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Updates to binary
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More discussion of SI interactions
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@ianw
ianw committed Jan 1, 2017
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<title>16, 32 and 64 bit computers</title>
</info>
<para>Numbers do not fit into bytes; hopefully your bank
balance in dollars will need more range than can fit into
one byte! Most modern architectures are <emphasis>32
bit</emphasis> computers. This means they work with 4 bytes
at a time when processing and reading or writing to memory.
We refer to 4 bytes as a <emphasis>word</emphasis>; this is
analogous to language where letters (bits) make up words in a
sentence, except in computing every word has the same size!
The size of a C <computeroutput>int</computeroutput>
variable is 32 bits. Newer architectures are 64 bits, which
doubles the size the processor works with (8 bytes).</para>
balance in dollars will need more range than can fit into
one byte! Modern architectures are at least <emphasis>32
bit</emphasis> computers. This means they work with 4 bytes
at a time when processing and reading or writing to memory.
We refer to 4 bytes as a <emphasis>word</emphasis>; this is
analogous to language where letters (bits) make up words in
a sentence, except in computing every word has the same
size! The size of a C <computeroutput>int</computeroutput>
variable is 32 bits. Modern architectures are 64 bits,
which doubles the size the processor works with to 8
bytes.</para>
</section>
<section>
<info>
<title>Kilo, Mega and Giga Bytes</title>
</info>
<para>Computers deal with a lot of bytes; that's what makes
them so powerful!</para>
<para>We need a way to talk about large numbers of bytes,
and a natural way is to use the "International System of
Units" (SI) prefixes as used in most other scientific areas.
So for example, kilo refers to
10<superscript>3</superscript> or 1000 units, as in a
kilogram has 1000 grams.</para>
them so powerful! We need a way to talk about large numbers
of bytes, and a natural way is to use the "International
System of Units" (SI) prefixes as used in most other
scientific areas. So for example, kilo refers to
10<superscript>3</superscript> or 1000 units, as in a
kilogram has 1000 grams.</para>
<para>1000 is a nice round number in base 10, but in binary
it is <computeroutput>1111101000</computeroutput> which is
not a particularly "round" number. However, 1024 (or
2<superscript>10</superscript>) is
(<computeroutput>10000000000</computeroutput>), and happens
to be quite close to the base ten meaning of kilo (1000 as
opposed to 1024).</para>
<para>Hence 1024 bytes became known as a
<emphasis>kilobyte</emphasis>. The first mass market
computer was the Commodore 64, so named because it had 64
kilobytes of storage.</para>
<para>Today, kilobytes of memory would be small for a wrist
watch, let alone a personal computer. The next SI unit is
"mega" for
<computeroutput>10<superscript>6</superscript></computeroutput>.
As it happens,
<computeroutput>2<superscript>20</superscript></computeroutput>
is again close to the SI base 10 definition; 1048576 as
opposed to 1000000.</para>
<para>The units keep increasing by powers of 10; each
time it diverges further from the base SI meaning.</para>
it is <computeroutput>1111101000</computeroutput> which is
not a particularly "round" number. However, 1024 (or
2<superscript>10</superscript>) is a round number &#x2014;
(<computeroutput>10000000000</computeroutput> &#x2014; and
happens to be quite close to the base 10 meaning value of
"kilo" (1000 as opposed to 1024). Thus 1024 bytes naturally
became known as a <emphasis>kilobyte</emphasis>. The next
SI unit is "mega" for
<computeroutput>10<superscript>6</superscript></computeroutput>
and the prefixes continue upwards by
10<superscript>3</superscript> (corresponding to the usual
grouping of three digits when writing large numbers). As it
happens,
<computeroutput>2<superscript>20</superscript></computeroutput>
is again close to the SI base 10 definition for mega;
1048576 as opposed to 1000000. Increasing the base 2 units
by powers of 10 remains functionally close to the SI base 10
value, although each increasing factor diverges slightly
further from the base SI meaning. Thus the SI base-10 units
are "close enough" and have become the commonly used for
base 2 values.</para>
<table>
<info>
<title>Bytes</title>
<title>Base 2 and 10 factors related to bytes</title>
</info>
<tgroup cols="2">
<tgroup cols="5">
<thead>
<row>
<entry>Name</entry>
<entry>Base 2 Factor</entry>
<entry>Bytes</entry>
<entry>Close Base 10 Factor</entry>
