Rational Numbers
ICSE Class 8 · Mathematics · Chapter 1
What this chapter covers
- What a rational number is
- Comparing rational numbers
- Representation on the number line
- Operations on rational numbers
- Properties of rational numbers
- Rational numbers between two rational numbers
- Word problems
Key points
- A rational number can be written as (p)/(q), where p and q are integers and q ≠ 0.
- Every integer and every fraction is a rational number.
- Standard form has a positive denominator and no common factor other than 1.
- To compare rational numbers, use positive denominators, then a common denominator, then compare numerators.
- Each rational number has one point on the number line; numbers to the right are greater.
- To subtract, add the additive inverse; to divide, multiply by the reciprocal.
- Rational numbers are closed under addition, subtraction and multiplication, and under division by any non-zero number.
- Addition and multiplication are commutative and associative; subtraction and division are not.
- 0 is the additive identity and 1 is the multiplicative identity.
- Every rational number has an additive inverse; every non-zero rational number has a reciprocal.
- Multiplication distributes over addition and subtraction: a(b + c) = ab + ac.
- Half the sum of two rational numbers lies between them, so there are infinitely many rational numbers between any two.
Questions answered
Write (-45)/(-60) in standard form.
Both signs are negative, so the number is (45)/(60). The HCF is 15, giving (3)/(4).
Which is greater, (-2)/(3) or (-3)/(5)?
With denominator 15, they are (-10)/(15) and (-9)/(15). Since -9 > -10, (-3)/(5) is greater.
Between which two integers does (-11)/(4) lie on the number line, and how would you mark it?
(-11)/(4) = -2(3)/(4), so it lies between -3 and -2. Divide each unit gap into 4 parts and count 11 parts to the left of 0.
Find (5)/(6) - (-3)/(4).
(5)/(6) + (3)/(4) = (10)/(12) + (9)/(12) = (19)/(12).
Find (-8)/(15) ÷ (-4)/(5).
(-8)/(15) × (5)/(-4) = (40)/(60) = (2)/(3).
Are rational numbers closed under division? Explain briefly.
No. Division by zero gives no number. If the divisor is not zero, the quotient is always rational.
Give one example to show that subtraction of rational numbers is not commutative.
(1)/(2) - (1)/(4) = (1)/(4), but (1)/(4) - (1)/(2) = (-1)/(4). The results differ.
Write the additive inverse and the multiplicative inverse of (-5)/(9).
The additive inverse is (5)/(9). The multiplicative inverse is (9)/(-5), which is (-9)/(5).