revision01 numbering systems

7/29/2019 Revision01 Numbering Systems

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Number Systems R1

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Humans and the importance of small electricalpulses...

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Input

Hearing electrical pulses

Smells electrical pulses

Tastes electrical pulses

Sight electrical pulses

Touch

electrical pulses


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Output

Electrical pulses

movement

Electrical pulses speech

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The importance of numbers Gives meaning to our lives...

Basis of our monetary system...

HumansInput: Electrical Pulses

Output: Electrical Pulses

MicroprocessorsInput: Numbers

Output: Numbers5


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Number SystemsThe most important number systems to

programmers of

Ps, are: The Binary Number System,

The Octal Number System,

The Hexadecimal Number System,

And The Binary Coded Decimal (BCD)Number System

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Rules of Counting

There are 3 rules.......

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Number SystemsRules for counting

Rule 1:The number of fundamental digits or discrete levels in a number

system equals the base or radix.

Eg.

16Hexadecimal

8Octal2Binary

10Decimal

Radix / BaseNumber System

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Rule 2:

Counting in discrete steps includes counting from the lowestfundamental digit to the highest. Once complete a column-placingtechnique is used, the digit is reset to the lowest fundamental digitand the digit to the left is incremented by a fundamental unit.

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Rule 3Each digits position in the column is of significant importance. The

digit to the extreme right is the digit of least significant or leastvalue, and the digit to the extreme left has the greatest value.

A digit always begins in the leftmost column.

DigitDigitDigitDigit

Base0Base1Base2Base3

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The Decimal Number SystemRevision

The decimal number system has a radix or baseof 10 and consists of the digits:

0 1 2 3 4 5 6 7 8 9Decimal numbers are constructed using only these

digits and each column in the number increases

by a power of 10.e.g.

5374 = 4 x 100 + 7 x 101 + 3 x 102 + 5 x 103

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Many mathematical operations can be applied todecimal numbers. However only the basicoperators are implemented in a P, such as:

Addition,

Subtraction,

Multiplication,

and Division


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Starting at the least significant digits of both numbers the top digit isincremented by the lower digit, if this increment exceeds the limit of

the digits then a carry condition occurs. The carry condition insertsan additional 1 to the column directly to the right that is introducedto the summation. This procedure is repeated until the two mostsignificant numbers are added.

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3

9

3+


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4

23

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4

23

9

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1

+


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4

23

9

38

1

+


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Starting at the least significant digits of both numbers the top digit isdecremented by the lower digit, if this decrement is lower than zerothen a borrow condition occurs. The borrow condition incrementsthe number by the base (10) and reduces the number to the right byone fundamental digit. The lower number is then subtracted from theupper number. This procedure is repeated until the two mostsignificant numbers are added.

2

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8

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2

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8-1

-

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2

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8-1

-

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4

2

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3

8-1

-

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Binary Number System

The binary number system has a radix or base of 2 andconsists of the digits

0 1 The beauty of the binary system is that it directly relates

to logical states of circuits, they are either ON or OFF.

Binary numbers are constructed using only thefundamental digits 0 and 1.

It is obvious that more binary digits must be used thanwould necessarily be needed to represent decimalnumbers.

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Binary Number SystemConversion to Decimal

Each column in a binary number increases by apower of 2, the least significant column is 20 the

next 2

1

and so forth. The process of converting a number from binary

to decimal is as simple as multiplying the binarynumber by the associated column weights andadding the results.

For example:1011001b = 1 x 20 + 0 x 21 + 0 x 22 + 1 x 23 + 1 x 24 + 0 x 25 + 1 x 26

= 89d

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Binary Number SystemConversion from Decimal

The conversion from Decimal to binary is achieved by dividing thedecimal number by 2 and noting the remainder, the process isrepeated until zero remains.

For example

145/2 = 72 remainder =172/2 = 36 remainder =0

36/2 = 18 remainder =0

18/2 = 9 remainder =0

9/2 =4 remainder =1

4/2 = 2 remainder =02/2 = 1 remainder =0

= 0 remainder = 1

Therefore 145 is 10010001 in binary.

Check:

1 x 20 + 0 x 21 + 0 x 22 + 0 x 23 + 1 x 24 +

0 x 25 + 0 x 26 + 1 x 27 = 145

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Binary Number SystemBinary Addition

Adding binary numbers is similar to adding decimal

numbers, you start by adding the least significantdigits, making your way to the most significant,carrying were necessary.

Example:

1110

+ 0110

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Example:

1110

+ 0110

0

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Example:

1110

+ 0110

001

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Example:

1110

+ 0110

00

1

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Example:

1110

+ 0110

00

1

11

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Example:

1110

+ 0110

00

1

1

1

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Example:

1110

+ 0110

00

1

1

1

10

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Binary Number SystemBinary Subtraction

The procedure for subtracting binary numbers issimilar to that for decimal numbers. If a digit is

smaller than the digit being subtracted, then aborrow occurs from the column to the right.

Example:

1110

- 0110

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Example:

1110

- 0110

0

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Example:

1110

- 0110

00

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Example:

1110

- 0110

000

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Example:

1110

- 0110

0001

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Octal Number System

The octal number system has a radix or base ofeight and consists of the fundamental digits,

0 1 2 3 4 5 6 7

Octal numbers are created using only thesedigits.

Counting with octal numbers commences at 0

and continues to 7, the least significant numberis reset to 0 and the digit 1 is inserted to theleft forming 10.

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Octal Number SystemConversion to Decimal

The conversion from octal to decimal dependshighly on the column placing model discussedearlier. Each column in the number is increased bya power of eight.

