Rational and Irrational Numbers
ICSE Class 9 · Mathematics · Chapter 1
What this chapter covers
- The real number system
- Rational numbers
- Irrational numbers
- Proofs of irrationality of root 2, root 3 and root 5
- Representing real numbers on the number line
- Surds
- Rationalisation of surds
Key definitions
Natural numbers are the counting numbers 1, 2, 3, 4, ldots, and the set of natural numbers is written N.
Whole numbers are the natural numbers together with zero, that is 0, 1, 2, 3, ldots, and the set is written W.
Integers are the whole numbers together with the negatives of the natural numbers, that is ldots, -3, -2, -1, 0, 1, 2, 3, ldots, and the set is written Z (or I).
Real numbers are all the rational and all the irrational numbers taken together, and the set of real numbers is written R.
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 density of rational numbers is the property that between any two different rational numbers there are infinitely many other rational numbers.
An irrational number is a real number that cannot be written in the form (p)/(q), where p and q are integers and q ≠ 0.
A non-terminating non-recurring decimal is a decimal that goes on for ever without any block of digits repeating, and every such decimal represents an irrational number.
The number line is a straight line on which every point stands for exactly one real number and every real number is marked by exactly one point.
A surd is an irrational root of a positive rational number; that is, sqrt[n]a is a surd when a is a positive rational number, n is a natural number greater than 1, and sqrt[n]a is irrational.
A pure surd is a surd whose whole rational factor is 1, such as √(18) or sqrt[3]7.
A mixed surd is a surd with a rational factor other than 1 in front of the root, such as 3√(2) or (1)/(2)sqrt[3]5.
Key points
- Natural numbers, whole numbers, integers and rational numbers sit inside one another: N subset W subset Z subset Q subset R.
- A rational number can be written as (p)/(q) with p, q integers and q ≠ 0; its decimal form terminates or recurs.
- A fraction in lowest terms has a terminating decimal exactly when its denominator has no prime factor other than 2 and 5.
- An irrational number cannot be written as (p)/(q); its decimal form is non-terminating and non-recurring.
- The real numbers are all the rational and irrational numbers together; every real number is exactly one of the two.
- Between any two real numbers there are infinitely many rational and infinitely many irrational numbers.
- A non-zero rational number added to, or multiplied by, an irrational number gives an irrational number; two irrationals may combine to give either kind.
- √(2), √(3) and √(5) are proved irrational by contradiction, using the fact that a prime dividing a2 must divide a.
- √(n) is marked on the number line by building a right triangle whose hypotenuse is √(n) and swinging an arc from 0.
- Every point of the number line is a real number, and every real number is a point of the line.
- A surd is an irrational root sqrt[n]a of a positive rational number a; n is its order and a its radicand.
- Only like surds can be added or subtracted; reduce to simplest form first.
- √(a) × √(b) = √(ab), but √(a) + √(b) ≠ √(a + b).
- To rationalise a denominator, multiply numerator and denominator by √(a) for a monomial surd, or by the conjugate for a binomial surd.
- (a + √(b))(a - √(b)) = a2 - b and (√(a) + √(b))(√(a) - √(b)) = a - b.
Questions answered
Define a rational number and an irrational number.
State, with a reason, whether each is rational or irrational: √(196), 3.overline27, √(8), (√(45))/(√(5)), 0.12112111211112ldots
Without dividing, state whether (13)/(125) and (7)/(15) have terminating or recurring decimals.
Express 0.overline45 and 1.overline3 in the form (p)/(q).
Insert two rational numbers and two irrational numbers between √(2) and √(3).
Prove that √(3) is irrational.
Prove that 5 + √(2) is irrational.
Why does the proof used for √(5) not show that √(9) is irrational?
Past board exam questions
ICSE 2019 sample paper · 3 marks
Rationalize the denominator: (14)/(5√(3)-√(5))