Content Curator
Number systems are a way of expressing numbers in different ways using a variety of bases. There are numerous types of number systems that serve several purposes and are equally important. Some number systems are binary number system, decimal number system, hexadecimal number system. The binary number system is useful when it comes to computer applications. As computers can only interpret 0 and 1, the binary number system is widely used in the field of Computer Technology. In order to convert binary, octal and hexadecimal to decimal number, we need to add the product of each digit to its positional value.The conversion between number systems is very important and useful in real-life applications.
| Table of Contents |
Key Terms: Binary number system, decimal number system, hexadecimal number system, octal number system, decimal, number system conversion.
What is a Number System?
[Click Here for Sample Questions]
Number systems are a way of number representation to serve different purposes of mathematical and other scientific calculations. It is a common way of representing different numbers with suitable bases and numeral digits. Number systems are also important for algebraic and arithmetic calculations. In other words, a number can be represented in a more unique manner using a particular number system. Number systems also define a set containing different numbers using mathematical notations. Number system conversions are useful in computer interpreters and several other real-life applications. The important factors to be noted while performing number system conversions are as follows.
- The digit.
- The base of the number system, for eg- the base in binary number system is 2, the base in octal number system is 8, the base in decimal number system is 10, the base in hexadecimal number system is 16.
- The position of the digits in the given number.
Types of Number System
[Click Here for Sample Questions]
There are various kinds of number systems where numbers are differently represented. These different kinds of number systems are as follows.
1. Decimal Number System or the Number System with base 10
The decimal number system is the number system that contains numbers with a base of 10. The decimal number system has numbers from 0 to 9. This number system has a base of 10 since it has 10 unique numbers to represent the various other numbers in the set. Each number in the decimal number system can be represented by a certain power of the base 10.
Let's say we take a number 592678,
= 5 x 10? + 9 x 10? + 2 x 103 + 6 x 10² + 7 x 10¹ + 8 x 10?
= 5 x 100000 + 9 x 10000 + 2 x 1000 + 6 x 100 + 7 x 10 + 8 x 1
= 500000 + 90000 + 2000 + 600 + 70 + 8
= 592678
Therefore, 592678, this number in the decimal number system could be represented as a power of the base 10 where the power is the position of the respective digits starting from zero and moving towards the left.
2. Binary Number System or the number system with base 2
The binary number system contains numbers with a base of 2. 0 and 1 are the only two numbers in the binary number system. All the numbers in the other number systems can be represented using the binary number system.
3. Octal Number System or the Number System with base 8
The octal number system has numbers from 0 to 7 to represent numbers. This number system has numbers with the base 8. The conversion of octal to decimal number system is similar to that of the conversion in the decimal number system, instead of a power of 10, in case of the octal number system we use a power of 8 where the power is the corresponding position of the respective digits. The octal number system is also useful in computer applications like the binary number system.
4. Hexadecimal Number System or the Number System with base 16
The Hexadecimal Number System contains numbers with a base of 16. The hexadecimal number system has a total of 16 characters where numbers from 0 to 9, like the decimal number system and alphabets A to F are present.
Number System Conversion
[Click Here for Sample Questions]
The different number system can be converted to other number systems with different bases. The different number system conversions are as follows.
-
Binary Number System to Decimal Number System
To convert a binary number to a decimal number we first write the position of the digits over the binary digits and then multiply that digit with a power of 2 where the power is the position of the respective number.
-
Decimal to Binary Number System
To convert a decimal number to a binary number we divide the decimal number by 2 and then write down the remainder, and continue the process until we reach 0.
-
Octal to Binary Number System
A simple way to convert an octal number to a binary number is to use the table below.
| Octal Number | Corresponding Binary Number |
|---|---|
| 0 | 000 |
| 1 | 001 |
| 2 | 010 |
| 3 | 011 |
| 4 | 100 |
| 5 | 101 |
| 6 | 110 |
| 7 | 111 |
-
Hexadecimal to Binary Number System
This conversion can also be done using this conversion table.
| Hexadecimal Number | Binary |
|---|---|
| 0 | 0000 |
| 1 | 0001 |
| 2 | 0010 |
| 3 | 0011 |
| 4 | 0100 |
| 5 | 0101 |
| 6 | 0110 |
| 7 | 0111 |
| 8 | 1000 |
| 9 | 1001 |
| A | 1010 |
| B | 1011 |
| C | 1100 |
| D | 1101 |
| E | 1110 |
| F | 1111 |
There are many more conversions from one number system to another.
