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<FONT SIZE=5><B> Numeral radixes</B></FONT><BR>
<FONT SIZE=4>Decimal, Hexadecimal and Octal </FONT>
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<P>
<H2>Numeral radixes</H2>
All of us we are accustomed since youngs to use decimal numbers to express quantities.
This nomenclature that seems so logical to us may not seems it to a Roman of the classic
Rome. For them each symbol that they put to express a number always represented the same
value:
<blockquote ><TT><PRE>
I <B> 1</B>
II <B> 2</B>
III <B> 3</B>
IV <B> 4</B>
V <B> 5</B>
</PRE></TT></BLOCKQUOTE>
If one pays attention all the <TT><B>I</B></TT> signs always represents the value
<TT><B>1</B></TT> (one) wherever they be placed, like the <TT><B>V</B></TT> sign always
represents our <TT><B>5</B></TT> (five). Nevertheless that does not take place in our
decimal system. When we write the decimal symbol <TT><B>1</B></TT> we are not always
talking about value 1 (<TT><B>I</B></TT> in Roman numbers). For example:
<blockquote ><TT><PRE>
<B> 1</B> I
<B> 10</B> X
<B>100</B> C
</PRE></TT></BLOCKQUOTE>
In these cases our symbol <TT><B>1</B></TT> does not always has a value of 1
(<TT><B>I</B></TT> in Roman numbers). For example, in the second case, the symbol
<TT><B>1</B></TT> represents the value 10 (ten, <TT><B>X</B></TT> in Roman) and in
the third one, <TT><B>1</B></TT> represents the value 100 (one hundred, <TT><B>C</B></TT>).
<P>
Another example:
<blockquote ><TT>
<B>275</B>
</TT></BLOCKQUOTE>
is not equivalent to <TT>2+7+5</TT>, rather could be decomposed as <TT>200+70+5</TT>:
<BLOCKQUOTE><TT><PRE>
200
+ 70
5
---
275
</PRE></TT></BLOCKQUOTE>
therefore, the first sign <TT><B>2</B></TT> is equivalent to 200 (2 x 100), the second
sign, <TT><B>7</B></TT> is equivalent to 70 (7 x 10) whereas the last sign corresponds
to value 5 (5 x 1).
<P>
All the previous can be mathematically represented in a very simple way. For example,
to represent the value 182736 we can assume that each digit is the product of itself
multiplied by 10 powered to its place as exponent, beginning from the right with
10<SUP><SMALL>0</SMALL></SUP>, following with 10<SUP><SMALL>1</SMALL></SUP>,
10<SUP><SMALL>2</SMALL></SUP>, and so on:
<BLOCKQUOTE><IMG SRC="hex1.gif"></BLOCKQUOTE>
<P>
<H2>Octal numbers (radix 8)</H2>
Like our "normal" numbers are radix 10 because we have 10 different digits
(from the 0 to the 9):
<BLOCKQUOTE><TT>0123456789</TT></BLOCKQUOTE>
the octals numbers include only the signs from the 0 to the 7:
<BLOCKQUOTE><TT>01234567</TT></BLOCKQUOTE>
and, therefore, its mathematical radix is 8. In C++ octal numbers have the peculiarity
that they always they begin by a <TT><B>0</B></TT> digit. Let us see how we would write
the first numbers in octal:
<BLOCKQUOTE><TT><PRE>
octal decimal
----- -------
<B> 0</B> 0 <I>(zero)</I>
<B> 01</B> 1 <I>(one)</I>
<B> 02</B> 2 <I>(two)</I>
<B> 03</B> 3 <I>(three)</I>
<B> 04</B> 4 <I>(four)</I>
<B> 05</B> 5 <I>(five)</I>
<B> 06</B> 6 <I>(six)</I>
<B> 07</B> 7 <I>(seven)</I>
<B>010</B> 8 <I>(eight)</I>
<B>011</B> 9 <I>(nine)</I>
<B>012</B> 10 <I>(ten)</I>
<B>013</B> 11 <I>(eleven)</I>
<B>014</B> 12 <I>(twelve)</I>
<B>015</B> 13 <I>(thirteen)</I>
<B>016</B> 14 <I>(fourteen)</I>
<B>017</B> 15 <I>(fifteen)</I>
<B>020</B> 16 <I>(sixteen)</I>
<B>021</B> 17 <I>(seventeen)</I>
</PRE></TT></BLOCKQUOTE>
Thus, for example, number 17 (seventeen, or <TT>XVII</TT> in Romans) it is expressed
<TT><B>021</B></TT> as an octal number.
