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 RQ, 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.