Questions on Decimal and Binary number system are frequently asked in Bank PO and Clerk exams. Here is a quick study on decimal and binary number systems.

## Binary to Decimal Computer notes for Bank PO Exam

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**Introduction of Computer Number System:**

*A*

*set of values used to represent different quantities is known as*

**Number System**

*.*For example, a number system can be used to represent the number of students in a class or number of viewers watching a certain TV program etc. The digital computer represents all kinds of data and information in binary numbers. It includes audio, graphics, video, text and numbers. The total number of digits used in a number system is called its base or radix. The base is written after the number as subscript.

Some important number systems are as follows.

- Decimal number system
- Binary number system
- Octal number system
- Hexadecimal number system

**Decimal number System**

The Decimal Number System consists of ten digits from 0 to 9. These digits can be used to represent any numeric value. The base of decimal number system is 10. It is the most widely used number system. The value represented by individual digit depends on weight and position of the digit.

**Binary Number System**

Digital computer represents all kinds of data and information in the binary system. Binary Number System consists of two digits 0 and 1. Its base is 2. Each digit or bit in binary number system can be 0 or 1. A combination of binary numbers may be used to represent different quantities like 1001. The positional value of each digit in binary number is twice the place value or face value of the digit of its right side. The weight of each position is a power of 2.

**Octal Number System**

Octal Number System consists of eight digits from 0 to 7. The base of octal system is 8. Each digit position in this system represents a power of 8. Any digit in this system is always less than 8. Octal number system is used as a shorthand representation of long binary numbers. The number 6418 is not valid in this number system as 8 is not a valid digit.

**Hexadecimal Number System**

The Hexadecimal Number System consists of 16 digits from 0 to 9 and A to F. The alphabets A to F represent decimal numbers from 10 to 15. The base of this number system is 16. Each digit position in hexadecimal system represents a power of 16. The number 76416 is valid hexadecimal number. It is different from 76410 which is seven hundred and sixty four. This number system provides shortcut method to represent long binary numbers.

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**Decimal System**

The decimal system consists of 10 numerals or symbols. These 10 symbols are 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9.

Right now you must be thinking, I learnt it by the first grade. But then there’s more to learn in simpler topics.

The decimal system is also called base-10 system.

It is a positional-value system. It means the value of a digit depends on its position.

Example: Consider a decimal number 736. The digit 7 actually represents 7 hundreds, 3 represents 3 tens and 6 represents 6 units. Then 7 carries the most weight of the three digits, it is referred to as the most significant digit (MSD). Then 2 carries the least weight, it is referred to as the least significant digit (LSD).

Note the concepts: most significant digit (MSD) and least significant digit (LSD)

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** Binary System**

Binary number system gained importance due to its application in the digital world. Computers run on digital binary data. In binary system there are only two symbols 0 and 1. Still, with only 0 and 1 any number, how so ever large can be represented.

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**Binary is also a positional number system.**

In the binary system the term binary digit is often abbreviated to the term, bit. In other words, bit is a short form for binary digit. The left most bit has the largest weight is the most significant bit (MSB). The right most bit has the smallest weight is the least significant bit (LSB).

**Example: 1010**

The leftmost 1 is the most significant bit (MSB)

The rightmost 0 is the least significant bit (LSB).

There are many methods or techniques which can be used to convert numbers from one base to another. We'll demonstrate here the following:

- Decimal to Other Base System
- Other Base System to Decimal
- Other Base System to Non-Decimal
- Shortcut method - Binary to Octal
- Shortcut method - Octal to Binary
- Shortcut method - Binary to Hexadecimal
- Shortcut method - Hexadecimal to Binary

### Decimal to Other Base System

steps

**Step 1 -**Divide the decimal number to be converted by the value of the new base.**Step 2 -**Get the remainder from Step 1 as the rightmost digit (least significant digit) of new base number.**Step 3 -**Divide the quotient of the previous divide by the new base.**Step 4 -**Record the remainder from Step 3 as the next digit (to the left) of the new base number.

Repeat Steps 3 and 4, getting remainders from right to left, until the quotient becomes zero in Step 3.

The last remainder thus obtained will be the most significant digit (MSD) of the new base number.

### Example

Decimal Number : 29

_{10}
Calculating Binary Equivalent:

Step | Operation | Result | Remainder |
---|---|---|---|

Step 1 | 29 / 2 | 14 | 1 |

Step 2 | 14 / 2 | 7 | 0 |

Step 3 | 7 / 2 | 3 | 1 |

Step 4 | 3 / 2 | 1 | 1 |

Step 5 | 1 / 2 | 0 | 1 |

As mentioned in Steps 2 and 4, the remainders have to be arranged in the reverse order so that the first remainder becomes the least significant digit (LSD) and the last remainder becomes the most significant digit (MSD).

