Chapter 1

Real Numbers

Natural Numbers:

• Natural numbers were the first to come. They are denoted by N. These numbers can be counted on fingers.

N = {1, 2, 3, 4, 5, …} 

Whole Number:

• Aryabahatta, famous Mathematician gave ‘0’ to the number system. It very powerful number. Anything multiplied by 0 becomes 0.
• This new number 0 (zero), when added to the Natural numbers gave a new set of numbers called Whole number.

                          W= { 0, 1, 2, 3, 4, 5, ….}

• It is denoted by W. It has all-natural numbers plus 0.
• All-Natural numbers are whole number but the reverse is not true.

Integers:

• As the field of Mathematics advanced & there was a need for negative numbers as well. If we make a set negative numbers to the whole number, we get Integers. It is denoted by “Z”.

Z={..,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,…}

• Z came from a German word Zahlen that means “to count”. It is used to express temperature, latitude, longitude etc which can have negative values.
• Integers contains all the whole number plus negative of all the natural numbers
• The natural numbers without zero are commonly referred to as positive integers
• The negative of a positive integer is defined as a number that produces 0 when it is added to the corresponding positive integer.
• Natural numbers with zero are referred to as non-negative integers
• The natural numbers form a subset of the integers.

Rational Numbers:

• Numbers that can be represented in the form \( \frac{p}{q} \) where q ≠ 0 and p, q are coprime, are  called Rational Number.
• Word Rational number came from Ratio. It is denoted by Q. Q letter is taken from word Quotient. e.g.1/2, 9/5 etc.
• Every integer, natural and whole number is a rational number as they can be expressed in terms of p/q
• There are infinite rational number between two rational number.
• They either have termination decimal expression or non-terminating repeating decimal expression
• The sum, difference and the product of two rational numbers is always a rational.
• The quotient of a division of one rational number by a non-zero rational number is a rational number.

Irrational Numbers:

• A number is called irrational if it cannot be expressed in the form of \( \frac{p}{q}\), where p and q are integers ( q ≠ 0).

Example: \(  \sqrt{2},\,\,\sqrt{3},\,\,\sqrt{5},\,\pi  \)  etc.

• They have non-terminating and non-repeating decimal expression.
• The sum, difference, multiplication and division of irrational numbers are not always irrational.

Real Numbers:

• All rational and all irrational number makes the collection of real numbers. It is denoted by the letter R.
• The sum or difference of a rational number and an irrational number is an irrational number.
• The product or division of a rational number with an irrational number is an irrational number.
• This process of visualization of representing a decimal expansion on the number line is known as the process of successive magnification.
• Real numbers satisfy the commutative, associative and distributive laws. These can be stated as :
• Commutative Law of Addition:

𝒂 + 𝒃 = 𝒃 + 𝒂

• Commutative Law of Multiplication:

𝒂 × 𝒃 = 𝒃 × 𝒂

• Associative Law of Addition:

𝒂 + (𝒃 + 𝒄) = (𝒂 + 𝒃) + 𝒄

• Associative Law of Multiplication

𝒂 × (𝒃 × 𝒄) = (𝒂 × 𝒃) × 𝒄

• Distributive Law:

𝒂 × (𝒃 + 𝒄) = (𝒂 × 𝒃) + (𝒂 × 𝒄)

Important Points

Even Number:

• A natural number which divisible by 2 is called as Even Number. It is in the form of 2n.

         Example: 2, 4, 6,8, …….

Odd Numbers:

• A natural number which is not divisible by 2 , called as Odd number. It is in the form of 2𝑛 − 1

         Example: 1, 3, 5, …….

Prime Numbers:

• The natural numbers greater than 1 which are divisible by 1 and the number itself are called prime numbers.
• Prime numbers have two factors i.e., 1 and the number itself

Example: 2, 3, 5, 7 & 11 etc.

• List of all prime numbers up to 100 is given below.

2            11      23      31      41      53      61      71      83      97

3            13      29      37      43      59      67      73      89

5            17      47      79

7            19

Note:1 is not a prime number as it has only one factor.

Composite Numbers:

• The natural numbers which are divisible by 1, itself and any other number or numbers are called composite numbers.

Example: 4, 6, 8, 9, 10 etc.

Note: 1 is neither prime nor a composite number.

Coprime Number:

• When two numbers have no common factors other than 1. Number a, b will be coprime if 𝑯𝑪𝑭(𝒂, 𝒃) = 𝟏

Example:

(i) 21 and 22 are coprime:

The factors of 21 are 1, 3, 7 and 21

The factors of 22 are 1, 2, 11 and 22

The only common factor is 1 therefore they are coprime

(ii)  But 21 and 24 are NOT coprime:

The factors of 21 are 1, 3, 7 and 21

The factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24

The common factors are 1 & 3 therefore they are not coprime

Fractions

Common fraction:        Fractions whose denominator is not 10.

Decimal fraction:          Fractions whose denominator is 10 or any power of 10.

Proper fraction:             Numerator < Denominator i.e. \( \displaystyle \frac{4}{7}\)

Improper fraction:       Numerator > Denominator i.e.\( \displaystyle \frac{9}{5}\)

Mixed fraction:              Consists of integral as well as fractional part i.e. \( \displaystyle 5\frac{4}{7} \)

Compound fraction:     Fraction whose number and denominator themselves are fractions i.e.\(\displaystyle \frac{{\frac{4}{7}}}{{\frac{5}{3}}}\)

Types of Decimal Numbers:

(a) Terminating Decimals: e.g  3.5, 12.345, 678.005, 0.0254 etc.

(b) Non-Terminating Repeating Decimals: e.g 1.3333…, 2.123123123…., \( \displaystyle 5.\overline{{026}}\),\( \displaystyle 6.23\overline{{69}}\) etc.

(i) Pure Recurring Decimals: All digits after decimal repeats e.g 1.235235235….., \( \displaystyle 78.\overline{{789}}\), \( \displaystyle 0.\overline{{006}}\)

(ii) Mix Recurring Decimals: All digits after decimal do not repeat. e.g \( \displaystyle 1.5\overline{{26}}\), \( \displaystyle 3.124\overline{{759}}\)

(c) Non-Terminating Non Repeating Decimals: e.g 1.202002000200002……. etc.