Study of Real Numbers Class 10

Let us learn some important definitions before we get into the concept of real numbers Class 10:

• A natural number is a counting number. Hence a set of natural number can be shown as N = {1, 2, 3, . .}
• A whole number is all the natural numbers including 0. Therefore a set of whole numbers is W = {0, 1, 2, ..}
• An integer is a set of all positive and negative whole numbers. The set of integers can be written as Z = { -4, -3, -2, -1, 0, 1, 2, . . }. The natural numbers (excluding zero) are known as positive integers. When a number gives zero on being added to its corresponding positive value is known as a negative integer. When we consider zero along with the natural numbers, we call it non-negative integers.
• A rational number is any number that can be expressed in p / q form where both the numerator and denominator are integers and the value of q is positive. This includes all integers, natural numbers and rational number. Any two rational numbers has infinite rational numbers between them. The rational number can either be terminating decimal or non-terminating decimal, which can again be recurring or non-recurring in nature. There are certain operations of rational numbers that need to be accounted for. The sum of two rational numbers is a rational number. The same hold for difference and product too. This may or may not be true for division.
• An irrational number is a number that cannot be expressed in p / q format. We can represent the set of irrational numbers on a number line with the help of Pythagoras theorem.

What are real numbers?

A collection of rational numbers and irrational numbers make up the set of real number. 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)

= 7

There are some laws of exponents as well that is demonstrated by real numbers. They are:

1. a^p x a^q = a ^(p+q)
2. a^p / a^q = a^(p-q)
3. (a^p)^q = a^pq
4. a^p x b^p = (ab)^p

Here a and b, both are real numbers.

What is Euclid’s division algorithm?

There is a value q and r for every set of positive integer such that

a = bq +r, where 0 ≤ r < b.

There are specific names for each term, a is dividend, b is divisor, q is quotient and r is remainder. It is a method used to find the highest common factor.

For example, when we need to use the Euclid’s algorithm to find the HCF of 135 and 225.

Since 225 > 135, we can write 225 = 135 x 1 +90

Since 90>0, the remainder is 45. We apply the division rule again to get
135 = 90 x 1 +45

We then consider 90 to be the new divisor with the remainder as 45 and on applying the algorithm again we get, 90 = 2 x 45 + 0. Here, since the remainder is 0, the HCF = 45, which is the divisor at this particular stage.

Some solved questions

1. Prove that √2 is irrational number

Answer: Let us assume that it is rational number

Then, √2=p/q

Where, p and q are co-primes.

Or, q√2=p

Squaring both sides, we get

2q²=p²

Hence, 2 divide p².

From theorem we know that,

2 will divide p also. p = 2c, where c is any constant

2q²=4c²

or q²=2c²

So q is divisible by 2 also

So 2 divide both p and q, which contradicts our assumption.

So √2 is an irrational number.

2. Show that a positive odd integer is in either of the following forms: 4q+1 or 4q+3 where q is an integer.

Answer: let a be any positive off integer. On applying the division algorithm with a, b = 4, we get that the remainder can be 0, 1, 2, or 3.

So, a can be 4q, 4q+1, 4q+2 or 4q+3 where the quotient is q.

Since a is odd, it cannot be 4q or 4q+2 (since both are divisible by 2)
hence, an odd integer is always of form 4q+1 or 4q+3.

3. Express 140 and 5005 as a product of its prime factor:

Answer:

140 = 2 x 2 x 5 x 7 = 2² x 5 x 7

5005 = 5 x 7 x 11 x 13

5. Find the LCM and HCF of 26 and 91

Answer: 26 = 2 x 13

91 = 7 x 13

HCF = 13

LCM = 2 x 7 x 13 = 182

Product of 2 numbers = 2366

HCF X LCM = 2366

Hence, proved

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