Digital number formats explained
Digital technology makes use of many different number systems, with binary, octal, decimal, and hexadecimal being the most prominent. These are used to better understand the ways in which computers perform the tasks assigned to them. The value of the numbers in these systems is represented by a base.
Binary number system
The binary system makes use of the base 2 and contains only two symbols 0 and 1. Each position in a binary number is represented by a power of 2.
For example, if a binary number is represented as (1011)2, its value in the decimal system can be calculates as follows:
1x23+0x22+1x21+1x20= 8+0+2+1 or 11
You can also represent any given decimal number in binary format by carrying out a simple, long division method. In order to do this, divide the given number repeatedly by 2, and list the remainder until the number reduces to one. The first remainder obtained is known as LSB or the least significant bit, and the last remainder obtained through long division is the MSB or the most significant bit. In such a division, the arrangement of remainders from MSB to LSB is itself the representation of the given number, in binary.
Octal number system
Octal number system as the name suggests makes use of 8 distinct digits or symbols to represent a value of any given magnitude. The symbols used in Octal number system include0,1,2,3,4,5,6,7. Each position in an octal number is represented with the help of the powers of 2.
For example, the decimal value of an octal number that is represented as (7536)8 can be calculated as follows:
7x83+5x82+3x81+6x80= 2688+320+24+6 or 3038
Any given octal number can also be easily represented in the form of an octal digit by using the long division method, which is similar to that used in the binary number system. The only difference would be that the divisor would be 8 instead of 2.
Decimal number system
Decimal number system is the most popularly used number system and makes use of 10 distinct digits or symbols to represent the value of any given magnitude. The symbols used in the decimal number system include 0,1,2,......9. The positions of decimal numbers are represented with the help of the powers of 10.
Hexadecimal number system
This number system is a complex type of number system used in computers. It makes use of 16 distinct digits or symbols to represent a value of any given magnitude. The symbols used in Hexadecimal number system include 0,1,2,.....,9,A,B,C,D,E,F. The alphabets are used to represent decimal numbers from 10 to 15. Each position in a hexadecimal number system is represented with the help of the powers of 16.
For example, if a hexadecimal number is represented as (1AC6)16, its value in decimal system can be expressed as follows:
1x163+Ax162+Cx161+6x160= 4096+2560+180+6 or 6842
Any given decimal number can also be easily represented in the form of a hexadecimal digit by making use of the long division method, which is similar to the one used in the binary number system. The divisor in this case would be 16. But take care to replace any remainders ranging between 10 and 15 with their appropriate hexadecimal representation.
Conversion between bases
Converting numbers between different bases can also be achieved conveniently by first using the exponent method to convert the number into decimal, and then use the long division method to change the base. In this case, the decimal number system acts as an intermediate for the conversion. You can also use external tools such as calculators or software for this purpose.
Floating Points in Number Systems
A floating point is a number that contains both an integer part as well as a fractional part. A floating point in the decimal number system is represented as 2543.986 or -7438.386, and so on. Similarly, floating points in binary, octal, and hexadecimal can also be represented as 1.0011, 64.25, and -ABC.93F respectively. So, we can generalize that the floating numbers can be represented in the form of exponents.
The general form for any floating point can be expressed as,
N = ±M x B±i
Here, 'M' is known as the mantissa, 'B' is known as the base, and 'i' is the exponent. The floating point is generally represented by two distinct parts, with the mantissa being the fixed point and the exponent being the binary or decimal position of the number.