Number System Conversion Table
[Click Here for Sample Questions]
The number system conversion table is given below.
| Binary Number System | Octal Number System | Decimal Number System | Hexadecimal Number System |
|---|---|---|---|
| 0000 | 0 | 0 | 0 |
| 0001 | 1 | 1 | 1 |
| 0010 | 2 | 2 | 2 |
| 0011 | 3 | 3 | 3 |
| 0100 | 4 | 4 | 4 |
| 0101 | 5 | 5 | 5 |
| 0110 | 6 | 6 | 6 |
| 0111 | 7 | 7 | 7 |
| 1000 | 10 | 8 | 8 |
| 1001 | 11 | 9 | 9 |
| 1010 | 12 | 10 | A |
| 1011 | 13 | 11 | B |
| 1100 | 14 | 12 | C |
| 1101 | 15 | 13 | D |
| 1110 | 16 | 14 | E |
| 1111 | 17 | 15 | F |
Things to Remember
- Number systems are a way of expressing numbers in different ways using separate bases.
- The binary number system is useful when it comes to computer applications.
- The hexadecimal number system has a total of 16 characters where numbers from 0 to 9, like the decimal number system and alphabets A to F are present.
- The important factors to be noted while performing number system conversions are-
- The digit.
- The base of the number system.
- The position of the digits in the given number.
- The different types of number systems are-
- Decimal Number System with base 10.
- Binary number System with base 2.
- Hexadecimal Number System with base 16.
- Octal Number System with base 8.
Sample Questions
Ques. Convert (89)16 into a binary number. [2 marks]
Ans. From the given table, the binary value of 8 and 9 can be found out,
Therefore,
(8)16 = (1000)2 (9)16 = (1001)2
Hence, the binary equivalent of the given hexadecimal number is
(89)16 = (10001001)2
Ques. Convert (100111)2 into the decimal system. [2 marks]
Ans. To convert the binary number to decimal we perform these steps.
Step 1: Specify the positions of the digits.
( 1 0 0 1 1 1 )2
5 4 3 2 1 0 ← position of the digits
Step 2: Multiply the digit with a power of 2 where the power is its respective position and add them to give the resultant decimal number.
1 x 25 + 0 x 24 + 0 x 23 + 1 x 22 + 1 x 21 + 1 x 20
= 1 x 32 + 0 + 0 + 1 x 4 + 1 x 2 + 1 x 1
= 32 + 4 + 2 + 1
= 39
Therefore, the decimal equivalent of the binary system 100111 is 39.
Ques. Convert 432010 into the equivalent octal number system. [3 marks]
Ans. The base of the octal number system is 8.
| Quotient | Remainder | |
| 8 | 432010 | |
| 8 | 54001 | 2 |
| 8 | 67 | 1 |
| 8 | 843 | 6 |
| 8 | 105 | 3 |
| 8 | 13 | 1 |
| 8 | 1 | 5 |
Therefore, (432010)10 = (1513612)8.
Ques. Convert (1010111100)2 to the equivalent decimal system. [3 marks]
Ans. (1010111100)2 is a binary number system. Now,
Step 1: Specify the positions of the digits.
( 1 0 1 0 1 1 1 1 0 0 )2
9 8 7 6 5 4 3 2 1 0 ← position of the digits
Step 2: Multiply the digit with a power of 2 where the power is its respective position and add them to give the resultant decimal number.
= 1 x 29 + 0 x 28 + 1 x 27 + 0 x 26 + 1 x 25 + 1 x 24 + 1 x 23 + 1 x 22 + 0 x 21 + 0 x 20
= 1 x 512 + 0 x 256 + 1 x 128 + 0 x 64 + 1 x 32 + 1 x 16 + 1 x 8 + 1 x 4 + 0 x 2 + 0 x 1
= 512 + 0 + 128 + 0 + 32 + 16 + 8 + 4 + 0 + 0
= 700
Thus, the equivalent decimal number system of (1010111100)2 is 70010.
Ques. Convert (300)10 into the binary system with base 2. [2 marks]
Ans. 300 is a decimal number. We divide 300 by base 2 and then write the quotient and the remainder beside it. This process is repeated until and unless we get a quotient that is less than 2.
| Quotient | Remainder | |
| 2 | 300 | |
| 2 | 150 | 0 |
| 2 | 75 | 0 |
| 2 | 37 | 1 |
| 2 | 18 | 1 |
| 2 | 9 | 0 |
| 2 | 4 | 1 |
| 2 | 2 | 0 |
| 1 | 0 |
→ ↑
Thus, the binary equivalent of 30010 is 1001011002.