<entry>Base 10 bytes</entry>
</row>
</thead>
<tbody>
<row>
<entry>1 Kilobyte</entry>
<entry>2<superscript>10</superscript></entry>
<entry>Kilobyte</entry>
<entry>1,024</entry>
<entry>10<superscript>3</superscript></entry>
<entry>1,000</entry>
</row>
<row>
<entry>1 Megabyte</entry>
<entry>2<superscript>20</superscript></entry>
<entry>Megabyte</entry>
<entry>1,048,576</entry>
<entry>10<superscript>6</superscript></entry>
<entry>1,000,000</entry>
</row>
<row>
<entry>1 Gigabyte</entry>
<entry>2<superscript>30</superscript></entry>
<entry>Gigabyte</entry>
<entry>1,073,741,824</entry>
<entry>10<superscript>9</superscript></entry>
<entry>1,000,000,000</entry>
</row>
<row>
<entry>1 Terabyte</entry>
<entry>2<superscript>40</superscript></entry>
<entry>Terabyte</entry>
<entry>1,099,511,627,776</entry>
<entry>10<superscript>12</superscript></entry>
<entry>1,000,000,000,000</entry>
</row>
<row>
<entry>1 Petabyte</entry>
<entry>2<superscript>50</superscript></entry>
<entry>Petabyte</entry>
<entry>1,125,899,906,842,624</entry>
<entry>10<superscript>15</superscript></entry>
<entry>1,000,000,000,000,000</entry>
</row>
<row>
<entry>1 Exabyte</entry>
<entry>2<superscript>60</superscript></entry>
<entry>Exabyte</entry>
<entry>1,152,921,504,606,846,976</entry>
<entry>10<superscript>18</superscript></entry>
<entry>1,000,000,000,000,000,000</entry>
</row>
</tbody>
</tgroup>
<caption><para>SI units compared in base 2 and base 10</para></caption>
</table>
<para>Therefore a 32 bit computer can address up to four
gigabytes of memory; the extra two bits can represent four
groups of <computeroutput>2<superscript>30</superscript>
bytes.</computeroutput>. A 64 bit computer can address up
to 8 exabytes; you might be interested in working out just
how big a number this is! To get a feel for how big that
number is, calculate how long it would take to count to
<computeroutput>2<superscript>64</superscript></computeroutput>
if you incremented once per second.</para>
<para>It can be very useful to commit the base 2 factors to
memory as an aid to quickly correlate the relationship
between number-of-bits and "human" sizes. For example, we
can quickly calculate that a 32 bit computer can address up
to four gigabytes of memory by noting the recombination of
2<superscript>2</superscript> (4) &#43;
2<superscript>30</superscript>. A 64-bit value could
similarly address up to 16 exabytes
(2<superscript>4</superscript> &#43;
2<superscript>60</superscript>); you might be interested in
working out just how big a number this is. To get a feel
for how big that number is, calculate how long it would take
to count to
<computeroutput>2<superscript>64</superscript></computeroutput>
if you incremented once per second.</para>
</section>
<section>
<info>
<title>Kilo, Mega and Giga Bits</title>
</info>
<para>Apart from the confusion related to the overloading of
SI units between binary and base 10, capacities will often
be quoted in terms of <emphasis>bits</emphasis> rather than
bytes.</para>
<para>Generally this happens when talking about networking
or storage devices; you may have noticed that your ADSL
connection is described as something like 1500
kilobits/second. The calculation is simple; multiply by
1000 (for the kilo), divide by 8 to get bytes and then 1024
to get kilobytes (so 1500 kilobits/s=183 kilobytes
per second).</para>
SI units between binary and base 10, capacities will often
be quoted in terms of <emphasis>bits</emphasis> rather than
bytes. Generally this happens when talking about networking
or storage devices; you may have noticed that your ADSL
connection is described as something like 1500
kilobits/second. The calculation is simple; multiply by
1000 (for the kilo), divide by 8 to get bytes and then 1024
to get kilobytes (so 1500 kilobits/s=183 kilobytes per
second).</para>
<para>The SI standardisation body has recognised these dual
uses, and has specified unique prefixes for binary usage.
Under the standard 1024 bytes is a
<computeroutput>kibibyte</computeroutput>, short for
<emphasis>kilo binary</emphasis> byte (shortened to KiB).
The other prefixes have a similar prefix (Mebibyte, for
example). Tradition largely prevents use of these terms,
but you may seem them in some literature.</para>
uses and has specified unique prefixes for binary usage.
Under the standard 1024 bytes is a
<computeroutput>kibibyte</computeroutput>, short for
<emphasis>kilo binary</emphasis> byte (shortened to KiB).
The other prefixes have a similar prefix (Mebibyte, MiB, for
example). Tradition largely prevents use of these terms,
but you may seem them in some literature.</para>
</section>
<section>
<info>
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