For example

1534o = 4 x 80 + 3 x 81 + 5 x 82 + 1 x 83

= 860d

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Octal Number SystemConversion from Decimal

The conversion of a decimal number to an octal numberis performed in the same way as converting a decimalnumber to binary. However, since the base is 8 instead of

2 the divisor is in fact 8 in this instance.For example:

145 / 8 = 18 remainder = 1

18 / 8 = 2 remainder = 2

2 / 8 = 0 remainder = 2

Therefore 145d is in fact 221 in octal.

Check:

221o = 1 x 80 + 2 x 81 + 2 x 82

=145d

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Octal Number SystemConversion to and from Binary

Since the radix of of the octal number system is a powerof 2, conversion two and from binary is made simple.

Since 8 is 23, segments of three bits are used in the

conversion to and from binary. To Binary:

For each octal digit the corresponding 3 bit binary number can befound and the resulting bit segments concatenated together.

For example:

516o = 5 1 6 = 101 001 110

= 101001110b

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Octal Number SystemConversion to and from Binary

From Binary

For each three bit segment thecorresponding octal digit can be found and

concatenated.

For example:

101001110b = 101 001 110 5 1 6

= 516o

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Hexadecimal Number

The hexadecimal number system has a radix orbase of 16 and consists of the followingfundamental digits,

0 1 2 3 4 5 6 7 8 9 A B C D E F

Hexadecimal number are created using thesefundamental digits.

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Hexadecimal Number SystemConversion to Decimal

The conversion from hexadecimal to decimal is simple ifyou keep in mind each column in the number increases bya factor of 16.

For example:

FC81h = 1 x 160 + 8 x 161 + 12(C) x 162 + 15(F) x 163

= 64641d

It is obvious from this example that less digits are need torepresent decimal numbers in hexadecimal.

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From decimal to hexadecimal.

64641 / 16 = 4040 remainder = 1

4040 / 16 = 252 remainder = 8

252 / 16 = 15 remainder = 12 (C)

15 /16 = 0 remainder = 15 (F)

Therefore 64641d is in fact FC81h

Hexadecimal Number SystemConversion to Hexadecimal

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Hexadecimal Number SystemConversion to and from Binary

To Binary:For each hexadecimal digit the corresponding four bit number can be

calculated and the resulting bit segments joined together.

For example:

FC81h = F C 8 1

= 1111 1100 1000 0001

= 1111110010000001b

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Hexadecimal Number SystemConversion to and from Binary

From Binary:

Segments of four bits are selected and a

hexadecimal digit found for each segment.For Example:

1011001011000101b = 1011 0010 11000101

= B 2 C 5 = B2C5h

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Binary Coded Decimal

Only the binary numbers that represent the decimalnumbers from 0 to 9 are valid in each 4 bit set.

101 100

XXXX XXXX

Using 8 bits a decimal number

between 0 and 99 can be

represented

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Representing Negative

Integers

The signed modulus / sign bit extensionThe ones complement method

The twos complement method

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Signed Modulus

Most Significant Bit represents the sign:

0 means positive,

1 means negative.

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Signed ModulusProblems

The signed modulus method contains two problems.

Problem # 1:

10000010-2

10000001-1

000000000

000000011

000000102

000000113

000001004

000001015

If 00000000 is the binary equivalent of zero

then what is 10000000, -0 perhaps.

The signed modulus method is inefficient,

it contains two different methods ofrepresenting zero.

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Signed ModulusProblems

Problem #2:

No additive Inverse

Example:

Does 34 + -34 = 0

00100010 + 10100010 = 1100010034 + -34 = - 68

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Ones Complement

The ones complement of 37 would becalculated as follows:

Firstly find the 8 bit binary equivalent of 37. 37d = 00100101b Then invert each digit, resulting in -37d = 11011010b

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Twos Complement

A method of representing negative numbers commonly usedin binary arithmetic is the twos complement.

The procedure for finding the twos complement is: Firstly, convert the number into its binary representation. 37d = 00100101b

Secondly, invert all the bits of the number.

= 11011010b

Finally, add 1 to this number.

11011010 + 1

-37d = 11011011b

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Twos Complement

The two complement is a successful method for finding an additivenegative binary number. This can be proven using a 3 bit example.

Check:

011

+ 101

(1)000

Check:

010+ 110

(1)000

100

011 3

010 2

001 1

000 0

111 -1

110 -2101 -3

100

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Multi-word Numbers

Microprocessors have a fixed word size. Limited range of numbers.

How do we represent bigger numbers?

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Multi-word Numbers

An 8 bit microprocessor

8 bit word length

numbers ranging value from 0 to 255

How could it represent a larger decimal number

such as 3867?3867d = 0000111100011011b

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Multi word Numbers


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Multi-word Numbers

By splitting the 16 bit number into two 8 bit

components.3867d = 0000111100011011b

location number 1 = 00001111 (15) MSB

location number 2 = 00011011 (27) LSB

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Multi word Numbers


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Multi-word Numbers

As mentioned earlier care must be made when dealing withmulti-word numbers. The significance of the numbersmust be taken into account.

Example: 4326 + 1276

16 bit Version:

0001000011100110

+ 00000100111111000001010111100010

8 bit Multi-word Version:

LSB first: MSB second:

11100110 00010000

+ 11111100

00000100(1)11100010 + 1

00010101

= 0001010111100010

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Multi-word Numbers


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Multi word NumbersBig Endian / Little Endian

When representing 16 bit numbers by two 8 bit ones theycan be stored in memory either MSB first or LSB first.

When the MSB is stored in a the memory locationpreceding the LSB, it is referred to asBig Endianrepresentation.

When the MSB is stored after the LSB in memory it isknown as Little Endian.

The S12X micro-controllers useBig Endian representationfor all 16 bit numbers used.

Big Endian Little Endian

0800:MSB

0801:LSB

0800:LSB

0801:MSB

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