## 10th Class Maths Notes Chapter 1 Real Numbers

→ “God made the integers. All else is the work of man” …… Leopold Kronecker

→ Euclid’s division lemma: Given positive integers, a, b there exists unique pair of integers q and r satisfying

a = bq + r; 0 ≤ r < b

This result was first published/recorded in book VII of Euclid’s “The Elements”.

→ Euclid’s division algorithm is a technique to compute the Highest Common Factor (H.C.F) of two given numbers.

E.g: HCF of 80 and 130

130 = 80 × 1 + 50 80 = 50 × 1 + 30

50 = 30 × 1 + 20 30 = 20 × 1 + 10

20 = 10 × 2 + 0 and H.C.F = 10

→ Euclid’s division algorithm can also be extended to all integers.

→ Numbers which can be expressed in the form p/q, where q ≠ 0 and ‘p and q’ are integers are called rational numbers; represented by Q.

Q = { pq ; q ≠ 0; p, q ∈ Z} .

→ Every rational number can be expressed either as a terminating decimal or as non-terminating recurring decimal.

→ Numbers that can’t be expressed in p/q form are called irrational numbers represented by S. You may notice that the first letter of surds is ‘S’.

Eg: √2, √3, √5, …….. etc,

→ The combined set of rationals and irrationals is called the set of Real numbers; represented by R.

R = Q ∪ S.

→ Diagramatic representation of the number system:

10th Class Maths Notes Chapter 1 Real Numbers

where N = the set of natural numbers; W = the set of whole numbers;

Z = the set of integers; Q = the set of rational numbers;

S = the set of irrational numbers; R = the set of real numbers.

→ Fundamental Theorem of Arithmetic: Every composite number can be expressed as a product of primes uniquely, (i.e.,) if x is a composite number, then

x = pl1⋅pm2⋅pn3 …… where p_{1}, p_{2}, p_{3}, ….. are prime numbers and l, m, n, …… are natural numbers.

Eg: 420 = 2 × 210 = 2 × 2 × 105 = 2 × 2 × 3 × 35 = 2 × 2 × 3 × 5 × 7

i. e. 420 = 2^{2} × 3^{1} × 5^{1} × 7^{1} and the factorization on the R.H.S is unique.

Note: R.H.S is called an exponential form of 420.

→ To find the H.C.F. of two or more numbers:

Step (i): Express given numbers in their exponential form.

Step (ii): Take the common bases.

Step (iii): Assign the respective smallest exponent from their exponential forms.

Step (iv): Take the product of the above.

Eg: H.C.E of 60 and 75 is

Step (i) 60 = 2^{2} × 3 × 5 ; 75 = 3 × 5^{2}

Step (ii) 3^{O} × 5^{O} [taking common bases]

Step (iii) 3^{1} × 5^{1} [∵ smallest exponent among 3^{1} and 3^{1} is 1]

Step (iv) 3 × 5 = 15 [smallest exponent among 5^{1} and 5^{2} is 1]

∴ H.C.F = 15

(i.e.) The highest common factor of the given set of numbers is the product of the com¬mon bases with the respective least exponents.

**10th Class Maths Notes Chapter 1 Real Numbers**

→ To find the L.C.M. of two or more numbers:

Step – 1: Express the given numbers in their exponential forms.

Step – 2: Take every base.

Step – 3: Assign the respective greatest exponent to each base.

Step – 4: Take the product of the above.

Eg: L.C.M. of 60 and 75 is

Step – 1: 60 = 2^{2} × 3 × 5 ; 75 = 3 × 5^{2}

Step – 2: 2^{O} × 3^{O} × 5^{O}

Step – 3: 2^{2} × 3^{1} × 5^{2}

Step – 4: 4 × 3 × 25 = 300

L.C.M = 300

## 10th Class Maths Notes Chapter 1 Real Numbers

→ We may notice that the product of any two numbers N_{1} and N_{2} is equal to the product of their L.C.M. (L) and H.C.F. (H).

i.e., N_{1} . N_{2} = L.H

→ Let x = p/q be a rational number. If the numerator p is divided by the denominator q, we get the decimal form of x. The decimal form of x may or may not be terminating, i.e., every rational number can be expressed either as a terminating decimal or a non-terminating decimal. This gives us the following theorems.

Theorem – 1: Let ‘x’ be a rational number when expressed in decimal form, terminates, then x can be expressed in the form p/q where p, q are co-primes and the prime factorization of q is of the form 2^{n} × 5^{m}, where n and m are non-negative integers.

Theorem – 2: Let x = p/q be a rational number, where q is of form 2^{n} × 5^{m} then x has a decimal expansion that terminates.

Theorem – 3: Let x = p/q be a rational number, where p, q are co-primes and the prime factorization of q is not of the form 2^{n} . 5^{m} (n, m ∈ Z^{+}) then x has a decimal expansion which is a non-terminating recurring decimal.

Theorem – 4: Let ‘p’ be a prime number. If p divides a^{2} then p divides a, where ‘a’ is a positive integer.

→ If a is a non-zero rational number and b is any irrational number, then (a + b), (a – b), a/b, and ab are all irrational numbers.

10th Class Maths Notes Chapter 1 Real Numbers

→ Properties of Real Numbers:

If a, b and c are any three real numbers we may notice that

- a + b is also a real number – closure property w.r.t. addition

a.b is also a real number – closure property w.r.t. multiplication - a + b = b + a – commutative property w.r.t. addition

a . b = b . a – commutative property w.r.t. multiplication - (a + b) + c = a + (b + c) – associative law w.r.t.

addition (a.b).c = a.(b.c) – associative law w.r.t. multiplication - a + 0 = 0 + a = a, where ‘0’ is the additive identity,

a × 1 = 1 × a = a, where 1 is the multiplicative identity, - a + (-a) = (-a) + a = 0 where (a) and (-a) are additive inverse of each other.

a × 1a = 1a × a = 1 where a and 1a are multiplicative inverse of each other.

→ If a^{n} = x, where a and x are positive integers and a ≠ 1, then we define log_{a}x = n read as logarithm of x to the base a is equal to n.

Eg.: 2^{4} = 16 ⇒ log_{2}16 = 4

→ log_{a}x + log_{a}y = log_{a}xy

→ log_{a}a = 1

→ log_{a}x – log_{a}y = log_{a}xy

→ log_{a}1 = 0

→ log_{a}x^{m} = m log_{a}x

→ In general, the bases in the logarithms are 10 (or) e, where e’ is approximated to 2.718.

→ If p is a prime number and p divides a^{2} then p divides ‘a’ also.

10th Class Maths Notes Chapter 1 Real Numbers