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Information Processing
Session 5B
Binary Arithmetic
Slide
000001
Objectives

After studying this week’s work, you
should:
gain insight into how the processor
deals with information at the bit level
 Understand numbers written in binary
form
 Be able to convert numbers from binary
to decimal notation and vice-versa
 Be able to add numbers in binary form

Slide
000010
Bits


A bit is the smallest piece
of information in the
computer
At a single location, the
information is either



ON
One - current is ON
Zero - current is OFF
OFF
There are no in-between
states
Slide
000011
Bytes



The way that
information is coded
is to use a sequence
of zeros and ones
It is usual to have a
sequence of 8 bits
collected together
This is called a byte
10110101
Slide
000100
Bits and Bytes


Depending upon the
design of the
computer, there could
be 4, 8 16, 32, 64 (or
even more!) bits
processed by the
computer at once
For the next few
slides we will look at a
simple 4-bit device.
Slide
000101
A 4-bit register


Reading from the
right, each bit is
worth double the one
preceding it.
The sequence,
reading from the right
is:


1,2,4,8, ...
If we had more bits, it
would continue:

... 16, 32, 64, etc.
8 4 2 1
4
1
is
ON
is
ON
Slide
000110
Binary Numbers



The register shown on
the right represents
the binary number
0101
This has ones in the
1 and 4 cells, and
zeroes in the others.
The number
represented is 5
8 4 2 1
0101
4
1
4+1 = 5
Slide
000111
Counting in Binary


Counting is an
automatic process
Follow the
sequence...
Slide
001000
Counting in Binary

A pulse enters on
the right
Slide
001001
Counting in Binary

To begin with the
first cell was OFF

It is flipped to ON
Slide
001010
Counting in Binary

Another pulse
enters on the right
Slide
001011
Counting in Binary



The first cell was
ON.
It is flipped to OFF
The pulse moves
to the next cell on
the left
Slide
001100
Counting in Binary


The next cell on
the left was OFF
That cell is flipped
to ON
Slide
001101
Counting in Binary

Another pulse
enters from the
right
Slide
001110
Counting in Binary

The first cell was
OFF

It is flipped to ON
Slide
001111
Counting in Binary

Another pulse
enters from the
right
Slide
010000
Counting in Binary


The first cell was
ON, and is flipped
to OFF
The pulse moves
to the second cell
Slide
010001
Counting in Binary


The second cell
was ON, and is
flipped to OFF
the pulse moves to
the third cell
Slide
010010
Counting in Binary

The third cell was
OFF and is flipped
to ON
Slide
010011
Counting in Binary


0001
0010
Follow the
sequence on the
right, and try to
continue it.
you will see that
the switching
creates a pattern
off ON/OFF in
each column
0011
0100
0101
0110
Slide
010100
Counting in Binary




The 1’s column
alternates 1,0,1,0 etc.
The 2’s column starts
at 2 and alternates
two 1’s, two 0’s
The 4’s column starts
at 4, and alternates
four 1’s, four 0’s
The 8’s column starts
at 8 and alternates
eight 1’s eight 0’s
0001
0010
0011
0100
0101
0110
Slide
010101
Decimal Numbers
By decimal, we simply mean that
the numbers are written in powers
of ten
 These are 1, 10, 100, 1000, etc.
 So that:

100
352 = 300 + 50 + 2
10
1
3 5 2
Slide
010110
Binary Numbers
By Binary, we mean that numbers
are written in powers of two
 These are 1, 2, 4, 8, 16 etc.
 So that:

16
8
4
1
0
1
10100 =
 Which is 16 + 4 = 20
2
1
0
0
Slide
010111
Converting Binary to
Decimal


Example: 101101
Reading from right to left the columns are
1,2,4,8 etc.
i.e.

32
1
16
0
8
1
4
1
2
0
1
1
So the number in decimal notation is:
32 + 8 + 4 + 1
=
45
Slide
011000
How do we convert
Decimal to Binary?


