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Arithematics

Arithematics

Types of number

Number Families Paper 1 + 2 9 min read

On this page
  1. In this note
  2. Formula bank
  3. The number families
  4. Testing for a prime
  5. Rational and irrational numbers
  6. Reciprocals; numbers in words
  7. Ways they test it
  8. Mistakes to avoid
  9. Summary
  10. Practice
  11. Can I do this?
  12. Quiz

In this note

Syllabus 1.1: types of number
  • 1.1Recognise natural numbers, integers, primes, squares and cubes; use factors and multiples.
  • 1.1Tell rational from irrational numbers, and find reciprocals.
  • 1.1Write numbers in words and in figures.

HCF and LCM in full: HCF and LCM.

Formula bank

Prime numberLearn itexactly two factors: 1 and itself1 is NOT prime; 2 is the only even prime.Use it when "write down a prime number".
Prime testLearn itdivide by primes up to \(\sqrt n\)none divides exactly → prime.Use it when is 91 prime?
Squares and cubesLearn it\(1^2\ldots15^2=225\); cubes 1, 8, 27, 64, 125, 1000learn them.Use it when picking from a list.
RationalLearn it\(\dfrac ab\) with integers \(a,b\) (\(b\ne0\))its decimal stops or recurs.Use it when rational or irrational?
IrrationalLearn ite.g. \(\pi\), \(\sqrt2\), \(\sqrt{12}\)\(\sqrt n\) is irrational unless n is a perfect square.Use it when rational or irrational?
ReciprocalLearn it\(\dfrac1n\); for a fraction, turn it upside downnumber × reciprocal = 1; 0 has none.Use it when "reciprocal of".

The number families

FamilyWhat it meansExamples
Natural numbersthe counting numbers1, 2, 3, 4, …
Integerswhole numbers: positive, zero or negative…, −2, −1, 0, 1, 2, …
Primesexactly two factors2, 3, 5, 7, 11, 13, … (not 1)
Squaresan integer × itself1, 4, 9, 16, …, 225
Cubesan integer × itself × itself1, 8, 27, 64, 125, 1000
Factors of ndivide n exactly (they come in pairs)12: 1, 2, 3, 4, 6, 12
Multiples of nn × 1, n × 2, …8: 8, 16, 24, 32, …

General methodHow to pick numbers from a list

  1. Go through the list once for each family asked.
  2. Odd-looking numbers (27, 51, 57, 91, 119): test them with 3, 7 and 13 - they are not prime.
  3. Give exactly as many answers as asked; an extra wrong one loses the mark.

Example From 21, 25, 27, 29, 33, 36 write down a prime, a cube and two squares.

Test each odd number for small factors.

  1. 21 = 3 × 7, 27 = 3³, 33 = 3 × 11, so 29 is the prime
  2. \(27=3^{3}\)
  3. \(25=5^{2}\), \(36=6^{2}\)

Answer 29; 27; 25 and 36

Try it yourself

1
Write down the prime numbers between 40 and 50.
Show answer

Answer 41, 43, 47

2
Which number between 50 and 100 is both a square and a cube?
Show answer

Answer 64

Testing for a prime

General methodHow to test whether any number is prime

  1. Work out roughly \(\sqrt n\).
  2. Divide by the primes 2, 3, 5, 7, 11, … up to \(\sqrt n\).
  3. If one divides exactly, it is NOT prime (write the factor pair). If none does, it IS prime.

Why stop at \(\sqrt n\)? Factors come in pairs, and one of each pair is at most \(\sqrt n\).

Example Is 91 prime?

\(\sqrt{91}\approx9.5\): try 2, 3, 5, 7.

  1. not ÷ 2, 3 or 5
  2. \(91\div7=13\)

Answer No: 91 = 7 × 13

Try it yourself

1
Is 51 prime? If not, give its factor pair.
Show answer

Answer No: 3 × 17

2
Find the only prime between 110 and 115.
Hint
111 = 3 × 37.
Show answer

Answer 113

Rational and irrational numbers

A rational number can be written as a fraction of two integers, so its decimal stops or recurs. An irrational number cannot: its decimal goes on forever with no pattern.

NumberRational?Why
\(\sqrt{16}\)rational= 4
\(\sqrt{12}\)irrational12 is not a square
\(0.\dot3\)rational\(=\tfrac13\)
\(\pi\)irrationalnever repeats
\(\tfrac{22}{7}\)rationala fraction of integers
\((\sqrt3)^{2}\)rational= 3

General methodHow to decide rational or irrational

  1. Can you simplify it to a whole number, a fraction, or a stopping/recurring decimal? → rational.
  2. A square root of a number that is not a perfect square (\(\sqrt{20}\)), or \(\pi\)? → irrational.

Try it yourself

1
Which is irrational: \(\sqrt{9}\), \(0.\dot7\), \(\sqrt{20}\), \(\tfrac3{11}\)?
Show answer

Answer \(\sqrt{20}\)

Reciprocals; numbers in words

The reciprocal of a number is 1 ÷ it. Reciprocal of 5 is \(\tfrac15=0.2\); of \(\tfrac23\) it is \(\tfrac32\). For numbers in words, write the digits in groups of three from the right (millions | thousands | units), with 0 in the empty places: four million, thirty thousand and five = 4 | 030 | 005.

