Transcript 0 - The Eclecticon of Dr French

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
How are numbers ‘best’ written down? Does it matter?
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.
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 using
a small set of memorized results (i.e. your times table)
123 x 456
=

400
50
6
100 40,000 5000 600
20 8000 1000 120
3
1200
150 18
=
40,000
8000
1200
5000
1000
150
600
120
18
--------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 =

300
90
4
100 30,000 9000 400
60 18,000 5,400 240
7
2,100
630
28
0.15 x 17.2
=
=

10
7
0.2
0 .1 1
0.7 0.02
0.05 0.5 0.35 0.01
30,000
18,000
2100
9000
5400
630
400
240
28
--------65,798
=
1.00
0.70
0.02
0.50
0.35
0.01
--------2.58
Binary numbers 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 manipulate
numbers by computers)
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
in binary?
(a) 64
(b) 73
Binary numbers 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 manipulate
numbers by computers)
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
in binary?
(a) 64 is 1000000 since 26 = 64
(b) 73 is 1001001 since 64 + 8 + 1 = 73
1
x 26
0
x 25
0
x 24
1
x 23
0
x 22
0
x 21
1
x 20
64
0
0
8
0
0
1
Hexadecimal numbers 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
A sixteen symbol system is HEXADECIMAL, which is typically used to describe computer
memory addresses.
Decimal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
Hexadecimal or ‘base 16’
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
What is
in hexadecimal?
(a) 31
(b) 117
Hexadecimal numbers 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F
A sixteen symbol system is HEXADECIMAL, which is typically used to describe computer
memory addresses.
Decimal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
Hexadecimal or ‘base 16’
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
What is
in hexadecimal?
(a) 31 is 1F
(b) 117 is 75
since
since
1 x 161 + 15 x 160 = 31
7 x 161 + 5 x 160 = 112 + 5 = 117
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
cuneiform digits
Note this wasn’t a
proper
‘place value’
system as there
was no zero!
Although it did appear later as