Lesson 1: Rational and Irrational Numbers | State Board · Class 8 · Mathematics

Rational numbers and the number line

Natural numbers: 1, 2, 3, … Whole numbers: 0, 1, 2, … Integers: …, −2, −1, 0, 1, 2, …
Rational numbers: numbers of the form mn, where m and n are integers and n ≠ 0. Examples: −253, 10−7, −4, 0, 8.
Every integer is rational, because n = n1. There are infinitely many rational numbers between any two rational numbers.
Showing a rational number on a number line: to show 73, divide each unit on the right of zero into 3 equal parts (the denominator). 73 = 7 × 13, so it is the 7th point from zero; also 73 = 2 + 13.
To show −23, first show 23; the point at the same distance to the left of 0 shows −23.

To show 73, each unit is divided into 3 equal parts. P is the 7th mark from 0, so P shows 73 = 2 + 13.

Natural numbers, whole numbers and integers are all rational numbers: the sets are nested, and a number like 37 is rational but not an integer.

On a number line, of two numbers the one on the left is smaller.
The value of a rational number does not change if numerator and denominator are multiplied by the same non-zero number: ab = kakb (k ≠ 0).
To compare, make the denominators equal. Example: 54 = 1512 and 23 = 812, so 54 > 23.
A negative number is always less than a positive number: −79 < 45.
Two negative numbers: if a < b (positive), then −a > −b. So 2 < 3 but −2 > −3; 54 < 74 but −54 > −74. Example: 73 < 52, therefore −73 > −52.
Rule: for ab and cd with b, d positive: if a × d < b × c then ab < cd; if a × d = b × c then they are equal; if a × d > b × c then ab > cd.

The number on the left is smaller: A shows 23 = 812 and B shows 54 = 1512, so 23 < 54.

Of two negative numbers, the one nearer to 0 is greater: B (−73) is on the right of A (−52), so −73 > −52.

Live figure

Comparison of rational numbers

In the app you can move this figure with your finger.

Divide the numerator by the denominator. If the remainder becomes zero, the decimal is a terminating decimal: 74 = 1.75.
Every rational number can be written as a non-terminating recurring decimal: 76 = 1.1666… ; 56 = 0.8333… ; −53 = −1.666… ; 227 = 3.142857142857… (block 142857 repeats); 2399 = 0.2323…
A terminating decimal can also be written as a recurring one: 1.75 = 1.75000…
Other results: 937 = 0.243243… ; 37 = 0.428571428571… ; −1035 = −20.6 (terminating) ; −1113 = −0.846153846153…
A fraction in lowest terms terminates when its denominator has only the prime factors 2 and 5 (e.g. 8, 20, 25).

Long division of 3 by 7: the remainders 3, 2, 6, 4, 5, 1 repeat, so 37 = 0.428571428571… (a recurring block of 6 digits).

Long division of 7 by 4: after the whole part 1 the remainders are 3, 2, 0. The remainder becomes 0, so the decimal terminates: 74 = 1.75.

A fraction in lowest terms terminates when the denominator has only the prime factors 2 and 5 (4 and 25 here); 6 and 15 have the factor 3, so 56 and 1115 recur.

Besides rational numbers there are many more numbers on the number line. Numbers that are not rational are irrational numbers. √2 is one.
The decimal form of an irrational number is non-terminating and non-recurring. (The proof that √2 is irrational is learnt in the next standard.)
To show √2: (1) point A shows 1; draw a line l perpendicular to the number line at A. (2) Take P on l with OA = AP = 1 unit. (3) Join OP; the right-angled triangle OAP gives OP² = 1² + 1² = 2, so OP = √2. (4) With centre O and radius OP draw an arc cutting the number line at Q. Q shows √2. The point at the same distance on the left of O shows −√2.
To show √3: from Q (showing √2) draw a perpendicular QR = 1. OR² = (√2)² + 1² = 3. The arc with centre O and radius OR cuts the line at C, which shows √3.
√5: take OA = 2 and AP = 1: OP² = 4 + 1 = 5. √7 can be shown similarly (7 = 6 + 1).
√4 = 2, √9 = 3, √16 = 4 … are rational; √2, √3, √5, √7 … are irrational.

Showing √2: OA = AP = 1, so OP² = 1² + 1² = 2. The arc with centre O and radius OP cuts the number line at Q, which shows √2.

Showing √3: with OA = √2 and AP = 1, OP² = 2 + 1 = 3. The arc cuts the number line at Q, which shows √3.

Live figure

Irrational numbers and showing √2 on the number line

In the app you can move this figure with your finger.

Numbers that can be shown by points on the number line are real numbers. All rational numbers are real numbers. There are infinitely many irrational numbers on the number line. Rational and irrational numbers together make the real numbers.
π is irrational. For calculation we take its value as 227 or 3.14, which are rational numbers close to π.
Numbers like 3√2, 7 + √2, 3 − √2 are irrational. Reason: if 3√2 were rational, then 3√23 = √2 would also be rational, which is not true.
Irrational numbers like √3, √5, … can be shown on the number line in the same way as √2.

Real numbers are rational (0.75, −4, 227) or irrational (√3, π, 5 + √2). The number 227 is only a rational value close to π.

√2 (A), √3 (B) and π (C) are real numbers shown on the number line; their decimals are non-terminating and non-recurring.

Rational = mn (n ≠ 0); infinitely many between two rational numbers; show on number line by dividing units into n parts.
Compare by common denominator; a × d versus b × c; negative numbers reverse the order.
Decimal: terminating (remainder 0) or non-terminating recurring.
Irrational: non-terminating non-recurring; √2 by right triangle with sides 1 and 1; √3 using √2 and 1.
Real numbers = rational + irrational; π is irrational; 3√2, 7 + √2 are irrational.