Number Systems Part2

7/31/2019 Number Systems Part2

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Number Systems Part 2Numerical Overflow

Right and Left Shifts

Storage Methods

Subtraction

Ranges


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Numerical Overflow

An overflow is when something doesntfit in a certain space

Numeric overflow is when the storagefor a calculation is too small to hold theresult

For example we have an 8 bits register, ifwe add two binary numbers and theresult turns out to be 9 bits it would not

fit in the register


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Example

Lets say we have an 8 bit register

Add the following;

Do we have an overflow?

11111111+

10101010


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Numerical Overflow

When we have a numeric overflow we

will have an error in our calculation

When we have an overflow we would

need to remove the extra bit at the start

of the number

Lets say we had a 7 bit register and the

result of a calculation is 11001100 the

actual answer would be 1001100


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Example

Lets say we have a 7 bit register

Add the following;

Do we have an overflow?

Actual answer =

1101111+

1101101


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Working

Suppose we have a 4 bit registers.

1. Perform the additions

2. Is there and overflow?

3. What is the actual answer?

1110+ 0101+ 1101+

1111 0110 0100


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Answers

1110+ 0101+ 1101+

1111 0110 0100

11101 1011 10001


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What is Bit Shifting?

Bit shifting is the process of moving all thebits in a binary number

We have two shifts1. A right shift

2. A left shift

The right shift would divide the numberwhile the left would multiply it


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Right Shift

The right shift is used to perform divisions

Hence if we where to perform three rightshifts we would be diving the number by 2

three times

For example if we shift 101100112, = 17910right by three places, we get 000101102 =

2210 179 /2 = 89

90/2 = 44

45/2 = 22


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How Right Shifts work

Lets say we have 11001012 and we wish

to perform 2 right shifts

1 1 0 0 1 0 11st Shift right

0 1 1 0 0 1 0 1

LOST

0 ADDED

2nd Shift right

0 0 1 1 0 0 1 0ANS =


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Working

Shift the numbers to the right by thenumber of shifts indicated in the brackets;

1. 1000110 (2)

2. 1110001 (4)

3. 1111000 (1)

4. 01010101 (4)

5. 0000111 (3)

Change to

decimal to check

your answers


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1000110 70 1110001 113 1111000 120

0100011 35 0111000 56 011110060

0010001 17 0011100 28

0001110 14

00001117

01010101 85 0000111 7

00101010 42 0000011 3

00010101 21

00001010 10

000001015

0000001 1

00000000


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Left Shift

The left shift is used to performmultiplications

Hence if we where to perform four left

shifts we would be multiplying the numberby 2 four times

If any bits are lost when shifting left the

result is no longer accurate

For example if we shift 011001012, = 10110left by one place, we get 110010102 = 20210

101 x 2 = 202


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How Left Shifts work

Lets say we have 011001012 and we wishto perform 2 left shifts

0 1 1 0 0 1 0 1

1 1 0 0 1 0 1 00

LOST

1 0 0 1 0 1 0 01Answer no

longer

correct as a

1 was lost

1st Left Shift

2nd Left Shift


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Working

Shift the numbers to the left by thenumber of shifts indicated in the brackets,also state which answers are not correct.

1. 0000110 (4)

2. 1110001 (3)

3. 0011110 (1)4. 01010101 (2)

5. 0000111 (3)

Change correct

answers to

decimal to checkyour answers


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0000110 6 1110001 0011110 30

0001100 12 1100010 011110060

0011000 24

0110000 48

110000096

1000100

0001000

Not correct

01010101 0000111 7

10101010 0001110 14

01010100

Not Correct

0011100 28

011100056


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Storage Methods

When we have a fixed register we mightwant to store out binary number in acertain way

The three ways are using;

1. sign and magnitude (first digit is 0 =positive, first digit is1 = negative)

2. ones complement

3. twos complement.


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Example 1

Lets say we have a register of 8-bits andwe wish to store the number 1410

Since the number is positive, all we needto do is convert it to binary and store it

It would be the same in the three

different storage methods, 1410 =

000011102


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Example 2

Lets say I have an 8 bit register and wantto store -1410

The number now changes since it is

negative, we first change it to binary =000011102

Sign and Magnitude 10001110 Note that the fist digit is 1 because in this

case 14 is a negative number.

Ones Complement 11110001 In this case we simply applied NOT on the

binary value of14.

Twos Complement 11110010 We change the binary number 14 into

twos complement


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Working

Convert the following to;a) Sign and magnitudeb) Ones Complementc) Twos Complement

Size of register is indicated in the brackets.1. 3310 (8)2. -6010 (9)

3. -4410 (7)4. 1010 (5)5. -7410 (10)


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Subtraction

We are able to subtract using binary numbersthanks to Twos Complement

When we need to subtract in binary we first

need to change the negative number to TwosComplement

if we have 2210 we have to change 10 to Twos

Complement then perform an addition

Binary numbers must be of the same length inorder to subtract in binary


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Example

We want to perform the followingsubtraction in binary;

22 - 10110 - 10110+

10 01010 10110

12 101100

Changeto BinaryTwos

Complement

Overflow


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Working

Perform the following subtractions inbinary;

1. 30-20

2. 10050

3. 5025

4. 52

5. 66- 60


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Ranges

A ranges determines the lowest and thehighest values which can be represented

by a certain register

We will be going through two types of

ranges using binary;

1. normal binary2. twos complement.


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Normal Binary

What is the range of numbers which can berepresented using 8-bits?

Since we have 8 Bits, the maximum valuerepresented in binary is 11111111 and

the smallest is 00000000.

If we convert these numbers to decimal

we end up with: 0 to 255


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Example

What is the range of numbers which can berepresented using 5-bits?

Since we have 5 Bits, the maximum valuerepresented in binary is 11111 and the

smallest is 00000.

If we convert these numbers to decimal

we end up with: 0 to 31


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Twos Complement

What is the range of numbers which can berepresented using 8-bits in twos complement?

The first number in twos complement

identifies the number as a negative number,the highest possible positive number is011111112 = 12710

To find the smallest number we convert thepositive number to negative so in this case-12710.

The range is: -127 to 126 (126 due to the 0)


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Example

What is the range of numbers which can berepresented using 10-bits in twos

complement?

01111111112 = 51110

51110 becomes negative to show thesmallest number helps = -51110.

The range is: 511 to 510

Number Systems Part2

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