Ques. Convert 5BC16 into its equivalent decimal number system. [2 marks]
Ans. 5BC16 is a hexadecimal number system. We know that B is also 11 and C is 12 in the hexadecimal system. Now,
Step 1: Specify the positions of the digits.
( 5 B C )16
2 1 0 ← position of the digits
Step 2: Multiply the digit with a power of 16 where the power is its respective position and add them to give the resultant decimal number.
5 x 162 + 11 x 161 + 12 x 160
= 1280 + 176 + 12
= 1468
Thus, the equivalent decimal number system of 5BC16 is 146810.
Ques. Convert 1448 into the decimal system. [3 marks]
Ans. To convert the given octal number 144 to a decimal number system we perform these steps.
Step 1: Specify the positions of the digits.
( 1 4 4 )8
2 1 0 ← position of the digits
Step 2: Multiply the digit with a power of 8 where the power is its respective position and add them to give the resultant decimal number.
1 x 82 + 4 x 81 + 4 x 80
= 1 x 64 + 4 x 8 + 4 x 1
= 64 + 32 + 4
= 100
Therefore, the decimal equivalent of the given octal number system 144 is 100.
Ques. Convert 22310 into its equivalent binary system. [2 marks]
Ans. To convert this decimal number to binary we divide it by 2 and then write down the quotient and the remainder is noted too. This division process is carried out until and unless the quotient becomes less than two. After this, the remainder column is written from bottom to top which is the binary number of the given decimal.
| Quotient | Remainder | |
| 2 | 223 | |
| 2 | 111 | 1 |
| 2 | 55 | 1 |
| 2 | 27 | 1 |
| 2 | 13 | 1 |
| 2 | 6 | 1 |
| 2 | 3 | 0 |
| 2 | 1 | 1 |
→ ↑
Thus, the binary equivalent of 22310 is 110111112.
Ques. Convert (13056)8 to its equivalent decimal system. [3 marks]
Ans. (13056)8 is an octal number system. Now,
Step 1: Specify the positions of the digits.
( 1 3 0 5 6 )8
4 3 2 1 0 ← position of the digits
Step 2: Multiply the digit with a power of 8 where the power is its respective position and add them to give the resultant decimal number.
=1 x 84 + 3 x 83 + 0 x 82 + 5 x 81 + 6 x 80
= 1 x 4096 + 3 x 512 + 0 x 64 + 5 x 8 + 6 x 1
= 4096 + 1536 + 0 + 40 + 6
= 5678
Thus, the equivalent decimal number system of (13056)8 is 567810.
Ques. Convert (1100100101)2 to the equivalent decimal system. [3 marks]
Ans. (1100100101)2 is a binary number system. Now,
Step 1: Specify the positions of the digits.
( 1 1 0 0 1 0 0 1 0 1 )2
9 8 7 6 5 4 3 2 1 0 ← position of the digits
Step 2: Multiply the digit with a power of 2 where the power is its respective position and add them to give the resultant decimal number.
= 1 x 29 + 1 x 28 + 0 x 27 + 0 x 26 + 1 x 25 + 0 x 24 + 0 x 23 + 1 x 22 + 0 x 21 + 1 x 20
= 1 x 512 + 1 x 256 + 0 x 128 + 0 x 64 + 1 x 32 + 0 x 16 + 0 x 8 + 1 x 4 + 0 x 2 + 1 x 1
= 512 + 256 + 0 + 0 + 32 + 0 + 0 + 4 + 0 + 1
= 805.
Thus, the equivalent decimal number system of (1100100101)2 is 80510.
Ques. Convert 547610 into its equivalent octal system. [2 marks]
Ans. 5476 is a decimal number. We divide the number by base 8 and then write the quotient and the remainder beside it. This process is repeated until and unless we get a quotient that is less than 8.
| Quotient | Remainder | |
| 8 | 5476 | |
| 8 | 684 | 4 |
| 8 | 85 | 4 |
| 8 | 10 | 5 |
| 8 | 1 | 2 |
→ ↑
Thus, the octal equivalent of 547610 is 125448.
Read Also:
Comments