📄 numbering_systems_tutorial.html
字号:
<!doctype HTML public "-//W3O//DTD W3 HTML 3.0//EN"><HTML><HEAD><TITLE>Numbering Systems Tutorial</TITLE><META name="description" content="Numbering Systems Tutorials"><META name="keywords" content="numbering systems tutorial, 8086, hex, oct, bin, binary, hexadecimal, octal, ocatal"><META name="robots" content="nofollow"></HEAD><BODY bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#007099" alink="#FF0000"><FONT FACE="Verdana" SIZE=3><TABLE WIDTH=80%><TR><TD><FONT SIZE=+2><B>Numbering Systems Tutorial</B></FONT><BR><HR><BR><FONT SIZE=+2><B>What is it?</B></FONT><BR><BR><!--There are many ways to represent the same numeric value, previouslyhomo sapiens was using sticks to count, so number 5 was represented as:<BR><B> | | | | |</B><BR>(five sticks).<BR><BR>Using sticks to count isn't a good idea, modern decimal system is much better,don't you think?-->There are many ways to represent the same numeric value. Long ago, humans usedsticks to count, and later learned how to draw pictures of sticks in the groundand eventually on paper. So, the number 5 was first represented as:<NOBR><STRONG>| | | | | </STRONG></NOBR>(for five sticks).<BR><BR>Later on, the Romans began using different symbols for multiple numbersof sticks:<NOBR><STRONG>| | | </STRONG></NOBR>still meant three sticks, but a <STRONG>V </STRONG>now meant five sticks,and an <STRONG>X </STRONG> was used to represent ten of them!<BR><BR>Using sticks to count was a great idea for its time. And usingsymbols instead of real sticks was much better. One of the best ways to represent a number today is by using the modern decimal system.Why? Because it includes the major breakthrough of using a symbol torepresent the idea of counting <I><STRONG>nothing</STRONG></I>. About 1500 years ago in India, <STRONG>zero</STRONG>(<STRONG>0</STRONG>) was first used as a number! It was later used in the Middle East as the Arabic,<I>sifr</I>. And was finally introduced to the West as the Latin, <I>zephiro</I>. Soon you'll see just how valuable an idea this is for all modern number systems.<BR><BR><HR><BR><FONT SIZE=+2><B>Decimal System</B></FONT><BR><BR><!--We, humans use decimal representation to count.In decimal system there are 10 digits:-->Most people today use decimal representation to count.In the decimal system there are 10 digits:<BR><BR><B>0, 1, 2, 3, 4, 5, 6, 7, 8, 9</B><BR><BR>These digits can represent any value, for example:<BR><B>754</B>.<BR>The value is formed by the sum of each digit, multiplied bythe <B>base</B> (in this case it is <B>10</B> because there are10 digits in decimal system) in power of digit position (counting from zero):<BR><BR><!-- 7 * 10^2 + 5 * 10^1 + 4 * 10^0 = 700 + 50 + 4 = 754 --><IMG SRC="num1.gif" WIDTH=448 HEIGHT=108><BR><BR>Position of each digit is very important!for example if you place "7" to the end:<BR><B>547</B><BR>it will be another value:<BR><BR><!-- 5 * 10^2 + 4 * 10^1 + 7 * 10^0 = 500 + 40 + 7 = 547 --><IMG SRC="num2.gif" WIDTH=448 HEIGHT=108><BR><BR><B>Important note:</B> any number in power of zero is 1, evenzero in power of zero is 1:<BR><IMG SRC="num3.gif" WIDTH=306 HEIGHT=60><BR><BR><HR><BR><FONT SIZE=+2><B>Binary System</B></FONT><BR><BR>Computers are not as smart as humans are (or not yet),it's easy to make an electronic machine with two states: <B>on</B> and <B>off</B>,or <B>1</B> and <B>0</B>.