Decimal Number : 29

_{10}= Binary Number : 11101_{2.}### Other base system to Decimal System

Steps

**Step 1 -**Determine the column (positional) value of each digit (this depends on the position of the digit and the base of the number system).**Step 2 -**Multiply the obtained column values (in Step 1) by the digits in the corresponding columns.**Step 3 -**Sum the products calculated in Step 2. The total is the equivalent value in decimal.

### Example

Binary Number : 11101

_{2}
Calculating Decimal Equivalent:

Step | Binary Number | Decimal Number |
---|---|---|

Step 1 | 11101_{2} | ((1 x 2^{4}) + (1 x 2^{3}) + (1 x 2^{2}) + (0 x 2^{1}) + (1 x 2^{0}))_{10} |

Step 2 | 11101_{2} | (16 + 8 + 4 + 0 + 1)_{10} |

Step 3 | 11101_{2} | 29_{10} |

Binary Number : 11101

_{2}= Decimal Number : 29_{10}### Other Base System to Non-Decimal System

Steps

**Step 1 -**Convert the original number to a decimal number (base 10).**Step 2 -**Convert the decimal number so obtained to the new base number.

### Example

Octal Number : 25

_{8}
Calculating Binary Equivalent:

### Step 1 : Convert to Decimal

Step | Octal Number | Decimal Number |
---|---|---|

Step 1 | 25_{8} | ((2 x 8^{1}) + (5 x 8^{0}))_{10} |

Step 2 | 25_{8} | (16 + 5 )_{10} |

Step 3 | 25_{8} | 21_{10} |

Octal Number : 25

_{8}= Decimal Number : 21_{10}### Step 2 : Convert Decimal to Binary

Step | Operation | Result | Remainder |
---|---|---|---|

Step 1 | 21 / 2 | 10 | 1 |

Step 2 | 10 / 2 | 5 | 0 |

Step 3 | 5 / 2 | 2 | 1 |

Step 4 | 2 / 2 | 1 | 0 |

Step 5 | 1 / 2 | 0 | 1 |

Decimal Number : 21

_{10}= Binary Number : 10101_{2}
Octal Number : 25

_{8}= Binary Number : 10101_{2}### Shortcut method - Binary to Octal

Steps

**Step 1 -**Divide the binary digits into groups of three (starting from the right).**Step 2 -**Convert each group of three binary digits to one octal digit.

### Example

Binary Number : 10101

_{2}
Calculating Octal Equivalent:

Step | Binary Number | Octal Number |
---|---|---|

Step 1 | 10101_{2} | 010 101 |

Step 2 | 10101_{2} | 2_{8} 5_{8} |

Step 3 | 10101_{2} | 25_{8} |

Binary Number : 10101

_{2}= Octal Number : 25_{8}### Shortcut method - Octal to Binary

Steps

**Step 1 -**Convert each octal digit to a 3 digit binary number (the octal digits may be treated as decimal for this conversion).**Step 2 -**Combine all the resulting binary groups (of 3 digits each) into a single binary number.

### Example

Octal Number : 25

_{8}
Calculating Binary Equivalent:

Step | Octal Number | Binary Number |
---|---|---|

Step 1 | 25_{8} | 2_{10} 5_{10} |

Step 2 | 25_{8} | 010_{2} 101_{2} |

Step 3 | 25_{8} | 010101_{2} |

Octal Number : 25

_{8}= Binary Number : 10101_{2}### Shortcut method - Binary to Hexadecimal

Steps

**Step 1 -**Divide the binary digits into groups of four (starting from the right).**Step 2 -**Convert each group of four binary digits to one hexadecimal symbol.

### Example

Binary Number : 10101

_{2}
Calculating hexadecimal Equivalent:

Step | Binary Number | Hexadecimal Number |
---|---|---|

Step 1 | 10101_{2} | 0001 0101 |

Step 2 | 10101_{2} | 1_{10} 5_{10} |

Step 3 | 10101_{2} | 15_{16} |

Binary Number : 10101

_{2}= Hexadecimal Number : 15_{16}### Shortcut method - Hexadecimal to Binary

steps

**Step 1 -**Convert each hexadecimal digit to a 4 digit binary number (the hexadecimal digits may be treated as decimal for this conversion).**Step 2 -**Combine all the resulting binary groups (of 4 digits each) into a single binary number.

### Example

Hexadecimal Number : 15

_{16}
Calculating Binary Equivalent:

Step | Hexadecimal Number | Binary Number |
---|---|---|

Step 1 | 15_{16} | 1_{10} 5_{10} |

Step 2 | 15_{16} | 0001_{2} 0101_{2} |

Step 3 | 15_{16} | 00010101_{2} |

Hexadecimal Number : 15

_{16}= Binary Number : 10101_{2}
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