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: 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
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: 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
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: 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
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: 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
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: 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
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: 101012
Calculating
hexadecimal Equivalent:
|
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
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: 1516
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