DIGITAL BASIC - 1.1 : NUMBER SYSTEM

Index	Chapter1	Chapter2	Chapter3	Chapter4
	Digital Background	Semiconductor Background	CMOS Processing

1.1	1.2	1.3a	1.3b	1.4	1.5	1.6
Number System	Digital Arithmetic	Logic Gates	Logic Gates	Combinational Circuits	Multiplex (MUX)

When I was talking to few of my friends, then they suggested me that I should write something about the basic also. Like Digital Basics / CMOS-MOS basics / VLSI basics.
In this series, I will try to add all the necessary basics from VLSI point of view in short and crispy manner. In few cases I have discussed the shortcut methods also.
Just in case you wanted to understand in details - you can refer the Books from the "Recommended books" section.

Let's start with the DIGITAL BASIC.

Electronics Circuits are of 2 types.

• Analog : Analog circuis are those in which voltages and currents vary continuously through the given range. They can take infinite value with in the specified range.
• Digital : Digital circuits are one in which the voltage levels assume a finite number of distinct value.

Digital circuits are also called

• Switching Circuits : because the voltage levels in a digital circuit are assumed to be swithed from one value to another simultaneously.
• Logic circuits : Because each type of digital circuits obey a certain set of logic rules.

Digital Circuits have many advantage over the analog circuits. 

• Easier to design: Switching circuits where there are only 2 states (HIGH/LOW , TRUE/FALSE, 0/1) are easier to design. Since these are state dependent so actual numerical value is not important only the range is important. In analog, signal has numerical significance, so it’s difficult to design.
• Information storage is easy: There are many types of memories which can store the data as long as required.
•  Accuracy and precisions are great: For obtaining the same accuracy and precision, analog circuits are more complex and costly then the digital circuits.
• Digital circuits are less effected by noise : Since in analog, exact value of voltage is important, so a small change in the voltage because of noise effect the finial output/functionality of the syste. In case of Digital, if the voltage is with in the range (even after effecting with Noise), there will be no change in the output.

Limitation of Digital Circuits.

• It has only one major Drawback, that REAL WORLD IS ANALOG.
• Just because of this Drawback, we have to design a lot of other system. Like – convert the analog info into digital and then finally the digital info to analog. Which usually increase the size of the circuit but this is also true that this dis-advantages are overweights by the other advantages.

Number System

Digital circuit use the binary number system. But to understand this system first we have to understand the other number system. There are many number systems present.  The most frequently used number systems in the applications of Digital Computers are Binary Number System, Octal Number System, Decimal Number System and Hexadecimal Number System.

Characteristics of Number system

• Base or radix is equal to the number of digits in the system.
• The largest value of digit is one (1) less than the radix.
• Each digit is multiplied by the base raised to the appropriate power depending upon the digit position.
•  The maximum value of digit in any number system is given by (Ω-1), where Ω is radix

Example 1:  Maximum value of digit in decimal number system = (10 – 1) = 9.

Base or Radix (r) of a Number System: The Base or Radix of a number system is defined as the number of different symbols (Digits or Characters) used in that number system.

Number System	Radix/Base	Number Sequence/ Possible symbols
Binary System	2	0 and 1
Ternary number system	3	0, 1 and 2
Quaternary	4	0, 1, 2 and 3
Quinary	5	0, 1, 2, 3 and 4
Octal System	8	0, 1, 2, 3, 4, 5, 6 and 7
Decimal System	10	0, 1, 2, 3, 4, 5, 6, 7, 8 and 9
Hexadecimal number system	16	0, 1, 2, 3, 4, 5, 6, 7, 8, 9,A, B, C, D, E and F

• To distinguish one number system from the other, the radix of the number system is used as suffix to that number.
• Eg:  10₂ Binary Numbers;   10₈ Octal Numbers.
• 10₁₀ Decimal Number; 10₁₆ Hexadecimal Number.

Positional Number Systems

• Binary, Octal, Decimal and Hexadecimal number systems are called positional number systems. 
• Any positional number system can be expressed as sum of products of place value and the digit value.

E.g 756­_10­ = 7 x 10² + 5 x 10¹ + 6 x 10⁰

156.24­­₈ = 1 x 8² + 5 x 8¹ + 6 x 8⁰ + 2 x 8^-1 + 4 x 8^-2

Number System	Place values or Weight of different Digits
Digit	X X X X X . X X X X
Digital number	104 103 102 101 100 . 10-1 10-2 10-3 10-4
Binary number	24 23 22 21 20 . 2-1 2-2 2-3 2-4
Octal Number	84 83 82 81 80 .  8-1 8-2 8-3 8-4
Hexadecimal number	164 163 162 161 160 . 16-1 16-2 16-3 16-4

Example 2:

Binary to Decimal Conversion (Short Cut Method)

Binary to Decimal = Binary → octal → Decimal

Eg. Convert 101110 into decimal

Solution (101 110)₂ = (56)₈ = 5 x 8 + 6 = (46)₁₀

Note: For converting Binary to octal make group of 3 bit starting from left most bit

  

Example 3:

Binary to Decimal Conversion (Equation Method)

S _i-1 = 2S _i + a _i-1

Where S_n = a_n and S₀ = the last sum term

Eg. (1101)2 to decimal

So  (1101)­₂  = (13)₁₀

Note: We can use calculator (scientific) but there is a limit of digit as input in calculator. We can use transitional way of multiplying each digit with 2^n-1 (where n is the position of digit in binary number) and adding in the last but for large binary digit it’s again a tedious task

In the next part we will discuss BASIC ARITHMETIC.