Binary Number System

When we type some letters or words, the computer translates them in numbers as computers can understand only numbers. A computer can understand the positional number system where there are only a few symbols called digits and these symbols represent different values depending on the position they occupy in the number.

The value of each digit in a number can be determined using −

• The digit
• The position of the digit in the number
• The base of the number system (where the base is defined as the total number of digits available in the number system)

Decimal Number System

The number system that we use in our day-to-day life is the decimal number system. Decimal number system has base 10 as it uses 10 digits from 0 to 9. In decimal number system, the successive positions to the left of the decimal point represent units, tens, hundreds, thousands, and so on.

Each position represents a specific power of the base (10). For example, the decimal number 1234 consists of the digit 4 in the units position, 3 in the tens position, 2 in the hundreds position, and 1 in the thousands position. Its value can be written as

(1 x 1000)+ (2 x 100)+ (3 x 10)+ (4 x l)

(1 x 103)+ (2 x 102)+ (3 x 101)+ (4 x l00)

1000 + 200 + 30 + 4

1234

As a computer programmer or an IT professional, you should understand the following number systems which are frequently used in computers.

 S.No. Number System and Description 1 Binary Number System Base 2. Digits used : 0, 1 2 Octal Number System Base 8. Digits used : 0 to 7 3 Hexa Decimal Number System Base 16. Digits used: 0 to 9, Letters used : A- F

Binary Number System

Characteristics of the binary number system are as follows −

• Uses two digits, 0 and 1
• Also called as base 2 number system
• Each position in a binary number represents a 0power of the base (2). Example 20
• Last position in a binary number represents a xpower of the base (2). Example 2x where x represents the last position – 1.

Example

Binary Number: 101012

Calculating Decimal Equivalent −

 Step Binary Number Decimal Number Step 1 101012 ((1 x 24) + (0 x 23) + (1 x 22) + (0 x 21) + (1 x 20))10 Step 2 101012 (16 + 0 + 4 + 0 + 1)10 Step 3 101012 2110

Note − 101012 is normally written as 10101.

Octal Number System

Characteristics of the octal number system are as follows −

• Uses eight digits, 0,1,2,3,4,5,6,7
• Also called as base 8 number system
• Each position in an octal number represents a 0power of the base (8). Example 80
• Last position in an octal number represents a xpower of the base (8). Example 8x where x represents the last position – 1

Example

Octal Number: 125708

Calculating Decimal Equivalent −

 Step Octal Number Decimal Number Step 1 125708 ((1 x 84) + (2 x 83) + (5 x 82) + (7 x 81) + (0 x 80))10 Step 2 125708 (4096 + 1024 + 320 + 56 + 0)10 Step 3 125708 549610

Note − 125708 is normally written as 12570.

Characteristics of hexadecimal number system are as follows −

• Uses 10 digits and 6 letters, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
• Letters represent the numbers starting from 10. A = 10. B = 11, C = 12, D = 13, E = 14, F = 15
• Also called as base 16 number system
• Each position in a hexadecimal number represents a 0power of the base (16). Example, 160
• Last position in a hexadecimal number represents a xpower of the base (16). Example 16x where x represents the last position – 1

Example

Calculating Decimal Equivalent −

 Step Binary Number Decimal Number Step 1 19FDE16 ((1 x 164) + (9 x 163) + (F x 162) + (D x 161) + (E x 160))10 Step 2 19FDE16 ((1 x 164) + (9 x 163) + (15 x 162) + (13 x 161) + (14 x 160))10 Step 3 19FDE16 (65536+ 36864 + 3840 + 208 + 14)10 Step 4 19FDE16 10646210

Note − 19FDE16 is normally written as 19FDE.

There are many methods or techniques which can be used to convert numbers from one base to another. In this chapter, we’ll demonstrate 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

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 the 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: 2910

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: 2910 = Binary Number: 111012.

Other Base System to Decimal System

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: 111012

Calculating Decimal Equivalent −

 Step Binary Number Decimal Number Step 1 111012 ((1 x 24) + (1 x 23) + (1 x 22) + (0 x 21) + (1 x 20))10 Step 2 111012 (16 + 8 + 4 + 0 + 1)10 Step 3 111012 2910

Binary Number: 111012 = Decimal Number: 2910

Other Base System to Non-Decimal System

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: 258

Calculating Binary Equivalent −

Step 1 – Convert to Decimal

 Step Octal Number Decimal Number Step 1 258 ((2 x 81) + (5 x 80))10 Step 2 258 (16 + 5)10 Step 3 258 2110

Octal Number: 258 = Decimal Number : 2110

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: 2110 = Binary Number: 101012

Octal Number: 258 = Binary Number: 101012

Shortcut Method ─ Binary to Octal

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 : 101012

Calculating Octal Equivalent −

 Step Binary Number Octal Number Step 1 101012 010 101 Step 2 101012 28 58 Step 3 101012 258

Binary Number: 101012 = Octal Number: 258

Shortcut Method ─ Octal to Binary

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: 258

Calculating Binary Equivalent −

 Step Octal Number Binary Number Step 1 258 210 510 Step 2 258 0102 1012 Step 3 258 0101012

Octal Number: 258 = Binary Number: 101012

Shortcut Method ─ Binary to Hexadecimal

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: 101012

 Step Binary Number Hexadecimal Number Step 1 101012 0001 0101 Step 2 101012 110 510 Step 3 101012 1516

Binary Number: 101012 = Hexadecimal Number: 1516

Shortcut Method – Hexadecimal to Binary

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

Calculating Binary Equivalent −

 Step Hexadecimal Number Binary Number Step 1 1516 110 510 Step 2 1516 00012 01012 Step 3 1516 000101012

Hexadecimal Number: 1516 = Binary Number: 101012

Reference:

https://www.electronics-tutorials.ws/binary/bin_5.html

https://www.tutorialspoint.com/computer_fundamentals/computer_number_conversion.htm

https://www.tutorialspoint.com/questions/index.php