1. INTRODUCTION
Digital techniques have found their way into numerous areas of technology, but the area of automatic digital computers is by far the most famous and most comprehensive. As you know, a computer is a system of hardware that performs arithmetic processes, exploits data, and makes decisions.
In science, technology, business, and, in fact, most other fields of endeavor, we regularly deal with quantities; so do computers. Quantities are estimated, monitored, recorded, manipulated arithmetically, observed, or used in some other way in most physical systems.
In digital systems like computers, the quantities are represented by symbols named digits. Many digit systems are used in digital technology that define the digits in various forms. The decimal, binary, octal, and hexadecimal systems are the most common. This chapter discusses these number systems and the physical representation of digits in computers.
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2. DIGITAL NUMBER SYSTEMS
In digital representation, various number systems are used. The most common digit systems used are decimal, binary, octal, and hexadecimal systems. Let us discuss these digit systems.
2.1. Decimal Number System (Base 10)
The decimal system is crafted of 10 numerals or symbols (Deca means 10, that is why this system is named decimal system). These 10 characters are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9; using these symbols as digits of a number, we can communicate any quantity. The decimal system, also called the base-10 system because it has 10 digits, has developed naturally as a result of the fact that man has 10 fingers.
The decimal system is a positional value system in which the value of a digit depends on its
position. For example, consider the decimal number 729. We know that the digit 7 actually means 7 hundred, the 2 represents 2 tens, and the 9 means 9 units. In essence, the
7 carries the most important of the three digits; it is directed to as the most significant digit (MSD). The 9 carries the least weight and is called the small significant digit (LSD).
Consider another example, 25.12. This number is actually equal to (2 tens plus 5 units plus
1 tenth plus 2 hundredths) i.e., 2x 10+5×1+ 1x 1/10+2×1/100 The decimal point is used to divide the integer and fractional portions of the number.
More precisely, the various positions close to the decimal point hold weights that can be
expressed as powers of 10. This is shown in Fig. 2.1 where the number 2512.1971 is
represented. The decimal point separates the positive powers of 10 from the opposing powers. The number 2512.1971 is hence equal to
2 x 10³ +5×10² +1×10¹ +2× 10° +1×10-1+9 × 10-2+7 x 10-31x 10-4
In general, any number is simply the sum of the products of per digit value and its positional value. The series of decimal numbers goes like 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21…. See after 9, each successive number is a combination of two (or more) (unique) symbols of this system.
2.2. Binary Number System (Base 2)
Unfortunately, the decimal digit system does not lend itself to suitable performance in digital systems. For example, it is very difficult to develop electronic equipment so that it can work with 10 different voltage levels (each one describing one decimal character, 0 via 9).
On the other hand, it is very easy to design easy, accurate electronic circuits that work with only two voltage levels. For this reason, almost every digital system uses the binary number system (base 2) as the basic number system of its functions, although other systems are often used in conjunction with binary.
In the binary system, there are only two signs or possible digit values, 0 and 1. Actually, this base-2 system can be used to define any quantity that can be defined in decimal or other number systems.
The binary technique is also a positional-value system, wherein per binary digit has its own importance or weight expressed as a power of 2. This is shown in Fig. 2.2.
Here, locations to the right of the binary point (counterpart of the decimal topic) are positive powers of 2, and zones to the right are negative powers of 2. The number 1010.0101 is shown represented in the constitution.
To find the decimal equivalent of an above shown binary number, we simply take the sum of the products of each digit value (0 or 1) and its positional value:
1010.0101₂ = (1 × 2³) + (0 × 2²) + (1 × 2¹) + (0 × 2º) + (0× 2 −¹) + (1 × 2 −²) + (0 × 2 −³) + (1 × 2 −4)
= 8+0+2+0+0+0.25 +0+0.0625 = 10.312510Notice in the initial operation that subscripts (2 and 10) were used to show the base in which the individual number is expressed. This pattern is used to bypass confusion whenever more than one number system is being operated.
In the binary system, the duration Binary digit is often shortened to the term bit, which we’ll use subsequently. As you see in Fig. 2.2, there are 4 bits to the left of the binary topic, representing the integer amount of the number, and 4 bits to the right of the binary point, representing the fractional parts.
The leftmost bit maintains the largest weight and consequently, is called the most significant bit (MSB). The rightmost bit maintains the smallest weight and is hence called the least significant bit (LSB).
The series of binary numbers runs as 00, 01, 10, 11, 100, 101, 110, 111, 1000, ——. The binary counting sequence has an important factor. The unit’s bit (LSB) changes either from 0 to 1 or 1 to 0 per count. The second bit (two’s (2¹) position) stays at 0 for two counts, then at 1 for two counts, then at 0 for two counts, and so on.
The third bit (four’s (2²) position) stays at 0 for four counts, then at 1 for four counts, and so on. The fourth bit (eight’s (23) position) stays at 0 for eight counts, then at 1 for eight counts. If we wanted to count further we would add more places, and this pattern would continue with Os and 1s alternating in groups of 2N-1.
2.3. Octal Number System (Base 8)
The octal number technique is very important in digital computer work. The octal number system has a base of eight, meaning that it has eight special symbols : 0, 1, 2, 3, 4, 5, 6, and 7. Thus, each digit of an octal number can have any weight from 0 to 7. The octal system is also a positional value technique, wherein each octal digit has its own value or weight expressed as a power of 8 (see Fig. 2.3).
The places to the left of the octal effectiveness (counter-part of decimal point and binary point) are positive powers of 8 and places to the right are negative powers of 8. The number 3721.2406 is shown represented in the figure To find the decimal equivalent of above shown octal number, simply take the sum of products of each digit value and its positional value:
3721.2406g = (3x8³) + (7x8²) + (2x8¹) + (1x80) + (2x 8-¹)+(4x8-2) + (0×2-³) +(6x8-4)
= 3 × 512 +7 x 64 +2× 8+1x1+2× 0.125+4x 0.015625 +0+ 6× 0.000244
= 1536 + 448 +16+1+0.25 +0.0625 +0+0.001464 = 2001.313964-10The sequence of octal numbers goes as 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22….. See each successive number after 7 as a combination of two or more unique symbols of the octal system.
2.4. Hexadecimal Number System (Base 16)
The hexadecimal system uses base 16. Thus, it has 16 possible digit symbols. It operates the digits 0 via 9 plus the letters A, B, C, D, E, and F as the 16-digit symbols.
Just like the above-discussed systems, the hexadecimal system is also a positional-value system, wherein each hexadecimal digit has its own value or weight expressed as a power of 16. (see Fig. 2.4).
The digit works in a Hexa- decimal number has importance as shown in Fig. 2.4. Following table 2.1 shows the connections between Hexa decimal, octal, decimal, and binary.
Note that per hexadecimal digit represents a group of four binary digits. It is important to remember that hex (abbreviation for hexadecimal) digits A through F are equivalent to the decimal values 10 via 15.
Applications of Hexadecimal Number System
The hexadecimal number system is used in computers to specify memory addresses (which are 16-bit or 32-bit long) e.g., a memory address 110101101010111 is a big binary address but with hex, it is D6AF, which is easier to remember.
Another application of the hexadecimal number system is to represent color codes e.g. (FF, FF, FF) represents White in RGB value and (80, 80, 80) symbolizes Grey in RGB value.
