Module 2 Section
3 Conversion Techniques
The goal for this module is that you be able to understand the binary,
octal, decimal and hexadecimal number systems, and that you be able to
convert a number in any of these systems to any of the other systems. We
still have one little problem to deal with. Consider the following number:
35
What is this number? What system is it written in? Obviously it cannot
be binary, because it has digits which are not allowed in binary numbers.
But it could be octal, decimal, or hexadecimal. How can you tell?
From this point forward we will adopt a convention of notation.
A convention is simply a way people have agreed to do something so that
everyone understands what everyone else has done. Our convention for identifying
the base of a number will be as follows:

The base can be given as a subscript. This will work for any base,
even the ones we have not studied. For example, the number 35_{6}
is in base 6.

A number which starts with a leading '0' (zero) is an octal number.
The C programming language does this, and it is easy to get used to. If
we write a leading zero then that means the rest of the number is octal.
So 035 is octal 35.

A number which starts with a $, a &, or '0x' is a hexadecimal number.
The C programming lanuage uses '0x'. Turbo pascal uses '$'. Some assemblers
use '&'. So $35, &35, and 0x35 are all hexadecimal 35.

A number which ends with 'H' is a hexadecimal number. Some assemblers
use this notation. 35H is hexadecimal 35.

A number which ends in 'B' is a binary number. An example is 0101011011B.
Some computer languages accept this notation.

A number which is written plainly, having no subscript, prefix, or suffix,
is assumed to be decimal.
We have already learned how to convert between binary and the other three
systems (octal, decimal, and hexadecimal). But what if you want to
convert between, say, octal and decimal? or decimal and hexadecimal?. There
are techniques for converting directly from one base to another. One way
is to use binary form as an intermediate form when converting between the
other bases.
For example, suppose you want to convert 24 to octal. First convert
24 to binary, then convert that to octal.
24 > 11000B > 030
What about converting hexadecimal number 0x3B to decimal? Again, first
convert to binary, then to decimal.
0x3B > 00111011B > 2^{5}+2^{4}+2^{3}+2^{1}+2^{0}
= 32+16+8+2+1 = 59
However, some conversion can be done even more easily than this. Here
are suggested ways to perform conversion between the bases:

Any base to decimal: Use expanded notation. Write down each
digit as a product of a power of the base, then sum them together.

Decimal to any base: Use the division method. Divide the base repeatedly
into the decimal number, writing down the remainder at each step. When
the number becomes zero, the string of remainders is the number as it would
be written in the new base.

Octal to hexadecimal or viceversa: Use binary as an intermediate
form.
Homework Questions
120. Complete the following table by filling in the empty cells. Each row is
the same number written in different bases. Each column is the same base.
Binary 
Octal 
Decimal 
Hexadecimal 
1 



10011011 




37 

1F 


58 



471 




2C 



8 
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Computer Integer Arithmetic
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