Counting numbers 1, 2, 3, 4, 5, .......etc. are called Natural numbers. Set of natural numbers is generally denoted by N.
All the natural numbers together with zero are called Whole numbers. The numbers 0, 1, 2, 3, 4, 5, ....... etc. are called Whole numbers. Set of Whole numbers is generally denoted by W. Every Natural number is a Whole number.
All natural numbers, zero and negatives of the natural numbers are called Integers, i.e. ......– 5, – 4, – 3, – 2, – 1, 0, 1, 2, 3, 4, 5,...........etc. are integers. Set of Integers is generally denoted by I or Z. Every Whole number is an Integer.
Numbers that can be written in the form of , where p and q are integers and q ≠ 0 are called Rational numbers. The collection of Rational numbers is denoted by Q. Between any two rational numbers there exists infinitely many rational numbers.
Numbers which cannot be expressed in the form of , where p and q are integers and q ≠ 0.
The set of irrational numbers is denoted by . , , are the examples of irrational numbers.
The ratio of the length of circumference of a circle to the length of its diameter is always constant. It is an irrational number and denoted by π. Decimal expansion of π is non-terminating and non-repeating. Value of π = 3.14159265......... Approximate value of π is , but not equal to the exact value.
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Irrational numbers can be represented on the number line using Pythagoras theorem.
The collection of all rational numbers and irrational numbers together is the set of real numbers. It is represented by R. A real number is either a rational number or an irrational number.
In a division, if remainder becomes zero after certain stage, then the decimal expansion is terminating.
If the remainder never becomes zero but repeats after certain stage, then the decimal expansion is non-terminating recurring.
A rational number can be expressed as its decimal expansion. The decimal expansion of a rational number is either terminating or non-terminating recurring. The decimal expansion of an irrational number is non-terminating non-recurring. Every real number can be represented on a number line uniquely. Conversely every point on the number line represents one and only one real number.
The process of visualisation of representing a decimal expansion on the number line is known as the process of successive magnification.
Operations on Real Numbers
The sum, difference and the product of two rational numbers is always a rational number. The quotient of a division of one rational number by a non-zero rational number is a rational number. Rational numbers satisfy the closure law under addition, subtraction, multiplication and division. They also satisfy the commutative law and associative law under addition and multiplication.
The sum, difference, multiplication and division of irrational numbers are not always irrational. Irrational numbers do not satisfy the closure property under addition, subtraction, multiplication and division.
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) or (a + b) × c = (a × c) + (b × c)
Real numbers can be represented on the number line. The square root of any positive real number exists and that also can be represented on number line.
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.
Some of the basic identities involving square roots are:
If a, b, c and d are positive real numbers,
( + )(– ) = a – b
(a + )(a – ) = a2 – b
( + )( + ) = + + +
( + )2 = a + 2 + b
If the product of two irrational numbers is a rational number, then each of the irrational numbers is called the rationalising factor of the other. The process of multiplying an irrational number with its rationalising factor to get the product as a rational number is called rationalisation of the given irrational number.
Laws of Exponents
An exponent is a mathematical notation that represents how many times a base is multiplied by itself. Other terms used to define exponents are ‘power’ or ‘index’. An exponential term is a term that can be expressed as a base raised to an exponent. For example, in an exponential expression an, 'a' is the base and ‘n' is the exponent.
The exponents can be numbers or constants; they can also be variables. Exponents are generally positive real numbers, but they can also be negative numbers.
Laws of exponents:
If a and b are any real numbers, and m and n are rational numbers then,
- am × an = am+n
- = am-n, m > n.
- (am)n = amn
- (am × bm) = (a × b)m
- = ()m
- a0 = 1
- a-n =