Information Processing

Session 5BBinary Arithmetic

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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

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Bits A bit is the smallest piece

of information in the computer

At a single location, the information is either One - current is ON Zero - current is OFF

There are no in-between states

ON

OFF

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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

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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.

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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.

148 2

4

is ON

1

is ON

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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

148 2

4 1

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0101

4+1 = 5

Counting in Binary Counting is an

automatic process Follow the

sequence...

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Counting in Binary A pulse enters on

the right

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Counting in Binary To begin with the

first cell was OFF

It is flipped to ON

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Counting in Binary Another pulse

enters on the right

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Counting in Binary The first cell was

ON. It is flipped to OFF The pulse moves

to the next cell on the left

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Counting in Binary The next cell on

the left was OFF That cell is flipped

to ON

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Counting in Binary Another pulse

enters from the right

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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

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Counting in Binary The second cell

was ON, and is flipped to OFF

the pulse moves to the third cell

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Counting in Binary The third cell was

OFF and is flipped to ON

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Counting in Binary 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

0001

0010

0011

0100

0101

0110

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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

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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:

352 = 300 + 50 + 2

100 10 1

3 5 2

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Binary Numbers By Binary, we mean that numbers

are written in powers of two These are 1, 2, 4, 8, 16 etc. So that:

10100 = Which is 16 + 4 = 20

16 8 4 2 1

1 0 001

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Converting Binary to Decimal Example: 101101 Reading from right to left the columns are

1,2,4,8 etc.

i.e. 32 16 8 4 2 11 0 1 1 0 1

So the number in decimal notation is:

32 + 8 + 4 + 1 = 45

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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.

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Converting Decimal to Binary:An example

Convert 117 to binary: 117÷ 2 = 58 remainder 1 58 ÷ 2 = 29 remainder 0 29 ÷ 2 = 14 remainder 1 14 ÷ 2 = 7 remainder 0 7 ÷ 2 = 3 remainder 1 3 ÷ 2 = 1 remainder 1 1 ÷ 2 = 0 remainder 1

In binary the number is: 1110101

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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

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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

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Adding in Binary Adding

10111 +11101

01

1 + 1 = 2This is 10 in BinaryPut 0 in the answer, carry 1

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Adding in Binary Adding

1 0 1 1 1 +1 1 1 0 1 0 0 1 1

1 + 0 + 1= 2This is 10 in BinaryPut 0 in the answer, carry 1

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Adding in Binary Adding

1 0 1 1 1 +1 1 1 0 1 1 0 0 1 1 1

1 + 1 + 1= 3This is 11 in BinaryPut 1 in the answer, carry 1

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Adding in Binary Adding

1 0 1 0 1 +1 1 1 0 1 1 0 0 1 1 1

Carry on with this…

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Adding in Binary The answer:

1 0 1 1 1 +1 1 1 0 1

1 1 0 1 0 0

1 1 1 1 1

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The Binary Adder We will add

0011 (3) 0110 (6)

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The Binary Adder Starting with the

end column, top cell is ON

This pulse enters into the bottom cell

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The Binary Adder The bottom cell

was OFF

The pulse causes it to flip to ON

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The Binary Adder The next top cell

was ON The pulse enters

into the bottom cell

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The Binary Adder The bottom cell

was ON The pulse flips it to

OFF The pulse moves

to the next cell

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The Binary Adder The next cell is ON The pulse flips it to

OFF The pulse moves

to the next cell

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The Binary Adder The next cell is

OFF The pulse flips it to

ON

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The Binary Adder The bottom line

now reads:

1001

This is 8 + 1= 9

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Bits and Bytes We have seen that

a 4-bit register can count from 0 [0000] to 15 [1111]

This means that it has 16 different states.

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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 2 x 2 x 2 x 2 = 16 states

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2222

2 x 2 x 2 x 2 = 24

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 22 44 168 25616 65,53632

4,294,987,296Slide 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

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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

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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 Hex0 01 12 23 34 45 56 67 78 89 910 A11 B12 C13 D14 E15 F

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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”

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