Natural Numbers
Counting numbers 1,2,3,4…. are called natural numbers and the collection of natural numbers is denoted by N.
Whole Numbers
Zero along with the natural numbers represent whole numbers and the collection of Whole number is denoted by ‘W’.
Integers
Integers are collection of positive and negative numbers including zero and the collection of integers is denoted by ‘Z’.
Rational Numbers
* A number ‘r’ is called a rational number, if it can be written in the form p/q, where p,q ε Z and q≠0.
* Between any two rational numbers there exists infinitely many rational numbers.
* The collection of rational numbers is denoted by Q.
Irrational Numbers
* A number which cannot be expressed in the form of p/q, where p,q ε Z and q≠0 is called an irrational number.
* The collection of irrational number is denoted by Q.
Pythagoras Theorem
In a right- angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
Real Numbers
• The collection of real numbers is the collection of all rational numbers and irrational numbers together. It is represented by R.
• A real number is either rational and irrational.
• A rational number p/q, where p and q are integers and q≠0 can be expressed as its decimal expansion.
In case of division:
• If remainder becomes zero after certain stage, then the decimal expansion is terminating.
• If the remainder never becomes zero but repeated after certain stage, then the decimal expansion is non- terminating recurring.
• The decimal expansion of a rational number is either terminating of non- terminating recurring.
• The decimal expansion of an irrational number is non- terminating non-recurring.
• There are infinitely many irrational numbers between two rational numbers.
• Every real number can e represented on a number line uniquely.
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 property under addition, subtraction, multiplication, and division.
• 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.
• Commutative Law of Addition: a+b = b+a
• Commutative Law of Multiplication: a x b = b x a
• Associative Law of Addition: a +(b x c) = (a + b) + c
• Associative Law of Multiplication: a x (b x c) + (a x b) x c
• Distributive Law : a x(b+c) = a x b + a x c or (a+b) x c = a x c + b x c
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.
Basic identities involving square roots are:
• √(ab) = √a √b
• √(a/b) = √a/ √b
• (√a + √b) (√a- √b)= a-b
• (a+ √b) (a- √b) = a² – b
• (√a + √b) (√c + √d)= √(ac) + √(ad) + √(bc) + √(bd)
• (√a + √b )² = a + 2 √(ab) + b
a,b,c and d are positive real numbers.
The process of converting the denominator into rational number is called rationalising the denominator.
Laws Of Exponents For Real Numbers
• 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 ‘n’ is the exponent.
• An exponent is a mathematical notation that represents how many times a base i multiplied by itself. Other terms used to define exponents are ‘ power ‘ or ‘index’.
• The exponents can be a number or a constant ; they can also be a variable. They are generally positive real numbers, but they can also be negative numbers.
Laws of exponents:
• am x an = am+n
• am/an = am-n
• (am)n= amn
• (am x bm)= (a x b )m
• am/bm=(a/b)m
• a0=1
• a-n = 1/an