Try it yourself

1
Reciprocal of 4?
Show answer

Answer 0.25

2
Reciprocal of \(\tfrac58\)?
Show answer

Answer \(\tfrac85\)

3
Ten thousand and seven in figures?
Show answer

Answer 10 007

Ways they test it

Type 1 From this list, write down …

"From the list, write down a prime number / a cube number / an irrational number." (1 mark each)

Spot it a list of numbers.

  1. Check each number against the family.
  2. Exactly the number of answers asked.

Type 2 Sum of the primes in a list

"Two of these numbers are prime: 27, 47, 57, 61, 75, 93. Find their sum."

Spot it a list full of odd numbers.

  1. Test each with 3 and 7.
  2. Add the ones that survive.

For example 27, 47, 57, 61, 75, 93

  1. 27, 57, 75, 93 are multiples of 3

Answer 47 + 61 = 108

Type 3 Show it is not prime

"Show that 119 is not prime."

Spot it "show that".

  1. Give a factor pair (119 = 7 × 17).

Type 4 Words ↔ figures; reciprocals

"Write four million, thirty thousand and five in figures."

Spot it a number in words.

  1. Groups of three digits, 0 in gaps.

Mistakes to avoid

  • ✗ 1 is prime✓ 1 is not primeA prime has exactly two factors.
  • ✗ 91, 51, 57 are prime✓ 7 × 13, 3 × 17, 3 × 19Test with small primes.
  • ✗ \(0.\dot3\) is irrational✓ rational\(=\tfrac13\).
  • ✗ \(\sqrt{16}\) is irrational✓ rational= 4.
  • ✗ reciprocal of \(\tfrac23\) is \(-\tfrac23\)✓ \(\tfrac32\)Reciprocal means 1 ÷, not negative.

Summary

  • 1 is not prime; 2 is the only even prime.
  • Squares to 225; cubes 1, 8, 27, 64, 125, 1000.
  • Prime test: primes up to \(\sqrt n\).
  • Rational = fraction of integers; its decimal stops or recurs.
  • Reciprocal = 1 ÷ number.

If you see …

  • a number ending in 1, 3, 7 or 9 that 'looks' primetest 3, 7, 13.
  • \(\sqrt{\ }\) of a non-squareirrational.
  • a recurring decimalrational.

Practice

Practice: easy, medium, hard

Easy

1
Write down the cube number: 125, 100, 36, 9.
Show answer

Answer 125

2
Reciprocal of 0.2?
Show answer

Answer 5

3
Write four million, thirty thousand and five in figures.
Show answer

Answer 4 030 005

Medium

1
Two of 27, 47, 57, 61, 75, 93 are prime. Find their sum.
Guide me

What is it asking? the sum of the primes

What do we know? six odd numbers

Plan test each with 3, 5, 7

Show working
  1. 27, 57, 75, 93 are multiples of 3
  2. 47 + 61 = 108

Answer 108

2
From \(\sqrt{36}\), \(0.\dot2\), \(\sqrt{30}\), 31, 64 write down an irrational number.
Guide me

What is it asking? an irrational number

What do we know? five numbers

Plan simplify each; which can't be a fraction?

Show working
  1. \(\sqrt{36}=6\); \(0.\dot2=\tfrac29\); \(\sqrt{30}\) is not a square root of a square

Answer \(\sqrt{30}\)

3
The integers in −2.5, −1, 0, 3.7, 4 are …
Guide me

What is it asking? the integers

What do we know? a mixed list

Plan whole numbers, including 0 and negatives

Show working
  1. −2.5 and 3.7 are not whole

Answer −1, 0, 4

Hard

1
Show that 119 is not prime, and find the prime between 110 and 115.
Guide me

What is it asking? a factor pair and a prime

What do we know? 119; 111 and 113

Plan try 7 on 119; test 111 and 113

Show working
  1. \(119=7\times17\)
  2. 111 = 3 × 37; 113 has no factor up to 10

Answer 119 = 7 × 17; 113

PartAnswer / workingMarksGuidance
119 = 7 × 17B1
113B1
Total: 2 marks
2
Find the number between 101 and 200 that is a square and odd, and less than 150.
Guide me

What is it asking? one number

What do we know? odd squares between 101 and 200

Plan list 11², 13², 15²

Show working
  1. \(11^{2}=121\), \(13^{2}=169\)
  2. 121 < 150

Answer 121

PartAnswer / workingMarksGuidance
121B1
Total: 1 mark
3
The reciprocal of a number is 2.5. What is the number?
Guide me

What is it asking? the number

What do we know? 1 ÷ n = 2.5

Plan the reciprocal of the reciprocal

Show working
  1. \(n=1\div2.5=0.4\)

Answer 0.4

PartAnswer / workingMarksGuidance
0.4B1
Total: 1 mark

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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