Rational Numbers
Focused last-minute revision with key concepts, formulas, worked examples, exam traps and practice.
Chapter Snapshot
- A rational number can be written as p/q where p and q are integers and q is not zero.
- 0 is the additive identity and 1 is the multiplicative identity.
- Addition and multiplication are commutative and associative; subtraction and division are not.
- Every rational number has an additive inverse; every non-zero rational number has a multiplicative inverse.
- There are infinitely many rational numbers between any two distinct rational numbers.
What You Will Learn
- Recognise rational numbers and write them in standard form
- Use closure, commutative, associative and distributive properties correctly
- Find additive and multiplicative inverses
- Perform the four operations on rational numbers
- Represent and compare rational numbers on a number line
Last-Minute Chapter Map
Use this chapter in three passes:
- Understand the core ideas.
- Memorise formulas and exam traps.
- Attempt the compact practice set without looking at the answers.
Core idea
A rational number is any number that can be written as p/q with integer numerator and denominator and denominator not equal to zero. Integers are also rational because an integer n can be written as n/1. In standard form, the denominator is positive and numerator and denominator have no common factor except 1.
Properties to remember
Rational numbers are closed under addition, subtraction and multiplication, and under division by a non-zero rational number. Addition and multiplication are commutative and associative. Subtraction and division are neither commutative nor associative. Multiplication distributes over addition and subtraction.
Operations
For addition or subtraction, first make denominators equal. For multiplication, multiply numerators together and denominators together and simplify. For division, multiply by the reciprocal of the second rational number.
Number line and numbers between
Positive rational numbers lie to the right of zero and negative rational numbers to the left. To place a fraction such as 3/4, divide the interval from 0 to 1 into four equal parts and mark the third point. A quick way to find one rational number between a and b is to take their average.
Rational numbers have exact positions on the number line.
Formula Sheet
Formula / Key Relations
Distributive property
Also use a(b-c)=ab-ac.
Additive inverse
Their sum is 0.
Multiplicative inverse
Their product is 1.
One rational between a and b
Repeat the method to obtain more.
Worked Examples
Change division to multiplication by the reciprocal, then simplify.
The average gives one valid rational number.
Exam Traps
Quick Revision
Quick Revision
- A rational number can be written as p/q where p and q are integers and q is not zero.
- 0 is the additive identity and 1 is the multiplicative identity.
- Addition and multiplication are commutative and associative; subtraction and division are not.
- Every rational number has an additive inverse; every non-zero rational number has a multiplicative inverse.
- There are infinitely many rational numbers between any two distinct rational numbers.
- Revise the formula sheet once more before attempting questions.
Compact Practice
Additive identity for rational numbers?
View Solution
0
Which is not commutative: addition, multiplication or subtraction?
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Subtraction
Write -12/18 in standard form.
View Solution
-2/3
Find the reciprocal of -7/9.
View Solution
-9/7
Find one rational number between 1/3 and 1/2.
View Solution
5/12
Final Chapter Summary
Chapter Summary
- A rational number can be written as p/q where p and q are integers and q is not zero.
- 0 is the additive identity and 1 is the multiplicative identity.
- Addition and multiplication are commutative and associative; subtraction and division are not.
- Every rational number has an additive inverse; every non-zero rational number has a multiplicative inverse.
- There are infinitely many rational numbers between any two distinct rational numbers.
- Prioritise correct method, clean working and final units where applicable.