There is a specific
technique which
allows us to do
this.
It involves
repeatedly dividing
a number by two
and noting the
remainder.
Slide
011001
Converting Decimal to Binary:
An example

Convert 117 to binary:








117÷ 2 = 58 remainder
58 ÷ 2 = 29 remainder
29 ÷ 2 = 14 remainder
14 ÷ 2 = 7 remainder
7 ÷ 2 = 3 remainder
3 ÷ 2 = 1 remainder
1 ÷ 2 = 0 remainder
1
0
1
0
1
1
1
In binary the number is:
1110101
Slide
011010
Adding In Binary


Addition in binary
is a direct
counterpart of
what happens at
the processor
level.
First of all we will
look at a numerical
example
Slide
011011
Adding in Binary



There are only four
possible
combinations.
The first three are
“obvious”
The last one is
special (remember
1 + 1 = 2, which is
10 in binary)

0+0=0
0+1=1
1+0=1

1 + 1 = 0, carry 1


Slide
011100
Adding in Binary

Adding
10111 +
11101
0
1
1+1=2
This is 10 in Binary
Put 0 in the answer, carry 1
Slide
011101
Adding in Binary

Adding
10111+
11101
00
1
1
1 + 0 + 1= 2
This is 10 in Binary
Put 0 in the answer, carry 1
Slide
011110
Adding in Binary

Adding
10111+
11101
100
1 1 1
1 + 1 + 1= 3
This is 11 in Binary
Put 1 in the answer, carry 1
Slide
011111
Adding in Binary

Adding
10101+
11101
100
1 1 1
Carry on with this…
Slide
100000
Adding in Binary

The answer:
10111+
11101
110100
1 1 1 1 1
Slide
100001
The Binary Adder

We will add


0011
0110
(3)
(6)
Slide
100010
The Binary Adder


Starting with the
end column, top
cell is ON
This pulse enters
into the bottom
cell
Slide
100011
The Binary Adder

The bottom cell
was OFF

The pulse causes it
to flip to ON
Slide
100100
The Binary Adder


The next top cell
was ON
The pulse enters
into the bottom
cell
Slide
100101
The Binary Adder



The bottom cell
was ON
The pulse flips it to
OFF
The pulse moves
to the next cell
Slide
100110
The Binary Adder



The next cell is ON
The pulse flips it to
OFF
The pulse moves
to the next cell
Slide
100111
The Binary Adder


The next cell is
OFF
The pulse flips it to
ON
Slide
101000
The Binary Adder

The bottom line
now reads:
1001

This is 8 + 1= 9
Slide
101001
Bits and Bytes

We have seen that
a 4-bit register can
count from



0
to 15
[0000]
[1111]
This means that it
has 16 different
states.
Slide
101010
Bits and Bytes


Each bit in the
register can be ON
or OFF. This means
that there are two
possibilities for
each cell
That is, altogether

2x2x2x2
= 16 states
2
2
2
2
2 x 2 x 2 x 2 = 24
Slide
101011
Bits and Bytes


The number of
possible states of
registers of other
sizes can be worked
out in the same way
For example an 8-bit
register (byte) has

2x2x2x2x2x2x2x2=
28 = 256 different
states.
Bits
States
1
2
4
8
16
32
2
4
16
256
65,536
4,294,987,296
Slide
101100
Megabits and Kilobytes


A Kilobyte is 210
bytes. This is the
nearest power of
two to 1000. In
fact 210 = 1024
A megabit is 220
bits. This is the
nearest power of 2
to 1 million. In fact
220 = 1048576
Slide
101101
Other Bases


Decimal and Binary
are two different
number bases used
by the computer, but
there are others
An important one is
Hexadecimal which
has 16 separate
characters:
0,1,2,3,4,5,6,7,
8,9,A,B,C,D,E,F
Slide
101110
Hexadecimal


The extra letters are so that
the numbers 10-15 can be
written using one character
each. This means that A8BC
is a number written in
Hexadecimal.
These numbers are written in
base 16, so that a number
like 9E means


the 9 is 9 x 16 = 144
the E is 14 x 1 = 14
Altogether this would be 158
Dec
0
1
2
3
4
5
6
7
8
9
10 
11 
12 
13 
14 
15 
Hex
0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F
Slide
101111
Summary




A bit has two states, ON or OFF, which
means that at the core of a computer we
need to use binary coding of numbers
(powers of two)
Registers count and add using in binary
code
There are algorithms for converting
decimal to binary and vice-versa
Binary addition has only four possible
addition pairs, and a “carrying rule”
Slide
110000