An introduction to Numbers

Dr Andrew French

You will need to consult your times table and your tables of integer powers

You will find it very useful to learn the powers up to 163 = 4,096

The DECIMAL system is when numbers are written right to left in powers of ten.

Only ten symbols are needed (0,1,2,3,4,5,6,7,8,9) plus a DECIMAL POINT to describe any number, of which there are infinitely many.

Not bad eh? In ancient cultures a different symbol is used for each integer, just like the way we say ‘one’, ‘two’, ‘three’ etc.

How are numbers ‘best’ written down? Does it matter?

So 1234.56 means

1 x 103 + 2 x 102 + 3 x 101 + 4 x 100 + 5 x 10-1 + 6 x 10-2

1000 + 200 + 30 + 4 + 0.5 + 0.06

The decimal system enables us to perform arithmetic calculations on numbers (i.e. addition, subtraction, multiplication and division) in a straightforward, systematic way.

You have been practising it for many years now!

Note we can use the decimal system to help us work out multiplications usinga small set of memorized results (i.e. your times table)

123 x 456 =

1815012003

1201000800020

6005000000,40100

650400

=

40,000800012005000100015060012018---------56,088

Use the ‘matrix decimal expansion’ to work out (NO CALCULATOR!)

167 x 394 =

0.15 x 17.2 =

Use the ‘matrix decimal expansion’ to work out (NO CALCULATOR!)

167 x 394 =

0.15 x 17.2 =

28630100,27

240400,5000,1860

4009000000,30100

490300

=

30,00018,00021009000540063040024028---------65,798

=

1.000.700.020.500.350.01---------2.58

01.035.05.005.0

02.07.011.0

2.0710

We don’t have to use the decimal system. In fact we can use any (integer!) number of symbols from two upwards.

A two symbol (0 or 1) system is BINARY (which is used to store and manipulatenumbers by computers)

Binary numbers 0, 1

Decimal Binary

17 10001 1 x 24 + 0 x 23 + 0 x 22 + 0 x 21 + 1 x 20

= 16 + 1 = 17

1234 10011010010 Note 1024 + 128 + 64 + 16 + 2 = 1234

1 x 210

0 x 29

0 x 28

1 x 27

1 x 26

0 x 25

1 x 24

0 x 23

0 x 22

1 x 21

0 x 20

1024 0 0 128 64 0 16 0 0 2 0

What are the decimal integers (a) 64 (b) 73

in binary?

Binary numbers 0, 1

Decimal Binary

17 10001 1 x 24 + 0 x 23 + 0 x 22 + 0 x 21 + 1 x 20

= 16 + 1 = 17

1234 10011010010 Note 1024 + 128 + 64 + 16 + 2 = 1234

1 x 210

0 x 29

0 x 28

1 x 27

1 x 26

0 x 25

1 x 24

0 x 23

0 x 22

1 x 21

0 x 20

1024 0 0 128 64 0 16 0 0 2 0

What are the decimal integers (a) 64 is 1000000 since 26 = 64 (b) 73 is 1001001 since 64 + 8 + 1 = 73

in binary?1 x 26

0 x 25

0x 24

1x 23

0 x 22

0x 21

1 x 20

64 0 0 8 0 0 1

We don’t have to use the decimal system. In fact we can use any (integer!) number of symbols from two upwards.

A two symbol (0 or 1) system is BINARY (which is used to store and manipulatenumbers by computers)

Hexadecimal numbers 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

Decimal Hexadecimal

17 11 1 x 161 + 1 x 160 = 16 + 1 = 17

1234 4D2 4 x 162 + 13 x 161 + 2 x 160 = 4 x 256 + 13 x 16 + 2 = 1234

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150 1 2 3 4 5 6 7 8 9 A B C D E F

A sixteen symbol system is HEXADECIMAL, which is typically used to describe computer memory addresses.

Decimal

Hexadecimal or ‘base 16’

What is (a) 31 (b) 117

in hexadecimal?

Hexadecimal numbers 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F

Decimal Hexadecimal

17 11 1 x 161 + 1 x 160 = 16 + 1 = 17

1234 4D2 4 x 162 + 13 x 161 + 2 x 160 = 4 x 256 + 13 x 16 + 2 = 1234

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150 1 2 3 4 5 6 7 8 9 A B C D E F

A sixteen symbol system is HEXADECIMAL, which is typically used to describe computer memory addresses.

Decimal

Hexadecimal or ‘base 16’

What is (a) 31 is 1F since 1 x 161 + 15 x 160 = 31 (b) 117 is 75 since 7 x 161 + 5 x 160 = 112 + 5 = 117

in hexadecimal?

Other ‘popular’ bases are:

12 Duodecimal 0,1,2,3,4,5,6,7,8,9,A,B

60 Sexagesimal Used by the ancient Babylonians around 3000BC

Note this wasn’t a proper‘place value’ system as there was no zero!

cuneiform digits

Although it did appear later as