. It may be expressed as p/q = 0/1. As a result, we may infer that 0 is a rational number.
Is 7 a logical number?
7 is a rational number because it can be expressed as a ratio, such as 7/1.
Is the number 3.14 rational?
Yes, 3.14 is a rational number since it ends. However, is not a rational number since its precise value is 3.141592653589793238…which is non-terminating and non-recurrent.
Find a number between 3 and 4 that is logical.
(3+4)/2 = 7/2 is a rational number between 3 and 4.
What is the rational number’s denominator?
The rational number’s denominator can be any real value except 0.
Is Pi(π) a rational number?
No, the number Pi(π) is not a rational number. It is an irrational number with a value of 3.142857.
