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The TCP/IP Guide  9  Networking Fundamentals       9  Backgrounder: Data Representation and the Mathematics of Computing

Decimal, Binary, Octal and Hexadecimal Numbers 1 2 3 4 Binary, Octal and Hexadecimal Arithmetic

Conversions From Decimal to Binary, Octal and Hexadecimal

Now let’s consider conversions from decimal. These require that you perform the opposite of the calculation above: you divide and subtract instead of multiplying and adding.

The easiest of the three conversions from decimal is to binary—since the maximum value of each digit is one, there is no dividing, just subtraction. All you do is the following:

1. Find the largest power of two that is smaller than the number.
   
   
2. Put a “1” in the digit place for that power of two and subtract that power of two from the decimal number.
   
   
3. Repeat steps #1 and #2 until you are reduced to zero.

This is easier to explain using an example and a table, of course. Let's convert the decimal number 689, as shown in Table 5. Again, read the table starting from the upper left, and going down and then across. We start by noticing that 1024 is not less than or equal to 689, so the “1024s” place gets a 0. In the next place, 512 is less than 689, so we make the “512s” place a 1 and subtract 512 from 689 to leave 177. The calculation continues, eventually showing shows that 689 decimal is 1010110001 binary.

The process for octal and hexadecimal is almost the same, except you must divide by powers of two instead of just subtracting:

1. Start with the highest power of 16 (hexadecimal) or 8 (octal) that is smaller than the number.
   
   
2. Divide the decimal number by that power, keeping only the integer part of the result.
   
   
3. Keep the remainder after the division is done, for the next step.
   
   
4. Repeat steps #1 to #3 until you get to the “ones” place, and then put there whatever is left after the higher digits were done.

Table 6 shows the same example as Table 5 but goes from decimal to hexadecimal instead of decimal to binary: 689 in decimal is 0x2B1 hexadecimal.

Table 6: Decimal to Hexadecimal Number Conversion

Decimal Value Before Considering This Digit Place	689	689	177	1
Power of 16	163	162	161	160
Value of Digit Place	4096	256	16	1
Value of Digit Place Smaller Than Current Decimal Number?	No	Yes	No	n/a
Division Step	skip	689/256 = 2.691; use “2” for this digit.	177/16 = 11.0625; use “B” for this digit.	n/a
Remainder After Division	skip	177	1	n/a
Hexadecimal Digits	0	2	B	1

Decimal, Binary, Octal and Hexadecimal Numbers 1 2 3 4 Binary, Octal and Hexadecimal Arithmetic

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