Introduction to Real Numbers
In mathematics, rational numbers and irrational numbers are together obtained from the set of real numbers. The set of real numbers is represented by the letter R. Therefore, it indicates that every real number is either a rational number or an irrational number. In either case, it contains a non–terminating decimal depiction. In the instance of rational numbers, the decimal depiction is repeating (including repeating zeroes) and if the decimal depiction is non–repeating, it is an irrational number.
Real Numbers
Real numbers in the number system are nothing but the combination of rational and irrational numbers. All the arithmetic operations are performed with these numbers and can be represented in the number line. Whereas imaginary numbers are the un-real numbers that cannot be represented in the number line and are commonly used to express a complex number.
Real numbers in Class 10 consist of some of the advanced concepts related to real numbers. Besides knowing what real numbers are, students can have a clear knowledge of the real numbers formulas and concepts like Euclid’s Division Lemma, Euclid’s Division Algorithm, and arithmetic fundamental theorem in class 10.
Euclid’s Division Lemma
Euclid’s Division Lemma states that, if there are two positive integers a and b, then there is an occurrence of unique integers q and r, such that it satisfies the condition a = (b x q) + r, (such that 0 ≤ r < b).
Where a, b, q, r are the dividend, divisor, quotient, and remainder respectively.
Fundamental Theorem of Arithmetic
According to the Fundamental Theorem of Arithmetic, every integer that is greater than 1 is either a prime number or is expressed in the form of primes. In other words, all natural numbers can be represented in the form of the product of its prime factors. Prime factors are the numbers that cannot be divisible by other numbers and are only divisible by 1 . For example, the number 56 can be written in the form of its prime factors as:
56 = 2³ × 7
For the number 56, the prime factors are 2 and 7.
Irrational Numbers
The real numbers which cannot be expressed as simple fractions are called irrational numbers. It cannot be expressed in the terms of a ratio, such as p/q, such that p and q are integers, q≠0, and is a contradiction of rational numbers.
Irrational numbers are generally represented as R\Q, such that the backward slash symbol represents ‘set minus’. it can also be denoted as R – Q, which is the difference between real numbers and rational numbers.
The calculations of irrational numbers are quite complicated. For example, √7, √13, √53, etc., are irrational.
Rational Numbers
The Rational numbers can be written in the form of p/q, where p and q are integers and q ≠ 0. If these numbers are solved further, it gives the result in decimals.
For example: 0.6, 7/3, -16.6, etc.
Solved Examples
Example1:
Find out the HCF of 867 and 255
Solution:
Using the Euclid's division algorithm, we have
867 = 255 x 3 + 102
255 = 102 x 2 + 51
102= 51 x 2 + 0
Therefore, HCF of (867, 255) = 51
Example2:
Find out if 1009 is a prime or a composite number
Solution:
Numbers are of two types - prime and composite. Prime numbers consist of only two factors namely 1 and the number itself while composite numbers consist of factors besides 1 and itself.
It can be observed that
7 x 11 x 13 + 13 = 13 x (7 x 11 + 1) (taking 13 out as common)
= 13 x (77 + 1)
= 13 x 78
= 13 x 13 x 6
The provided expression consists of 6 and 13 as its factors. Thus, it is a composite number.
= (7 x 6 x 5 x 4 x 3 x 2 x 1) + 5 = 5 x (7 x 6 x 4 x 3 x 2 x 1 + 1)
= 5 x (1008 +1)
= 5 x 1009
1009 cannot be factorized any further. Thus, the given expression consists of 5 and 1009 as its factors. Therefore, it is a composite number.
FAQs on Real Numbers for Class 10
1. Where Can I Find the Best Real Numbers Class 10 Solutions?
To prepare for board Class 10 Maths Chapters 1, you can refer to Vedantu. You will find complete Class 10 Maths Chapter 1 Solutions as well as NCERT Solutions Chapter 1 Real Numbers which are exclusively prepared by the expert faculty at Vedantu. These Class 10 Maths Ch 1 Solutions will help students in their board exam preparations. Vedantu provides step-by-step Solutions for Maths so as to aid the students in solving the problems easily.
Real Numbers Class 10 Solutions are designed in a way that will allow students to focus on preparing the solutions in a manner that is easy to understand. A detailed and step-wise explanation of each answer to the questions is provided in the exercises of these solutions.
2. What are the Benefits of Referring to Class 10th Maths Chapter 1?
Answers for the questions are provided for Real Numbers or the first chapter of Maths. Moreover, you will have access to detailed step-by-steps solutions provided in free PDF available at Vedantu. At Vedantu, students are introduced to ample important concepts which will be useful for those who wish to pursue Mathematics as a subject in their Class 11. Based on these solutions, students can prepare excellently for their upcoming Board Exams. These solutions are of great help and benefit to Class 10 Board students as the syllabus covered here follows NCERT guidelines.
All the positive integers are natural numbers, starting from 1 to infinity. Most Importantly, all the natural numbers are integers but all integers are not necessarily natural numbers. These are the sets of all positive counting numbers such as 1, 2, 3, 4, 5, 6, 7, 8, 9, ……..∞.
Real numbers are numbers that have both rational and irrational numbers. Rational numbers are integers (-2, 0, 1), fractions (1/2, 2.5), and irrational numbers (√3, 22/7 ), etc.
As per the Fundamental Theorem of Arithmetic, every integer greater than 1 is said to be either a prime number or is expressed in the form of primes. In other words, all natural numbers can be represented in the form of the product of its prime factors. Prime factors are numbers that cannot be divisible by other numbers and are only divisible by 1. For example, the number 24 can be written in the form of its prime factors as:
24 = 2³ × 3
Here, 2 and 3 are the prime factors of 24.
The real numbers which cannot be expressed as simple fractions are called irrational numbers. It cannot be expressed in terms of fractions or ratios p/q, where p and q are integers, q ≠ 0, and is a rational number contradiction.
Irrational numbers are generally represented as R\Q, such that the backward slash symbol represents ‘set minus’. it can also be denoted as R – Q, which is the difference between real numbers and rational numbers.
Euclid’s Division Lemma states that, if two positive integers a and b are present, then there is a possible occurrence of unique integers q and r, such that it satisfies the condition
Where a = (b x q) + r, (such that 0 ≤ r < b).
Where a, b, q, r are the dividend, divisor (or HCF), quotient, and remainder respectively.
This method is repeated until the remainder becomes zero. The divisor is the H.C.F of the given set of numbers.
The steps to represent the real numbers on the number line is as follows
Step 1: Draw a horizontal line with arrows at the extreme ends and mark the center of the line as 0. The number 0 is referred to as the origin.
Step 2: Make a mark at equal intervals on both sides of the origin and label it with a definite scale.
Step 3: The positive numbers lie on the right side of the origin and the negative numbers lie on the left side of the origin.
