numbering system

Malaysian Institute of Aviation Technology


LEARNING OUTCOME Binary, octal and hexadecimal Conversion between : decimal and binary, octal and

hexadecimal systems and vice-versa BCD Signed numbers

Conversion between positive and negative numbers into 1s and 2s compliment

Addition of numbers in the 2s compliment

Digital calculation Addition and subtraction in BCD and HEX forms Conversion of HEX numbers into 2s compliment form


Comprise of 10 digits from 0 to 9

Base 10 system. Example 25410

POSITIONAL VALUE SYSTEM


Example : 25410 consists of 2 HUNDREDS, 5 TENS and 4 ONE units.Written as : (2 X 100) + (5 X 10) + (4 X 1)

= (2 X 102) + (5 X 101) + (4 X 100)

Digit 2 carries the MOST weight and is known as MOST SIGNIFICANT DIGIT (MSD)

Digit 4 carries the LEAST weight and is known as LEAST SIGNIFICANT DIGIT (LSD)


DISADVANTAGE for usage in digital computer example transistor due to :-Having 10 Discrete Value Level which is extremely difficult to operate due to:a. Any VARIATION of POWER SUPPLY would cause errorb. Component TOLERANCE MUST be ZEROc. Component VALUE will change with AGE


Comprise of 2 digits (0 & 1) known as BITS Base 2 system. Example : 10112 POSITIONAL value system

10112=(1X23)+(0X22)+(1X21)+(1X20)

BINARY to DECIMAL Conversion10112=(1X23)+(0X22)+(1X21)+(1X20)

= 8+0+2+1= 11


DECIMAL to BINARY Conversioni)SUCCESSIVE POWER OF 2.

Example: 27 =16+8+2+1 =24+23+21+20

=(1x24)+(1x23)+(0X22)+(1X21)+(1X20) =110112

ii)SUCCESSIVE divide by 2 and record any remainder of division.

Suitable for SMALL number


Any number converted into BINARY form, the binary numbers is known as a WORD.

Each word is formed of a numbers of BITS(BINARY DIGITS) and this represents the WORD LENGTH Example : 34710 = 1010110112. So 1010110112 is WORD. Word length is 9

because there is 9bits


Base 8 systems Composed of 8 digits from 0 to 7 OCTAL to DECIMAL conversion

Successive power of 8

Example : 2758=(2x82)+(7x81)+(5x80)=128+56+5 = 18910

DECIMAL to OCTAL conversion Divide by 8 and Record any REMAINDER of division


OCTAL to BINARY conversion- Convert each OCTAL number into 3 bits BINARY equivalent.

Example : 6358 TO BINARY.6 3 5110 011 101

Thus, 6358= 1100111012


BINARY to OCTAL conversion- Divide BINARY number into groups of 3 BITS starting from LSB.

Example :1001110112 TO OCTAL.100 111 0114 7 3

Thus, 1001110112=4738

If the FINAL group of MSB does NOT have 3 BITS, ADD enough ZERO to make up 3 BITS.


BASE 16. Composed of 16 digit Symbols0 1 2 3 4 5 6 7 8 9 A B C D E F

Example: 85D1B16 HEX to DECIMAL conversion

- SUCCESSIVE POWER OF 16Example :B2F16=(11x162)+(2x161)+(15x160)

=2816+32+15=286310


DECIMAL to HEX ConvertionDivide with 16 and take the REMAINDER of division

HEX to BINARY ConversionConvert each HEX digit into 4 bits BINARY equivalent.i.e. B2F16 TO BINARY

B 2 F1011 0010 1111

THUS, B2F16= 1011001011112


BINARY to HEX Conversion- Divide BINARY number into groups of 4 bits STARTING at LSB.

i.e. 1101101010012 TO HEX1101 1010 1001

13 10 9D A 9

Thus, 1101101010012=DA916


HEX to OCTAL Conversion and vice versai) Convert HEX to BINARY ii)Convert BINARY to OCTAL

To Convert OCTAL to HEX, just REVERSE the processExample : 3D16 convert to OCTALi)Convert HEX to BINARY, 3 D

0011 1101 3D16=1111012

ii)Convert BINARY to OCTAL, 111 1017 5 1111012=758

Thus, 3D16=758


Binary Coded Decimal Number represented in 4 bits binary code Leaving a space between each group of 4

digits Example :

a) 1110 to BCD is 0001 0001b) 1000 0101 in BCD to Decimal is 8510


Exercise:a) Convert from decimal to BCDi. 94ii. 529iii. 2947

b) Convert from BCD to decimali. 011100001001ii. 001101100100


Rules :a) 0 + 0 = 0b) 0 + 1 = 1c) 1 + 0 = 1d) 1 + 1 = 0 with 1 to carry

Example : 10112 + 11102 = 110012


Rules :a) 0 0 = 0b) 1 0 = 1c) 1 1 = 0d) 0 1 = 1 borrow 1

Example : 110112 - 101012 =001102


Rulesa) 0 x 0 = 0b) 0 x 1 = 0c) 1 x 0 = 0d) 1 x 1 = 1 Example : 11002 x 112 = 1001002

In computer, it is achieved by repeated addition. Ex:In decimal 2 x 4 is computed as 2+2+2+2 = 8


Rulesa) 1 1 = 1b) 0 1 = 0 Example : 1111002 1102 (60 6) = 10102

(10)

Computer CAN NOT divide. It divide by repeated subtraction


Computer need to distinguish between positive and negative numbers.

For storage purpose, additional bit added to identify positive or negative

0 for positive and 1 for negative


Example: Using 8 bit binary word, the sign bit is added on the front

a) Decimal : - 10.25 Binary : 101010.010-ve sign

b) Decimal : +10.25 Binary : 001010.010 +ve sign


Additional bit ONLY for storing numbers but DOES NOTallow direct subtraction of one number from another

Use TWOs COMPLEMENT Process of inverting each bit in a word and adding 1

Example : Find negative binary number of +510i) +5 as a 4 bit word 0 0101ii) Invert (change 0 to 1 & 1 to 0) 1 1010iii)Add 1 1iv)Answer 1 1011


When numbers are represented as NEGATIVE, subtraction is achieved by addition

Adding a negative number is the SAME as subtracting a positive number

Example : 5 7 = 5 + (-7)


5 7 = 5 + (-7) +7 = 00111= 00101 + 11001 Invert = 11000=11110 Add 1 = 11001= -2

+2 = 00010Invert = 11101Add 1 = 11110


Conclusion Both addition and subtraction CAN be done

by the SAME circuits in a computer which considerably REDUCES the hardwareinvolved

LEARNING OUTCOME

numbering system

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