MATHEMATICS CLASS X
CHAPTER 1
REAL NUMBERS
Number Line:
Number line is a straight line in which numbers are placed in at equal distance.
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9
Different Groups of Numbers:
1. Natural Numbers
1, 2, 3, 4, 5, ……..
2. Whole Numbers
0, 1, 2, 3, 4, 5, …….
3. Integers
……………., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ………
4. Rational Numbers
𝑝
A number which can be written in the form of 𝑞
,where p and q are integers and q≠0.
2 1
Example: 3 , 2 , 3, 5, etc
𝑝
Note: All integers can be written in the form of 𝑞 , where q=1.
5. Irrational Numbers
𝑝
A number which cannot be written in the form of , where p and q are integers and q≠0.
𝑞
Example: 2 , 3 , 𝜋 , etc 1.4142135623730, .......
6. Real Numbers
All rational and irrational numbers which can be represented on a number line is called real
numbers.
, 7. Imaginary Numbers
Numbers which cannot be represented on the number line is called Imaginary Numbers
Example : −1 , −4 , i
Note: i2 = -1
Factors
Factors of a particular number are that numbers which can divide the given number and give reminder
0.
Example : Factors of 12 are 1, 2, 3, 4, 6, 12
Factors of 20 are 1, 2, 4, 5, 10, 20
Prime Numbers
A number whose factors are only 1 and the number itself is called prime numbers.
2, 3, 5, 7, 11, 13, 17, 23, etc
Example : 2 is a prime number because it has only two factors i.e. 1 and 2 only.
3 is a prime number because it has only two factors i.e. 1 and 3 only.
11 is a prime number because it has only two factors i.e. 1 and 11 only.
Etc
Prime Factors
Prime factors of a number are those factors of a number which are prime number.
Example :
Prime factors of 12 are 2 and 3 because the factors of 12 are 1, 2, 3, 4, 6 and 12, but among this only 2
and 3 are prime numbers thus 2 and 3 are the prime factors of 12.
Prime factors of 30 are 2, 3 and 5 because the factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30, but among
this only 2, 3 and 5 are prime numbers.
, How to find the prime factors of a Number
2 30 2 120
3 15 2 60
5 5 2 30
1 3 15
5 5
1
Prime factors of 30 are 2, 3 and 5.
Prime factors of 120 are 2, 3 and 5.
Steps
1. Divide the given number with the lowest prime number and write down the result.
2. Divide the result again with the lowest prime number possible.
3. If the number does not divide with the lowest prime number then try to divide with the next
prime number
4. Continue the steps till the results comes to 1.
5. Note: Only divide with the prime numbers and follow the order of prime number such as 2, 3, 5,
7, 11, 13, 17 etc
Composite Numbers:
A number which has more than two factors are called composite number.
Example:
2 is not a composite number because it has only two factors i.e. 1 & 2.
3 is not a composite number because it has only two factors i.e. 1 & 3.
4 is a composite number because it has more than two factors i.e. 1, 2 & 4.
5 is not a composite number because it has only two factors i.e. 1 & 5.
6 is a composite number because it has more than two factors i.e. 1, 2, 3 & 6.
Theorem 1.1 (Fundamental Theorem of Arithmetic)
Every composite number can be expressed (factorised) as a product of primes, and this factorization is
unique, apart from the order in which the prime factors occur.
Example:
120 can be written as the product of its primes i.e.
120 = 2 x 2 x 2 x 3 x 5 = 23 x 31 x 51
120 = 3 x 2 x 5 x 2 x 2 = 31 x 23 x 51
In both the cases the only the order is different but the prime and it number of time the prime occurred
is same.
Note: Writing a composite number as a product of its prime is called factorization.
Note: Prime factors of the number can divide the given number exactly and give the reminder 0.
CHAPTER 1
REAL NUMBERS
Number Line:
Number line is a straight line in which numbers are placed in at equal distance.
-8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9
Different Groups of Numbers:
1. Natural Numbers
1, 2, 3, 4, 5, ……..
2. Whole Numbers
0, 1, 2, 3, 4, 5, …….
3. Integers
……………., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ………
4. Rational Numbers
𝑝
A number which can be written in the form of 𝑞
,where p and q are integers and q≠0.
2 1
Example: 3 , 2 , 3, 5, etc
𝑝
Note: All integers can be written in the form of 𝑞 , where q=1.
5. Irrational Numbers
𝑝
A number which cannot be written in the form of , where p and q are integers and q≠0.
𝑞
Example: 2 , 3 , 𝜋 , etc 1.4142135623730, .......
6. Real Numbers
All rational and irrational numbers which can be represented on a number line is called real
numbers.
, 7. Imaginary Numbers
Numbers which cannot be represented on the number line is called Imaginary Numbers
Example : −1 , −4 , i
Note: i2 = -1
Factors
Factors of a particular number are that numbers which can divide the given number and give reminder
0.
Example : Factors of 12 are 1, 2, 3, 4, 6, 12
Factors of 20 are 1, 2, 4, 5, 10, 20
Prime Numbers
A number whose factors are only 1 and the number itself is called prime numbers.
2, 3, 5, 7, 11, 13, 17, 23, etc
Example : 2 is a prime number because it has only two factors i.e. 1 and 2 only.
3 is a prime number because it has only two factors i.e. 1 and 3 only.
11 is a prime number because it has only two factors i.e. 1 and 11 only.
Etc
Prime Factors
Prime factors of a number are those factors of a number which are prime number.
Example :
Prime factors of 12 are 2 and 3 because the factors of 12 are 1, 2, 3, 4, 6 and 12, but among this only 2
and 3 are prime numbers thus 2 and 3 are the prime factors of 12.
Prime factors of 30 are 2, 3 and 5 because the factors of 30 are 1, 2, 3, 5, 6, 10, 15 and 30, but among
this only 2, 3 and 5 are prime numbers.
, How to find the prime factors of a Number
2 30 2 120
3 15 2 60
5 5 2 30
1 3 15
5 5
1
Prime factors of 30 are 2, 3 and 5.
Prime factors of 120 are 2, 3 and 5.
Steps
1. Divide the given number with the lowest prime number and write down the result.
2. Divide the result again with the lowest prime number possible.
3. If the number does not divide with the lowest prime number then try to divide with the next
prime number
4. Continue the steps till the results comes to 1.
5. Note: Only divide with the prime numbers and follow the order of prime number such as 2, 3, 5,
7, 11, 13, 17 etc
Composite Numbers:
A number which has more than two factors are called composite number.
Example:
2 is not a composite number because it has only two factors i.e. 1 & 2.
3 is not a composite number because it has only two factors i.e. 1 & 3.
4 is a composite number because it has more than two factors i.e. 1, 2 & 4.
5 is not a composite number because it has only two factors i.e. 1 & 5.
6 is a composite number because it has more than two factors i.e. 1, 2, 3 & 6.
Theorem 1.1 (Fundamental Theorem of Arithmetic)
Every composite number can be expressed (factorised) as a product of primes, and this factorization is
unique, apart from the order in which the prime factors occur.
Example:
120 can be written as the product of its primes i.e.
120 = 2 x 2 x 2 x 3 x 5 = 23 x 31 x 51
120 = 3 x 2 x 5 x 2 x 2 = 31 x 23 x 51
In both the cases the only the order is different but the prime and it number of time the prime occurred
is same.
Note: Writing a composite number as a product of its prime is called factorization.
Note: Prime factors of the number can divide the given number exactly and give the reminder 0.
