GCSE Mathematics
Higher Tier
11
, Number skills
We use numbers every day of our lives. You need to be confident in the basic number skills.
Rational and irrational numbers
You are expected to know the difference between a rational and an irrational number.
A rational number is any number that can be written as a fraction.
Rational numbers include 5, -7, 3, 5·628, 81, 3 125, 7·3, 5·217
7
3
81 = 9 125 = 5
An irrational number is any number that cannot be written as a fraction.
Irrational numbers include:
Most square roots, eg 7, 8, (not 9 because 9 is a square number).
Most cube roots, eg 3 18, 3 20 (not 3 64 because 64 is a cube number). 6
Almost anything involving , eg 2 , 7 , but not 76 because it cancels to give 7
Note: (2 + 3)2 = (2 + 3) (2 + 3) Note: Expansion of
=4+2 3+2 3+3 3x 3=3 brackets is explained
=7+4 3 on page 16.
Recurring decimals are rational: Write the following recurring numbers as fractions:
7
0·7 = 0·777777 = 1 recurring number therefore 9 Note: 0·07 = 7
9 90
0·67 = 0·676767 = 67 2 recurring numbers therefore 99
99
0·427 = 0·427427 = 427 3 recurring numbers therefore 999
999
Questions
1 In each question state whether the number is rational or irrational. If the number
is rational write the number in the form a .
b
b 7
a 7 c 7x 7
4 7
d 0·4 e 0·287 f 0·67
g 5 h 7+ 7
2 5 x a is rational. Find a possible value of a.
Answers
1 4
1 a Irrational b c 7 d
4 9
e 287 f 67
g Irrational h Irrational
999 100
2 a = 5 or a = N x 5 where N is a square number eg 16 x 5 = 80 (because 5 x 80 = 20)
(Other answers a = 20, 45, 80, 125...)
viii
,
, Calculator skills
Your calculator can be your best friend or your worst enemy. Spend some time learning to
use it. You must buy a calculator with a fraction keyabc . You will be shown how to use a
calculator but if the keys shown do not work ask your teacher for help.
Using a calculator: Brackets, memory and fractions
Most calculators automatically use algebraic logic and can work out the answers. Your
task is to use the correct keys. You need to know how to use brackets – ‘Method A’ –
and how to use the memory – ‘Method B’.
Use of brackets
3(6+8) this means 3x(6+8)
Calculator keys: Answer 42
(8–5)3 this means (8–5)x3
Calculator keys: Answer 9
Questions involving division
Method A: Using brackets
3·86 – 4·23 Place brackets at the start and end of the top line (3·86 – 4·23)
7·25 x 3·68 Place brackets at the start and end of the bottom line (7·25 x 3·68)
Calculator keys:
Answer – 0·013868065
Method B: Using the memory
First work out the answer to the bottom line (remember to press =).
Place this number in memory. Work out the answer to the top line. Divide by memory
recall. Calculator keys: (look at your calculator instruction booklet if you do not know
how to use the memory) your memory key could say
Answer -0·013868065
Fractions abc key
3 4
x 21 Calculator keys:
5 4
11
3 ab c 4 abc 5 x 2 abc 1 abc 4 = Answer = 8 20
viii
Higher Tier
11
, Number skills
We use numbers every day of our lives. You need to be confident in the basic number skills.
Rational and irrational numbers
You are expected to know the difference between a rational and an irrational number.
A rational number is any number that can be written as a fraction.
Rational numbers include 5, -7, 3, 5·628, 81, 3 125, 7·3, 5·217
7
3
81 = 9 125 = 5
An irrational number is any number that cannot be written as a fraction.
Irrational numbers include:
Most square roots, eg 7, 8, (not 9 because 9 is a square number).
Most cube roots, eg 3 18, 3 20 (not 3 64 because 64 is a cube number). 6
Almost anything involving , eg 2 , 7 , but not 76 because it cancels to give 7
Note: (2 + 3)2 = (2 + 3) (2 + 3) Note: Expansion of
=4+2 3+2 3+3 3x 3=3 brackets is explained
=7+4 3 on page 16.
Recurring decimals are rational: Write the following recurring numbers as fractions:
7
0·7 = 0·777777 = 1 recurring number therefore 9 Note: 0·07 = 7
9 90
0·67 = 0·676767 = 67 2 recurring numbers therefore 99
99
0·427 = 0·427427 = 427 3 recurring numbers therefore 999
999
Questions
1 In each question state whether the number is rational or irrational. If the number
is rational write the number in the form a .
b
b 7
a 7 c 7x 7
4 7
d 0·4 e 0·287 f 0·67
g 5 h 7+ 7
2 5 x a is rational. Find a possible value of a.
Answers
1 4
1 a Irrational b c 7 d
4 9
e 287 f 67
g Irrational h Irrational
999 100
2 a = 5 or a = N x 5 where N is a square number eg 16 x 5 = 80 (because 5 x 80 = 20)
(Other answers a = 20, 45, 80, 125...)
viii
,
, Calculator skills
Your calculator can be your best friend or your worst enemy. Spend some time learning to
use it. You must buy a calculator with a fraction keyabc . You will be shown how to use a
calculator but if the keys shown do not work ask your teacher for help.
Using a calculator: Brackets, memory and fractions
Most calculators automatically use algebraic logic and can work out the answers. Your
task is to use the correct keys. You need to know how to use brackets – ‘Method A’ –
and how to use the memory – ‘Method B’.
Use of brackets
3(6+8) this means 3x(6+8)
Calculator keys: Answer 42
(8–5)3 this means (8–5)x3
Calculator keys: Answer 9
Questions involving division
Method A: Using brackets
3·86 – 4·23 Place brackets at the start and end of the top line (3·86 – 4·23)
7·25 x 3·68 Place brackets at the start and end of the bottom line (7·25 x 3·68)
Calculator keys:
Answer – 0·013868065
Method B: Using the memory
First work out the answer to the bottom line (remember to press =).
Place this number in memory. Work out the answer to the top line. Divide by memory
recall. Calculator keys: (look at your calculator instruction booklet if you do not know
how to use the memory) your memory key could say
Answer -0·013868065
Fractions abc key
3 4
x 21 Calculator keys:
5 4
11
3 ab c 4 abc 5 x 2 abc 1 abc 4 = Answer = 8 20
viii
