By:Iqra Sundip Yasin1450-213001

NUMBER SYSTEM &

DEFINATIONS

Digital Number System

Many number system are in use in digital technology. The most common are the decimal, binary, octal, hexadecimal system.

OTHER NUMBER SYSTEM

Base-2 binary system Base3 tritary system Base-11

undecimal system

Base-4 quaternary system  Base-12 duodecimal system

Base-5 quinary system  Base-13 tridecimal system

Base-6 senary system Base-14 tetradecimal system

Base-7 septenary system  Base-15 pentadecimal system

Base-8 octal system  Base-16 hexadecimal system

Base-9 nonary system 

Base-10 decimal system Base-20 The vigesimal system

Base-36 hexatridecimal system

Decimal System

The decimal system is composed of 10 symbols. These 10 symbols are 0,1,2,3,4,5,6,7,8,9; using these symbols as digits of a number of a number. The decimal system also called base 10 system because it has 10 digits, has evolved naturally as a result of the fact that people have 10 finger .

In fact, the word “digit” is derived from the Latin word for “finger.”

Positional Notation The decimal system is a positional –value system

in which the value of a digit depends on its position….

642 in base 10 positional notation is:

6 x 102 = 6 x 100 = 600 + 4 x 101 = 4 x 10 = 40 + 2 x 10º = 2 x 1 = 2 = 642 in base 10

The power indicates

the position of

the number

MSD & LSD

Stands for Most significant digit and least significant digit respectively e.g. 2735.214.2 carries the most weight or value so 2 is MSD and 4 carries the least weight or value so 4 is LSD .

CONVERSIONDecimal to binary , octal and hexadecimal

Conversion from decimal system into other system. It involves using successive division by the radix until the dividend reaches 0. At each division, the remainder provides a digit of the converted number, starting with the least significant digit.

For example: convert 3710 to binary

37 / 2 = 18 remainder 1 (least significant Bit)

18 / 2 = 9 remainder 0

9 / 2 = 4 remainder 1

4 / 2 = 2 remainder 0

2 / 2 = 1 remainder 0

1 / 2 = 0 remainder 1 (most significant Bit)

The resulting binary number is: 100101 THAT WE START FROM MSB TO LSB

EXAMPLE: CONVERT (177)10 TO OCTAL

Conversion of decimal fraction to octal fraction is carried out in the same manner as decimal to binary except that now the multiplication is carried out by 8.SOLUTION:177 / 8 = 22 remainder is 122 / 8 = 2 remainder is 6 2 / 8 = 0 remainder is 2

The resulting binary number is: (261)8

Example: convert (4768)10 to hex.

Solution:= 4768 / 16 = 298 remainder 0= 298 / 16 = 18 remainder 10 (A)= 18 / 16 = 1 remainder 2= 1 / 16 = 0 remainder 1The resulting binary number is: (12A0)16

What if 642 has the base of 13?

6 x 132 = 6 x 169 = 1014+ 4 x 131 = 4 x 13 = 52 + 2 x 13º = 2 x 1 = 2 2+52+1014 = 1068 in base 10

642 in base 13 is equivalent to 1068 in base 10

Binary Number System

Base = 2 2 digits { 0, 1 }, called binary digits or “bits”

Weights Weight = (Base) Position

Formal NotationGroups of bits 4 bits = Nibble

8 bits = Byte

Converting Binary to Decimal What is the decimal equivalent of the binary number 1101110?

1 x 26 = 1 x 64 = 64 + 1 x 25 = 1 x 32 = 32 + 0 x 24 = 0 x 16 = 0 + 1 x 23 = 1 x 8 = 8 + 1 x 22 = 1 x 4 = 4

+ 1 x 21 = 1 x 2 = 2 + 0 x 2º = 0 x 1 = 0 = 110 in base 10

Conversion of binary to octal and hex Conversion Conversion of binary numbers to octal and hex simply

requires grouping bits in the binary numbers into groups of three bits for conversion to octal and into groups of four bits for conversion to hex.

