Real Numbers – CBSE Class 10th

Natural Numbers

Natural numbers are defined as the numbers that are used for counting and ordering. Hence the set of Natural numbers is shown as N ={1,2,3,4………}

Whole Numbers

All-natural numbers including zero(0) are called whole numbers. Hence the set of whole numbers is shown as W ={0,1,2,3,4………}

Integer

An integer is a set of all positive and negative numbers, including zero, are called integers. All the numbers before zero are called negative integers. All the numbers that present after zero are called positive integers.

I ={-4,-3,-2,-1,0,1,2,3,4….}

Rational Numbers

A Rational Number is a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p, and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number.

For example, 1/7  and  −3/4 are rational numbers.

Irrational Numbers

The irrational numbers are all the real numbers which are not rational numbers, the latter being the numbers constructed from ratios of integers.

Some Examples: √2, √42, √8.3333….. are irrational numbers.

What are Real Numbers?

A collection of Rational numbers and Irrational numbers make up the set of real numbers. A real number can be expressed on the number line and has some specific properties. They satisfy:

• The commutative law of addition. That is when a and b are two real numbers then a + b = b + a. For example 1 + 3 = 3 + 1 = 4
• The commutative law of multiplication. That is, when a and b are two real numbers then a x b = b x a. For example 1 x 3 = 3 x 1 = 3
• The associative law of addition. That is, when a, b and c are three real numbers then a + (b + c) = (a + b) + c.
• For example 1 + (3 + 4) = (1 + 3) + 4 = 8 .
• The associative law of multiplication. That is, when a, b and c are three real numbers then a x (b x c) = (a x b) x c. For example, 1 x (3 x 4) = (1 x 3) x 4 = 12.
• The law of distribution. That is, when a, b, c are three real numbers then a x (b +c) = (a x b) + (a x c). For example 1 x (3 + 4) = (1 x 3) + (1 x 4).

What is Euclid’s division Lemma?

According to Euclid’s Division Lemma if we have two positive integers a and b, then there exist unique integers q and r which satisfies the condition a = bq + r where 0 ≤ r ≤ b.

The basis of the Euclidean division algorithm is Euclid’s division lemma. To calculate the Highest Common Factor (HCF) of two positive integers a and b we use Euclid’s division algorithm. HCF is the largest number which exactly divides two or more positive integers. By exactly we mean that on dividing both the integers a and b the remainder is zero.

For Example, Let us now get into the working of this euclidean algorithm.

Consider we have two numbers 78 and 980 and we need to find the HCF of both of these numbers. To do this, we choose the largest integer first, i.e. 980 and then according to Euclid Division Lemma, a = bq + r where 0 ≤ r ≤ b;

980 = 78 × 12 + 44

Now, here a = 980, b = 78, q = 12 and r = 44.

Now consider the divisor as 78 and the remainder 44 and apply the Euclid division method again, we get

78 = 44 × 1 + 34

Similarly, consider the divisor as 44 and the remainder 34 and apply the Euclid division method again, we get

44 = 34 × 1 + 10

Following the same procedure again,

34 = 10 × 3 + 4

10=4×2+2

4=2×2+0

As we see that the remainder has become zero, therefore, proceeding further is not possible and hence the HCF is the divisor b left in the last step which in this case is 2. We can say that the HCF of 980 and 78 is 2.

LCM (Least Common Multiple)

The least number which is exactly divisible by each of the given numbers is called the least common multiple of those numbers.

For example, consider the numbers 3, 31 and 62 (2 x 31). The LCM of these numbers would be 2 x 3 x 31 = 186.

HCF (Highest Common Factor) or Greatest Common Divisor (GCD)

The largest number that divides two or more numbers is the highest common factor (HCF) for those numbers. For example, consider the numbers 30 (2 x 3 x 5), 36 (2 x 2 x 3 x 3), 42 (2 x 3 x 7), 45 (3 x 3 x 5). 3 is the largest number that divides each of these numbers, and hence, is the HCF for these numbers.

Fundamental Theorem of Arithmetic:

Fundamental Theorem of Arithmetic states that every integer greater than 1 is either a prime number or can be expressed in the form of primes. In other words, all the natural numbers can be expressed in the form of the product of its prime factors. To recall, prime factors are the numbers which are divisible by 1 and itself only. For example, the number 35 can be written in the form of its prime factors as:

For Example:

• 35 = 7 × 5 Here, 7 and 5 are the prime factors of 35
• 140 = 2 × 2 × 5 × 7 = 22 × 5 × 7

Proof of Irrationality of √2:

As √2 is an irrational number we prove that by using the Contradiction technique. Now let us assume that

√2 = a/b a/bis a fraction in its lowest form

Now squaring on both sides

(√2)² = (a/b)² 2 = a ²/ b ² 2b² = a² (by multiplying with b² on both sides)

So a 2 is an even number ⇒ a is an even number. We can, therefore, express a as 2c where c is also an integer.

2b² = a² 2b² = (2c)² substituting 2c for a 2b² = 4c² getting rid of brackets b² = 2c² cancelling with on 2 both sides

Hence we proved that √2 is an irrational number.