<P>
We can apply the scheme that we saw previously with the decimal numbers to the octal
numbers simply by considering that its radix is 8. For example, taking
the octal number 071263:
<BLOCKQUOTE><IMG SRC="hex2.gif"></BLOCKQUOTE>
therefore the octal numbers <TT><B>071263</B></TT> it is expressed as
<TT><B>29363</B></TT> in decimal numbers.
<P>
<H2>Hexadecimal numbers (radix 16)</H2>
Like decimal numbers have 10 different digits to be represented (<TT>0123456789</TT>)
and octal numbers have 8 (<TT>01234567</TT>), hexadecimal numbers have 16: numbers
from <TT>0</TT> to <TT>9</TT> and letters
<TT>A, B, C, D, E</TT> and <TT>F</TT> that together they serve us to represent
the 16 different symbols that we need:
<BLOCKQUOTE><TT><PRE>
hexadecimal decimal
----------- -------
<B> 0</B> 0 <I>(zero)</I>
<B> 0x1</B> 1 <I>(one)</I>
<B> 0x2</B> 2 <I>(two)</I>
<B> 0x3</B> 3 <I>(three)</I>
<B> 0x4</B> 4 <I>(four)</I>
<B> 0x5</B> 5 <I>(five)</I>
<B> 0x6</B> 6 <I>(six)</I>
<B> 0x7</B> 7 <I>(seven)</I>
<B> 0x8</B> 8 <I>(eight)</I>
<B> 0x9</B> 9 <I>(nine)</I>
<B> 0xA</B> 10 <I>(ten)</I>
<B> 0xB</B> 11 <I>(eleven)</I>
<B> 0xC</B> 12 <I>(twelve)</I>
<B> 0xD</B> 13 <I>(thirteen)</I>
<B> 0xE</B> 14 <I>(fourteen)</I>
<B> 0xF</B> 15 <I>(fifteen)</I>
<B>0x10</B> 16 <I>(sixteen)</I>
<B>0x11</B> 17 <I>(seventeen)</I>
</PRE></TT></BLOCKQUOTE>
Once again we can use the same method to translate a number from a base to another one:
<BLOCKQUOTE><IMG SRC="hex3.gif"></BLOCKQUOTE>
<P>
<H2>Binary representations</H2>
Octal and hexadecimal numbers have a considerable advantage over our decimal numbers
in the world of bits, and is that their bases (<TT><B>8</B></TT> and <TT><B>16</B></TT>)
are perfect multiples of 2 (<TT><B>2<SUP><SMALL>3</SMALL></SUP></B></TT> and
<TT><B>2<SUP><SMALL>4</SMALL></SUP></B></TT>) which allows us to make easier conversions
to binary than with decimal numbers (whose base is <TT><B>2x5</B></TT>). For example,
suppose that we want to pass the following binary sequence to numbers of other bases:
<BLOCKQUOTE><TT>110011111010010100</TT></BLOCKQUOTE>
In order to pass it to decimal we would need to conduct a mathematical operation similar
to the one we have used previously to convert from hexadecimal or octal, which would give
us the decimal number <TT><B>212628</B></TT>.
<P>
Nevertheless to pass this sequence to octal it will only take to us some seconds and we
can do it just seeing it: Since 8 is <TT><B>2<SUP><SMALL>3</SMALL></SUP></B></TT>,
we will separate the binary value in groups of 3 numbers:
<BLOCKQUOTE><TT>110 011 111 010 010 100</TT></BLOCKQUOTE>
and now we just have to translate to octal numberal radix each group separately:
<BLOCKQUOTE><TT><PRE>
110 011 111 010 010 100
6 3 7 2 2 4
</PRE></TT></BLOCKQUOTE>
giving the number <TT>637224</TT> as result. This same process can be inversely performed
to pass from octal to binary.
<P>
In order to
conduct the operation with hexadecimal numbers we only have to perform the same process
but separating the binary value in groups of 4 numbers
(<TT><B>16 = 2<SUP><SMALL>4</SMALL></SUP></B></TT>):
<BLOCKQUOTE><TT><PRE>
11 0011 1110 1001 0100
3 3 E 9 4
</PRE></TT></BLOCKQUOTE>
Therefore, the binary expression <TT>110011111010010100</TT> can be represented in C++
as <TT><B>212628</B></TT> (decimal), as <TT><B>0637224</B></TT> (octal) or as
<TT><B>0x33e94</B></TT> (hexadecimal).<P>
The hexadecimal code is specially interesting in computer science since nowadays
computers are based on bytes composed of 8 binary bits and therefore each byte
matches with the rank that 2 hexadecimal numbers can represent. For that reason is the
most used type when representing values translated from binary.
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