Rational and Irrational Numbers
Real Numbers
The collection of real numbers is the collection of all the rational numbers and irrational numbers together .
It is represented by R.
A real number is either rational or 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 the 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.
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.
There are infinitely many irrational numbers between two rational numbers.
Every real number can be represented on a number line uniquely.
Introduction to Rational Numbers
Whole numbers, integers, fractions and decimal numbers together from the group of rational numbers.
A rational number is a number that can be written in the form p/q, where p and q are integers and q#0.
The denominator of a rational number can never be zero.
A rational number is positive if both its numerator and denominator are positive integers or negative integers.
If either the numerator or the denominator of a rational number is a negative integer, then the rational number is called a negative rational number.
The rational number zero is neither negative nor positive.
On the number line:
• Positive rational numbers are represented to the right of 0.
• Negative rational numbers are represented to the left of 0.
By multiplying or dividing both the numerator and the denominator of a rational number by the same non-zero integer, we can get another rational number that is equivalent to the given rational number.
A rational number is said to be in its said to be in its standard from if its numerator and denominator have no common factor other than 1, and its denominator is a positive integer.
To reduce a rational number to its standard from, divide its numerator and denominator by their Highest Common Factor or HCF.
To find the standard form of a rational number with a negative integer as the denominator, divide its numerator and denominator by their HCF with a minus sign.
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 law.
• Commutative Law of Addition: a+b=b+a
• Commutative Law of Multiplication: a x b = b x a
• Associative Law of Addition: a + (b + c) = (a x 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:
• √ = √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 a rational number is called rationalising the denominator.
Expressing Decimals as Rational Numbers
Numbers of the form p/q, where p and q are integers and q#0 , are called rational numbers.
Decimal numbers formed by a division operation in which the remainder is zero are called terminating decimal numbers. Decimal numbers formed by a division operation in which the remainder is never zero, and a digit or a set of digits repeats itself endlessly after the decimal point, are called nonterminating or recurring decimal numbers.
Recurring decimal numbers in which all the digits after the decimal point get repeated are called pure recurring decimal numbers. Recurring decimal numbers in which one or more digits immediately to the right of the
decimal point do not get repeated, but the digits following them get repeated, are called mixed recurring decimal numbers.
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 represents whole numbers and the collection of Whole numbers 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 numbers is denoted by Q.
Pythagoras Theorem
In a right-angles triangle, the square of the hypothenuse is equal to the sum of the squares of the other two sides.