MELJUN CORTES Binary Calculations

7/29/2019 MELJUN CORTES Binary Calculations

1/21

Lesson 2 - 1

Year 1

CS113/0401/v1

LESSON 2

COMPUTER-BASEDCALCULATIONS

Computers store numbers inbinary

They calculate in binary

Principles of binary calculation

are essentially similar to those for

decimal

When using computers, make

sure you know what the binary

data means!


2/21

Lesson 2 - 2

Year 1

CS113/0401/v1

BINARY ADDITION

0 + 0 = 0 carry 0

0 + 1 = 1 carry 0

1 + 1 = 0 carry 1

1 + 1 + 1 = 1 carry 1

1010102+ 1110012

11000112 Result = 11000112

111 carry

1 + 1 + 1 + 1 = 0 carry 10 (binary)

1101102

+ 1011012

+ 11110102

11011101

1101111 carry


3/21

Lesson 2 - 3

Year 1

CS113/0401/v1

BINARY SUBTRACTION

1 1 10 1 10 1 0 12

- 101 1 10 1 1 0 02

0 1 1 0 1 0 0 12

Result 11010012

This is not the way computers do it


4/21

Lesson 2 - 4

Year 1

CS113/0401/v1

SHIFT OPERATIONS

In reality, multiplication and division

are done using Shifting

Circular, Logical and Arithmetic shifts

exit

We will consider the Arithmetic Shift


5/21

Lesson 2 - 5

Year 1

CS113/0401/v1

SHIFTING FOR

MULTIPLICATION (1)

Shift LEFT

73510 shifted left 1 place

7 3 5

7 3 5 0

= 735010 = 73510 X 1010

11012 shifted left 1 place

1 1 0 1

1 1 0 1 0

= 110102 = 11012 X 102


6/21

Lesson 2 - 6

Year 1

CS113/0401/v1

SHIFTING FOR

MULTIPLICATION (2)

Binary shift left n places multiplesby 2n

Fill in right hand side with zeros

Beware the sign bit!

Repeated doubling cannotchange a numbers sign, but it

can send it out of range

Computers have a specialregister to detect this


7/21

Lesson 2 - 7

Year 1

CS113/0401/v1

SHIFTING FOR DIVISION


8/21

Lesson 2 - 8

Year 1

CS113/0401/v1

NUMBER STORAGE IN THE

COMPUTER

Computer store is arranged inwords

Words are fixed length groups of

binary digits (bits)

Words vary in length on different

types of computer

Common word lengths are 8, 12,

16, 24 and 32 bits

We shall use 12 bit words in

examples

NB : An 8-bit word is called a byte


9/21

Lesson 2 - 9

Year 1

CS113/0401/v1

SIZE LIMITS ON DATA

Computer stores are of finite size

This limits the range of values

which can be stored and the

accuracy of fractions

Example

16-bit words are common and

can hold integers in the range

from -32768 to +32767


10/21

Lesson 2 - 10

Year 1

CS113/0401/v1

STOREGE OF INTEGERS (1)

735 DECIMAL = 1011011111 (10 bits)

Add 2 bits padding in a 12-bit word

By convention, the first (left-hand) bit is

the SIGN BIT

Therefore only 11 bits are left for the

number value

For sign Modulus method : 0 Positive

0 0 1 0 1 1 0 11111

0 0 1 0 1 1 0 11111


11/21

Lesson 2 - 11

Year 1

CS113/0401/v1

Small numbers stored in 12-bitwords contain mostly padding

bits

Padding is not really wasteful, it is

necessary to make the

calculation method work properly

Sign bit Padding Decimal 9

0 0 0 0 0 0 0 10010



12/21

Lesson 2 - 12

Year 1

CS113/0401/v1


The upper limit for storing anumber in a 12-bit word is

011111111111 = 2047 decimal

Larger numbers need double

length


13/21

Lesson 2 - 13

Year 1

CS113/0401/v1

STORAGE OF FRACTIONS

(1)

First bit still sign bit

Decimal point not stored

Implied after sign bit


14/21

Lesson 2 - 14

Year 1

CS113/0401/v1


(2)

if 11 bits is not enough for exactrepresentation

Truncate

Round off

Extend to double length (or more)


15/21

Lesson 2 - 15

Year 1

CS113/0401/v1


(3)

Double length fraction

0.7323 decimal

Double length fractions have

increased accuracy, not range


16/21

Lesson 2 - 16

Year 1

CS113/0401/v1

Usual convention is one word for

integral part, the other for fraction

STORAGE OF MIXED

NUMBERS (FIXED POINTNOTATION)


17/21

Lesson 2 - 17

Year 1

CS113/0401/v1

TYPES OF NUMBERS

Even if you know that the data isnumeric, make sure you have the

right format


18/21

Lesson 2 - 18

Year 1

CS113/0401/v1

STORAGE OF NEGATIVE

VALUES (1)

Sign-and-modulus method is

unsuitable for calculation

Computers usually use twos

complement method

Three stages to finding a twos

complement

Example : -837 decimal


19/21

Lesson 2 - 19

Year 1

CS113/0401/v1

STORAGE OF NEGATIVE

VALUES (2)

Conversion from twos

complement to decimal


20/21

Lesson 2 - 20

Year 1

CS113/0401/v1

STORAGE OF NEGATIVE

VALUES (3)

Ranges of numbers

Largest negative number in 12

bits

= 100000000000 = -2048

Total range of 12 bits is -2048 to

+ 2047

Multiple length works with

negative values as well


21/21

Lesson 2 - 21

Year 1

CS113/0401/v1

TWOS COMPLEMENT

SUBTRACTION

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MELJUN CORTES Binary Calculations