CS10110: Number Bases
Decimal, Binary & Hexadecimal
(and a little bit of Octal..)
Laurence Tyler
lgt@aber.ac.uk
November 2012
The Digital Computer
► Computers
1's and 0's
●
(as we know them today) work on
and nothing else!
► All
commands to the CPU are just strings of
these
► All
output from the CPU is in strings of these
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So we need to understand binary?
► Yes!
●
●
To properly understand how a computer works, you
have to understand how it represents and uses
numbers
Every bit of data a computer uses is eventually
represented by a binary number
► You
●
●
should be able to:
Convert from binary to decimal, and vice-versa
Do simple sums in binary
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Binary
► Otherwise
► Each
●
●
0 or 1 is called a bit (binary digit)
The bit is the fundamental unit of information
► We
●
known as base 2
can string bits together to form a byte
A byte is usually the smallest addressable unit of
computer storage
Usually 8 bits, eg: 10011101 (sometimes called an
'octet')
► We
are not good at reading binary - we have
been trained to work with decimal
► But
to understand the computer, we need to
understand binary
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How the decimal system works
In decimal (base 10), each higher digit position is worth
ten times the next lower one - powers of 10 - and the
digits used are 0..9 (0 to 10-1)
E.g: what's the value of 201810 in decimal? ☺
← Most significant digit
Least →
Units
103
1000's
102
100's
101
10's
100
1's
Digit
2
0
1
8
Value
2000
0
10
8
Result: 2000 + 10 + 8 = 2018
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Binary to Decimal
In binary, it's just the same - except that each higher
digit position is worth twice the next lower one - powers
of 2 - and the only digits are 0 and 1 (0 to 2-1)
Taking our example: 100111012
← Most
Units
significant bit
Least →
27
128's
26
64's
25
32's
24
16's
23
8's
22
4's
21
2's
20
1's
1
0
0
1
1
1
0
1
128
0
0
16
8
4
0
1
Bit
Value
Result: 128 + 16 + 8 + 4 + 1 = 157
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Decimal to Binary
One method: try to subtract powers of two, starting
with the highest. Each successful subtraction is a 1 bit,
each failure is a 0 bit. E.g. to convert 20110 to binary:
Remainder
201
73
9
9
9
1
1
1
Subtract 2n
-128
-64
-32
-16
-8
-4
-2
-1
Does it go?
Yes
Yes
No
No
Yes
No
No
Yes
Result bit
1
1
0
0
1
0
0
1
Result: 11001001 (read highest to lowest)
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Powers of the Number Base
Binary
10000000
1000000
100000
10000
1000
100
10
1
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Decimal Power of 2
128
27
64
26
32
25
16
24
8
23
4
22
2
21
1
20
Decimal
Power of 10
1000
103
100
102
10
101
1
100
Successive digit
positions are powers of
the number base whatever that number is
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But Decimal is clumsy!
► Decimal
(base 10) doesn't translate easily in our
heads to and from binary
► e.g.
in 8 bits (1 byte) we can represent 28 = 256
different numbers
► To
know the bit pattern for a byte we would
have to convert it, or else learn it by heart
► Instead,
we often use another number base that
fits better into bytes - Hexadecimal (hex for
short)
► Hexadecimal
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is base 16
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Hexadecimal Digits
► We
only have ten symbols to represent
numbers, but for hexadecimal we need 16
► Q:
How do we represent the numbers 10 to 15
as single digits?
► A:
By convention, we use the letters A to F
► So
our counting goes:
0123456789ABCDEF
(then 10 11 12 ... 19 1A 1B 1C 1D 1E 1F 20 ...)
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Powers of 16
Hex
November 2012
Decimal
Power of 16
10000
65536
164
1000
4096
163
100
256
162
10
16
161
1
1
160
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Why is Hex better?
► One
byte = 8 bits
► 256
= 28 = 162 = 102.40824
► 16
= 24
► This
means each four bits can be represented
by one hex digit, and one byte is two hex digits
► e.g.
101100102 = B216
Binary
Hex
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1011
B
A group of 4 bits is
often called a "nybble"!
0010
2
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Octal
► We
can - and sometimes do - use other number
bases as well
► Octal
(base 8) arises at times. You've seen it
already in this course:
$ chmod 751 myfile
$ ls -l myfile
-rwxr-x--x
111101001
7
► Now
5
1
try converting 751 into binary!
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Octal beware!
► Many
programming languages let you write
numbers in different bases in your source code:
●
► In
●
●
●
Decimal, Hex, Octal and sometimes binary
C, C++ and Java (and possibly others):
Decimal - just the digits e.g. 12857
Hexadecimal - 0xB16F or 0xb16f
Binary - 0b101010 (if supported)
► BUT
NOTE: Any plain "number" that starts with
just a '0' is treated as OCTAL, not DECIMAL!
●
●
so 017 in your program is not seventeen but fifteen!
writing e.g. 019 will get you a compiler error!
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Arithmetic in other bases
► It's
just like decimal, except you have to
remember to "carry" or "borrow" at the right
point
1 0
+ 1 1
=====
1 0 1
1 0 1
- 1 1
=====
1 0
Binary
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3 F
+ 8 4
=====
C 3
4 3
- 2 E
=====
1 5
Hexadecimal
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Summary
► Make
●
●
sure you can:
Convert between binary, decimal and hex (without a
calculator!)
Do simple sums in binary and hex (just addition and
subtraction)
► Tip:
Learn to recognise the first few powers of 2
(2, 4, 8, 16, 32, 64, 128, 256, 512, 1024...) and
powers of 16 (16, 256, 4096, 65536...) rather
than having to work them out each time you
need them
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