Number families and reciprocals
Understand what a number represents before choosing a calculation. A number can belong to several families at once.
Core and Extended. Continue to the next lesson for factors and multiples in the same syllabus section.
Before you begin
- Recognise place value in whole numbers.
- Know that a fraction represents division.
Work on paper. Try each question before revealing help, and explain why your method works.
Quick readiness check
Which fraction equals 0.6?
What you will learn
- Classify integers, rational numbers and irrational numbers with reasons.
- Read large numbers using place value.
- Find a reciprocal and check it by multiplication.
Start with the quantity
A count of books, a temperature and a share of a cake need different kinds of number. Natural numbers are counting numbers: 1, 2, 3, and so on. Some conventions include zero; state your convention if it matters. Integers include zero and the negative whole numbers as well. A temperature of −4°C is an integer; a length of 4.5 cm is not.
Place value tells you the size of each digit. Group digits in threes from the right: units, thousands, millions, billions. The number 4 030 006 is four million, thirty thousand and six. The zeros hold empty places. To write two million and forty, begin with 2 000 000 and add 40: 2 000 040.
Rational means an exact ratio of integers
A rational number can be written as a/b, where a and b are integers and b is not zero. This includes every integer because, for example, −4 = −4/1. It also includes terminating decimals, such as 0.125 = 125/1000 = 1/8, and recurring decimals, such as 0.333… = 1/3. Recurring means a fixed block of digits repeats forever.
An irrational number cannot be written as such a fraction. Its decimal expansion neither ends nor settles into a repeating block. Examples include π and √2. The symbol √ means square root: √2 is the non-negative number whose square is 2. A calculator shows a finite approximation; that display does not make the exact number rational. Also, a root symbol does not guarantee an irrational number: √49 = 7 is rational.
Special families overlap
A positive prime number has exactly two positive factors: 1 and itself. Thus 2 is prime, but 1 is not. A square number is an integer squared; a cube number is an integer cubed. The positive numbers 1, 4, 9 and 16 are squares, while 1, 8, 27 and 64 are cubes. The number 64 is both a square and a cube. Zero is also a square and a cube.
Do not treat these labels as separate boxes that cannot overlap. The number 9 is a natural number, an integer, rational and a square. It is not prime because 3 is an additional factor. Being odd does not make a number prime: 9 and 15 are counterexamples.
A reciprocal undoes multiplication
The reciprocal of a non-zero number is the number that multiplies it to give 1. For a/b with a non-zero numerator, it is b/a. The reciprocal of 5 is 1/5; of −3/4 it is −4/3. The sign stays negative because the product of two negatives is positive.
Convert a mixed number into an improper fraction first: two wholes contain six thirds, so 2 1/3 = 7/3 and its reciprocal is 3/7. Zero has no reciprocal because zero multiplied by any real number remains zero. A reciprocal is different from an additive opposite: the opposite of 5 is −5, which adds to 5 to give zero.
Pause and explain
Which statement is always true?
Worked example
Classify −2.5 as integer, rational or irrational. Find its reciprocal.
Show the worked solution
- Write the decimal exactly as a fraction: −2.5 = −25/10 = −5/2.
- It is rational because its numerator and denominator are integers. It is not an integer, and cannot also be irrational.
- Reverse the fraction: the reciprocal is −2/5. Check: (−5/2) × (−2/5) = 1.
Answer Rational, not an integer; reciprocal −2/5.
From guided practice to a new situation
Write your answer and reasoning first. Use a hint only if you are stuck. The model solution is for self-checking; your written work is not automatically marked.
Find the reciprocal of 1 3/4. Show the check.
Give me a hint
First count the total number of quarters.
Compare my reasoning
- 1 3/4 = 7/4.
- The reciprocal is 4/7.
- (7/4) × (4/7) = 1.
4/7
Look for these in your work
- Converted the whole number and fractional part together.
- Checked the product is 1.
Write five million, eight thousand and nine in digits. Classify −12 and √7 as rational or irrational.
Give me a hint
Build the large number as 5 000 000 + 8 000 + 9.
Compare my reasoning
- Place values give 5 008 009.
- −12 = −12/1, so it is rational.
- Seven is not a perfect square; √7 is irrational.
5 008 009; −12 rational; √7 irrational.
Look for these in your work
- Kept the zero placeholders.
- Used exact definitions rather than calculator digits.
A square has area 2 cm². A screen reports its side as 1.414 cm. A student says the exact side must be rational because 1.414 = 707/500. Explain the error.
Give me a hint
Separate the exact length from the screen's approximation.
Compare my reasoning
- The exact side is √2 cm, since side × side = area.
- 1.414 is a rational approximation, not the exact side.
- For a check, 1.414² = 1.999396, which is not exactly 2.
The displayed approximation is rational; the exact length √2 is irrational.
Look for these in your work
- Identified which number is exact.
- Explained why a finite display cannot decide the exact number's family.
Common mistakes
- A negative sign does not make a number irrational.
- A rounded calculator display is not the exact value of an irrational number.
- Do not confuse a reciprocal with an opposite.
What can you explain now?
Is √81 irrational? What is its reciprocal?
Compare with the explanation
No. √81 = 9, and its reciprocal is 1/9.
Classify the value, not the appearance of its notation. Nine is an integer and therefore rational.
After trying it yourself, choose your next review. This is your self-assessment.
Your review choice appears on Today. Sign in to sync it across devices.
Make it stick
Tomorrow, explain why −8 is rational and zero has no reciprocal without opening these notes.
If you needed the solution, close it and solve the problem again from a blank page. A correct answer is stronger when you can explain the reason for each step.
Original StudyForA teaching content · AI-assisted checks · Human teacher review pending.
Number contains 19 sequenced lessons; Algebra and graphs contains 21; Coordinate geometry contains 10 teaching lessons and a mixed checkpoint. Nine earlier overviews support selected topics across the syllabus. The other six syllabus areas, full cumulative assessment and human teacher review are not yet complete. Written lesson practice is self-checked, not automatically graded. The checkpoint samples skills and does not save an exam grade or certify mastery.
Check the official syllabus 2025–2027 Open related practice and resources