Computer - Number Conversion

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Computer - Number Conversion
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
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
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