Number
systems are systems in mathematics that are used to represent or express
numbers in various forms and are understood by computers. The number is a mathematical
value used for counting and measured the objects, and to performing arithmetic
calculations. The numbers having various categories like whole numbers, natural
numbers, irrational and rational numbers, so on. Similarly, there are various
types of number systems that have different properties, but we would like to
discuss the digital electronics relevant number systems like the decimal number
system, binary number system, octal number system and the hexadecimal number
system.
Types of Number Systems
In this post, we will explore different types of number systems that we use such as the binary number system, the octal number system, the decimal number system, and the hexadecimal number system. We will discuss and learn about the conversions between in all of these number systems and solve examples for a better understanding purpose.
What are Number Systems?
A number system is a system representing or express the numbers. It is also called the system of number or numeration and it defines a set of values to represent a quantity. These numbers are utilized as digits and the most common example is ones are 0 and 1, that are used to represent the binary numbers. and the Digits from 0 to 9 are used to represent decimal number systems.
Types of Number Systems
There are these four different types of number systems used in Digital Electronics which are follows.
1. Binary number system (Base - 2)
2. Octal number system (Base - 8)
3. Decimal number system (Base - 10)
4. Hexadecimal number system (Base - 16)
We will study each of these systems one by one in detail
after going through the following number system chart.
Number System Table
Given below is a table to understand the main four types of
number systems that we are using to represent numbers in Digital Electronics systems.
and circuit.
Binary Number System
The binary number system having just two digits: 0 and 1. The
numbers in this system having a base of just 2. Digits 0 and 1 are called bits
the four bits are equals to nibble 1111 and 8 bits together or two nibbles
together make a byte like this 11111111. Always the data in computers is accept
and stored in terms of bits and bytes. The binary number system does not deal
with other numbers such as 2,3,4,5,6,7,8,9 and so on. For example: 101012,
1101012, 0101012 are some examples of numbers in the binary number system
Octal Number System
The octal number system uses eight digits: 0,1,2,3,4,5,6 and
7 with the base of 8. The advantage of this system is that it has lesser digits
when compared to several other systems, hence, there would be fewer
computational errors. Digits like 8 and 9 are not included in the octal number
system. Just as the binary, the octal number system is used in minicomputers
but with digits from 0 to 7. For example: 358, 238, 1418 are some
examples of numbers in the octal number system.
Decimal Number System
The decimal number system uses ten digits: 0,1,2,3,4,5,6,7,8 and 9 with the base number as 10. The decimal number system is the system that we generally use to represent numbers in real life. If any number is represented without a base, it means that its base is 10. For example: 72310, 3210, 425710 are some examples of numbers in the decimal number system.
Hexadecimal Number System
The hexadecimal number system uses sixteen digits/alphabets: 0,1,2,3,4,5,6,7,8,9 and A,B,C,D,E,F with the base number as 16. Here, A-F of the hexadecimal system means the numbers 10-15 of the decimal number system respectively. This system is used in computers to reduce the large-sized strings of the binary system. For example, 7B316, 6F16, 4B2A16 are some examples of numbers in the hexadecimal number system.
Conversion Rules of Number Systems
A number can be converted from one number system to another
number system using number system formulas. Like binary numbers can be
converted to octal numbers and vice versa, octal numbers can be converted to
decimal numbers and vice versa, and so on. Let us see the steps required in converting
number systems.
Step by Step Conversion
of Binary to Decimal Number System
To convert a number from the binary to the decimal system,
we use the following steps.
Let us understand the steps with the help of the following
example in which we need to convert a number from binary to decimal number
system.
Example: Convert 1011012 into the decimal system.
Solution:
Step 1: Identify the
base of the given number. Here, the base of 1011012 is
2.
Step 2: Multiply each
digit of the given number, starting from the rightmost digit, with the exponents of the base. The exponents should start with 0 and
increase by 1 every time as we move from right to left. Since the base is 2
here, we multiply the digits of the given number by 20, 21, 22, and so on from right to left.
Step 3: We just
simplify each of the above products and add them.
Here, the sum is the equivalent number in the decimal number system of the given number. Or, we can use the following steps to make this process simplified.
Step by Step Conversion
of Decimal Number System to Binary / Octal / Hexadecimal Number System
To convert a number from the decimal number system to a
binary/octal/hexadecimal number system, we use the following steps. The steps
are shown on how to convert a number from the decimal system to the octal
system.
Example: Convert 298010 into the octal system.
Solution:
Step 1: Identify the
base of the required number. Since we have to convert the given number into the
octal system, the base of the required number is 8.
Step 2: Divide the
given number by the base of the required number and note down the quotient and
the remainder in the quotient-remainder form. Repeat this process (dividing the
quotient again by the base) until we get the quotient less than the base.
Step 3: The given number in the octal number system is obtained just by reading
all the remainders and the last quotient from bottom to top.
Conversion from One
Number System to Another Number System
To convert a number from one of the binary/octal/hexadecimal
systems to one of the other systems, we first convert it into the decimal
system, and then we convert it to the required systems by using the
above-mentioned processes.
Example: Convert 1111101002 to the hexadecimal system.
Solution:
Step 1: Convert this number to the decimal number system as explained in the above
process.
Thus, 1111101002 = 50010 → (1)
Step 2: Convert the
above number (which is in the decimal system), into the required number system
(hexadecimal).
Here, we have to convert 50010 into the
hexadecimal system using the above-mentioned process. It should be noted that
in the hexadecimal system, the numbers 15 is written as F respectively.
Thus, 50010 = 1F416 → (2)
From the equations (1) and (2), 1111101002 = 1F416
Number Systems Examples
Example 1: Convert 20010 into the binary number system with base 2.
Solution: 20010 is in the decimal system. We divide 200 by 2 and note down the quotient and the remainder. We will repeat this process for every quotient until we get a quotient that is less than 2.
11001000
The equivalent number in the binary system is obtained by reading all the remainders and just the last quotient from bottom to top as shown above.
Thus, 20010 = 110010002
Example 2: Convert 5BC16 into the decimal system.
Solution: 5BC16 is in the hexadecimal system. We know that B = 11 and C = 12 in the hexadecimal system. So we get the equivalent number in the decimal system using the following process:
Thus, 1228 = 8210 → (1). Now we will convert this into the hexadecimal system as follows:
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