Lesson 2 - 1

Year 1

CS113/0401/v1

LESSON 2COMPUTER-BASED

CALCULATIONS

Computers store numbers in binary

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!

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

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

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

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

Lesson 2 - 6

Year 1

CS113/0401/v1

SHIFTING FOR MULTIPLICATION (2)

Binary shift left n places multiples by 2n

Fill in right hand side with zeros

Beware the sign bit!

Repeated doubling cannot change a number’s sign, but it can send it out of range

Computers have a special register to detect this

Lesson 2 - 7

Year 1

CS113/0401/v1

SHIFTING FOR DIVISION

Lesson 2 - 8

Year 1

CS113/0401/v1

NUMBER STORAGE IN THE COMPUTER

Computer store is arranged in words

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

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

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

1 Negative

0 0 1 0 1 1 0 11111

0 0 1 0 1 1 0 11111

Lesson 2 - 11

Year 1

CS113/0401/v1

Small numbers stored in 12-bit words 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

STOREGE OF INTEGERS (2)

Lesson 2 - 12

Year 1

CS113/0401/v1

STOREGE OF INTEGERS (3)

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

011111111111 = 2047 decimal

Larger numbers need double length

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

Lesson 2 - 14

Year 1

CS113/0401/v1

STORAGE OF FRACTIONS (2)

if 11 bits is not enough for exact representation

Truncate

Round off

Extend to double length (or more)

Lesson 2 - 15

Year 1

CS113/0401/v1

STORAGE OF FRACTIONS (3)

Double length fraction

0.7323 decimal

Double length fractions have increased accuracy, not range

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 POINT

NOTATION)

Lesson 2 - 17

Year 1

CS113/0401/v1

TYPES OF NUMBERS

Even if you know that the data is numeric, make sure you have the right format

Lesson 2 - 18

Year 1

CS113/0401/v1

STORAGE OF NEGATIVE VALUES (1)

Sign-and-modulus method is unsuitable for calculation

Computers usually use two’s complement method

Three stages to finding a two’s complement

Example : -837 decimal

Lesson 2 - 19

Year 1

CS113/0401/v1

STORAGE OF NEGATIVE VALUES (2)

Conversion from two’s complement to decimal

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

Lesson 2 - 21

Year 1

CS113/0401/v1

TWO’S COMPLEMENT SUBTRACTION