Introduction: Number System

Real numbers

All rational and irrational numbers are combined to form real numbers. On the number line, any real number can be represented.

Euclid division lemma

• This lemma is essentially equivalent to : dividend = divisor × quotient + remainder.

Fundamental theorem of arithmetic

• The Fundamental Theorem of Arithmetic states that the prime factorisation for a given number is unique if the arrangement of the prime factors is ignored.
• Example: 36=2×2×3×3 OR, 36=2×3×2×3

Prime factorization

The method of representing a natural number as a product of prime numbers is known as prime factorization.

Method for finding LCM

1.Example: To find the Least Common Multiple (L.C.M) of 36 and 56 36=2×2×3×3
2.56=2×2×2×7
3. The common prime factors are 2×2
4. The uncommon prime factors are  3×3 for 36 and 2×7 for 56.
5.LCM of 36 and 56 = 2×2×3×3×2×7  which is 504

Method for finding HCF

H.C.F can be found using two methods – Prime factorisation and Euclid’s division algorithm.

Prime Factorisation :

Given  two numbers, we express both of them as products of their respective prime factors. Then, we select the prime factors that are common to both the numbers
Example: To find the H.C.F of 20 and24 20=2×2×5   and   24=2×2×2×3
The Factor  common to 20 and 24 is 2×2, which is 4, which in turn is the H.C.F of 20 and 24.

Euclid’s Division Algorithm:

It is the repeated use of Euclid’s division lemma to find the H.C.F of two numbers.
Example: To find the HCF of 18 and 30

Example : Find the HCF of 96 and 404 by the prime factorisation method. Hence, find their LCM.
96 = 25 × 3, 404 = 22 × 101
Therefore, the HCF of these two integers is 22 = 4.
LCM = 96*404/HCF(96,404) = 96*404/4 = 9696.

Example : Find the HCF and LCM of 6, 72 and 120, using the prime factorisation method.

Solution :
We have : 6 = 2 × 3, 72 = 23 × 32, 120 = 23 × 3 × 5
Here, 21 and 31 are the smallest powers of the common factors 2 and 3, respectively.
So, HCF (6, 72, 120) = 21 × 31 = 2 × 3 = 6 23, 32 and 51 are the greatest powers of the prime factors 2, 3 and 5 respectively involved in the three numbers.
So, LCM (6, 72, 120) = 23 × 32 × 51 = 360.

Example: Find the largest number that divides 70 and 125 leaving the remainder 5 and 8 respectively.

Solution: First, subtract the remainder from the number.   70-5 = 65     125-8 = 117.
To find the largest number, take the HCF of 65 and 117.
65 = 5×13              117 = 3×3×13.
Hence, HCF (65, 117) = 13.

Important results-

•HCF*LCF= one number *another number.
•LCM of fractions = LCM of numerator/HCF of denominator
•HCF of fractions = HCF of numerator/LCM of denominator

Irrational numbers-

Any number that cannot be expressed in the form of p/q (where p and q are integers and q≠0.) is an irrational number. Example – 4.965458.. , 3.14592….

Terminating and non-terminating decimals-

Terminating decimals are decimals that end at a certain point. Example: 0.2, 2.56 and so on.
Non-terminating decimals are decimals where the digits after the decimal point don’t terminate. Example: 0.333333….., 0.13135235343…

Complex numbers-

a complex number is a simple representation of addition of two numbers, i.e., real number and an imaginary number. One part of it is purely real and the other part is purely imaginary.

Integers –

Integers are made up of both positive and negative numbers. Integers, on the other hand, are numbers that can be positive, negative, or zero but not a fraction.

On integers, we can do all arithmetic operations such as addition, subtraction, multiplication, and division.

Integers include numbers such as 1, 2, 5,8, -9, -12, and so on. Integers are denoted by the letter “Z.”

Fractions-

it is defined as the parts of a whole. The whole can be an object or a group of objects. In real life, when we cut a piece of cake from the whole of it, then the portion is the fraction of the cake. 

1.Proper fraction – The proper fractions are those where the numerator is less than the denominator. For example, 8/9 will be a proper fraction since “numerator < denominator”. 

2.Improper fraction – The improper fraction is a fraction where the numerator happens to be greater than the denominator. For example, 9/8 will be an improper fraction since “numerator > denominator”.

3. Mixed fraction – mixed fraction is a combination of the integer part and a proper fraction. These are also called mixed numbers or mixed numerals. For example:

4. Compound fraction- a fraction of a fraction is called the compound fraction.
5. Complex fraction- any complicated combination of the other type of fractions.

Prime numbers – the number that have only two factors 1 and number itself. Eg-2,3,5 etc.