Groups are formed beginning with the LSB and progressing to the MSB.

10101011 10 101 011 2 5 3

10101011 1010 1011 A(10) B(11)

Conversion of binary to octal

Conversion of binary to hexadecimal

Convert the binary number 0110101110001100 to hexadecimal

Divide into groups of 4 digits 0110 1011 1000 1100

Convert each group to hex digit 6 B(11) 8 C(12) 6B8C in hex

Calculator Hint

If you use a calculator to perform the divisions by 2, you can tell whether the remainder is 0 or 1 by whether or not the result has a fractional part. For instance, 25/2 would produce 12.5. Since there is a fractional part (.5 ), the remainder is a 1. If there were no fractional part, such as 12/2=6 then the remainder would be 0.

Octal refers to a numbering system that has a base of eight. This means it only uses the eight numerals 0,1,2,3,4,5,6,7 for each digit of a number.

Conversion:Converting Octal to Decimal:

What is the decimal equivalent of the octal number 642? 6 x 82 = 6 x 64 = 384 + 4 x 81 = 4 x 8 = 32 + 2 x 8º = 2 x 1 = 2

= 418 in base 10 Example: convert (632)8 to decimal= (6 x 82) + (3 x 81) + (2 x 80)= (6 x 64) + (3 x 8) + (2 x 1)= 384 + 24 + 2= (410)10

1) Convert (634)8 to binary equivalent? Sol. 6 3 4 110 011 100 . Binary number = (110011100)2

2) Convert (615)8 to hexadecimal equivalent?Sol. Step1 octal to binary 6 1 5 110 001 101 Binary number = 110001101 Step2 binary to hexadecimal 0001 1000 1101 1 8 D.: Hexadecimal number = (18D)16

Calculator Hint

If a calculator is used to perform the division , the result will include a decimal fraction instead of a remainder. The remainder can be obtained, However, by multiplying the decimal fraction by 8, for example 266/8 produces 33.25.The remainder becomes 0.25 * 8 =2.

We need this hint when we convert decimal system into octal system..

Hexadecimal number system

Convert (BEEF)16 into ( )10

CONVERT (9F2)16 INTO ( )2 ?

Shorthand method to representing binary numbers…(8421)(9F2)16 = 9 F 2 1001 1111 0010 SO binary number is= 100111110010

Convert (25B)16 to octal equivalent?Sol. Step1 hexadecimal to binary 2 5 B 0010 0101 1011 Binary number = 001001011011 Step2 binary to octal 001 001 011 011 1 1 3 3 Octal number = (1133)8

http://www.translatorscafe.com/cafe/EN/units-converter/numbers/2-1/octal-binary/

link:

DEFINATIONS:

Digital Signal: A digital signal is a physical 

signal that is a representation of a sequence of discrete values. Analog Signal: Analog signal

is a continuous signal w.r.t time. An Analog signal have its changing value at every instant of time

Active high and low:

Active high: It can call as positive logic. The high voltage +5v represent logic high that is 1 and lower 0v represent the logic low that is 0.Active low: it can call as negative logic . The higher voltage represent as logic low that is o and lower voltage represent as logic high that is 1..

Duty CycleA duty cycle is the percentage of one period in which a signal is active. A period is the time it takes for a signal to complete an on-and-off cycle. As a formula, a duty cycle may be expressed as: D=T/P *100where   is the duty cycle,   is the time the signal is active, and   is the total period of the signal. Thus, a 50 % duty cycle means the signal is on 50 % of the time but off 50% of the time.

Flip FlopMemory device capable of storing a logic level.Types of FF:i. S-R FFii. J-K FFiii.D FFiv.T FF

Fan in and out

Fan-in is a term that defines the maximum number of digital inputs that a single logic gate can accept. A typical logic gate has a fan-in of 1 or 2.

Fan-out is a term that defines the maximum number of digital inputs that the output of a single logic gate can feed.

OR The maximum number of logic inputs that an

output can drive reliably called fan out.

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