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.
Try a topic test
Choose Easy, Normal or Difficult. Save your answers and open worked review after finishing.
Test this topic →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.
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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.