1.2 Numbering Systems and Conversions

Number System:

A number system is a way of writing and showing numbers. It uses specific digits or symbols to represent numbers. It helps us organize and understand numbers within a certain group. Number systems also make it easier to do basic math operations like addition, subtraction, multiplication, and division.

A Number:

A number is a mathematical value used for counting or measuring or labelling objects. Numbers are used to perform arithmetic calculations. Examples include: 20, 45, - 10, 3.4, 11.5, - 75.6, etc.

1.2.1 Types of Number Systems

There are various types of number systems in mathematics. The four most common number system types are:

Decimal number system (Base- 10) Binary number system (Base- 2) Octal number system (Base- 8) Hexadecimal number system (Base- 16)

Decimal Number System

The decimal (also called Denary) number system is composed of the 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. Using these symbols we can express any quantity. The decimal number system is called base 10 system because it has 10 digits. The decimal number system is a positional- value system, in which the value of a digit depends on its position in the number. For example, consider the decimal number 964. Here:

9 represents 9 hundreds 6 represents 6 tens 4 represents 4 units

The 9 carries the most weight. It is called the Most Significant Digit (MSD). The 4 carries the least weight and is called the Least Significant Digit (LSD). Each digit of a number carries weight that can be expressed as powers of 10.

In general, any number is simply the sum of the products of each digit value times its positional value.

Binary Number System

In binary number system there are only two symbols or digits, 0 and 1. The base of binary number system is 2. All the statements made earlier about the decimal number system are equally applicable to the binary number system as well. The binary system is also a positional- value system, wherein each bit has its own value or weight expressed as power of 2.

The binary number $(1011.101)_{2}$ is represented above and its equivalent decimal value can be found by taking the sum of the products of each bit value (0 or 1) times its positional value. The left most bit carries the most weight and it is called the Most Significant Bit (MSB). The positions to the right of the binary point are negative powers of 2. The right most bit carries the least weight and is referred to as the least significant bit (LSB).

Octal Number System

The Octal number system has a base of 8, meaning it has eight possible digits; 0, 1, 2, 3, 4, 5, 6, and 7. Thus, each digit of an octal number system can have any value from 0 to 7. The digit positions in an octal number system have weight with powers of 8. An octal number can be easily converted to its decimal equivalent by multiplying each digit by its positional weight.

Hexadecimal Number System

The hexadecimal number system uses base 16. Thus it has 16 possible digits. It uses the digits 0 - 9 and letters A, B, C, D, E and F as the 16 digit symbols. The table shows the relationship among hexadecimal and decimal numbers. It is important to remember that hexadecimal digits A through F are equivalent to the decimal values 10 through 15.

A hexadecimal number can be converted to its decimal equivalent by using the fact that each digit position has a weight that is power of 16. The LSD has a weight of $16^{0} = 1$ the next higher digit has a weight of $16^{1} = 16$ , the next higher digit has a weight of $16^{2} = 256$ , and so on.

The conversion process is shown below.

1.2.2 Conversions from one Number system to the other

i. Conversion from Binary to Decimal

Example1: (10111)2 to ( )10(10111)2 = (1×2²)+(0×2³)+(1×2²)+(1×2³)+(1×2²)= 16+0+4+2+1= (23)10

Example2: (100101.1011)2 to ( )10(100101.1011)2 = (1×2²)+(0×2³)+(0×2²)+(1×2²)+(0×2²)= 32+0+0+4+0+1+(1/2)+(0)+(1/8)+(1/16)= 37+(11/16)= 37+0.6875= (87.6875)10

1i. Conversion from Octal to Decimal

iii. Conversion from Hexadecimal to Decimal

Conversion from Base-10(Decimal) to any other Base

To convert a decimal number to any other number:

Divide the decimal number to be converted by the value of the new base and record the remainder as the LSD of the new base number.

Divide the quotient of the previous division by the new base again and record the remainder as the next digit to the left of the new base number.

Repeat this process, until the quotient becomes zero or not divisible by the base.

Note that the last remainder obtained will be the MSD of the new base number.

The following examples will demonstrate the above method by converting the decimal numbers to their binary, octal and hexadecimal equivalents, respectively.

1. From Decimal to Binary Converting the Integer Part:

> Divide the integer part by 2. >> Record the remainder (0 or 1). >> Update the integer part to the quotient. >> Repeat steps 1- 3 until the integer part is 0. >> The binary representation is the remainders read in reverse order.

Converting the Fractional Part:

> Multiply the decimal part by 2. >> Record the integer part (0 or 1). This becomes the next binary digit. >> Remove the integer part from the product; leaving only the decimal part. >> Repeat the process with the new decimal part. >> Continue until the decimal part becomes 0 or until you have reached the desired precision. >> The binary representation is the sequence of recorded integers.

1. From Decimal to Octal

iii. From Decimal to Hexadecimal

Conversion from Octal to Binary and vice versa

Using these conversions, any octal number can be converted to binary and vice versa.

1. From Octal to Binary

ii. From Binary to Octal

Conversion from Hexadecimal to Binary and vice versa

Using these conversions, any Hexadecimal number can be converted to binary and vice versa.