<BR>Computers use binary system, binary system uses 2 digits:<BR><BR><B>0, 1</B><BR><BR>And thus the <B>base</B> is <B>2</B>.<BR><BR>Each digit in a binary number is called a <B>BIT</B>, 4 bits form a <B>NIBBLE</B>,8 bits form a <B>BYTE</B>, two bytes form a <B>WORD</B>, two words forma <B>DOUBLE WORD</B> (rarely used):<BR><BR><IMG SRC="num4.gif" WIDTH=339 HEIGHT=126><BR><BR>There is a convention to add <B>"b"</B> in the end of a binary number, this way wecan determine that 101b is a binary number with decimal value of 5.<BR><BR>The binary number <B>10100101b</B> equals to decimal value of 165:<BR><BR><!--10100101b = = 1 * 2^7 + 0 * 2^6 + 1 * 2^5 + 0 * 2^4 + 0 * 2^3 + 1 * 2^2 + 0 * 2^1 + 1 * 2^0 = = 128 + 0 + 32 + 0 + 0 + 4 + 0 + 1 = 165--><IMG SRC="num5.gif"><BR><BR><HR><BR><FONT SIZE=+2><B>Hexadecimal System</B></FONT><BR><BR>Hexadecimal System uses 16 digits:<BR><BR><B>0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F</B><BR><BR>And thus the <B>base</B> is <B>16</B>.<BR><BR>Hexadecimal numbers are compact and easy to read.<BR>It is very easy to convert numbers from binary system to hexadecimal system and vice-versa,every nibble (4 bits) can be converted to a hexadecimal digit using this table:<BR><BR><A NAME="hextable"> </A><TABLE BORDER=0 COLS=2 CELLSPACING=10><TR><TD> <TABLE BORDER=1 COLS=3> <TR> <TD><B>Decimal<BR>(base 10)</B></TD> <TD><B>Binary<BR>(base 2)</B></TD> <TD><B>Hexadecimal<BR>(base 16)</B></TD> </TR> <TR> <TD>0</TD> <TD>0000</TD> <TD>0</TD> </TR> <TR> <TD>1</TD> <TD>0001</TD> <TD>1</TD> </TR> <TR> <TD>2</TD> <TD>0010</TD> <TD>2</TD> </TR> <TR> <TD>3</TD> <TD>0011</TD> <TD>3</TD> </TR> <TR> <TD>4</TD> <TD>0100</TD> <TD>4</TD> </TR> <TR> <TD>5</TD> <TD>0101</TD> <TD>5</TD> </TR> <TR> <TD>6</TD> <TD>0110</TD> <TD>6</TD> </TR> <TR> <TD>7</TD> <TD>0111</TD> <TD>7</TD> </TR> <TR> <TD>8</TD> <TD>1000</TD> <TD>8</TD> </TR> <TR> <TD>9</TD> <TD>1001</TD> <TD>9</TD> </TR> <TR> <TD>10</TD> <TD>1010</TD> <TD>A</TD> </TR> <TR> <TD>11</TD> <TD>1011</TD> <TD>B</TD> </TR> <TR> <TD>12</TD> <TD>1100</TD> <TD>C</TD> </TR> <TR> <TD>13</TD> <TD>1101</TD> <TD>D</TD> </TR> <TR> <TD>14</TD> <TD>1110</TD> <TD>E</TD> </TR> <TR> <TD>15</TD> <TD>1111</TD> <TD>F</TD> </TR> </TABLE></TD><TD><IMG SRC="num7.gif"></TD></TR></TABLE><BR><BR>There is a convention to add <B>"h"</B> in the end of a hexadecimal number, this way wecan determine that 5Fh is a hexadecimal number with decimal value of 95.<BR>We also add <B>"0"</B> (zero) in the beginning of hexadecimal numbers that begin with a letter (A..F),for example <B>0E120h</B>.<BR><BR>The hexadecimal number <B>1234h</B> is equal to decimal value of 4660:<BR><BR><!--1 * 16^3 + 2 * 16^2 + 3 * 16^1 + 4 * 16^0 = 4096 + 512 + 48 + 4 = 4660--><IMG SRC="num6.gif" WIDTH=592 HEIGHT=134><BR><BR><HR><BR><FONT SIZE=+2><B>Converting from Decimal System to Any Other</B></FONT><BR><BR>In order to convert from decimal system, to any other system, it is requiredto divide the decimal value by the <B>base</B> of the desired system, each timeyou should remember the <B>result</B> and keep the <B>remainder</B>, thedivide process continues until the <B>result</B> is zero.<BR><BR>The <B>remainders</B> are then used to represent a value in that system.<BR><BR>Let's convert the value of <B>39</B> (base <U>10</U>) to<I>Hexadecimal System</I> (base <U>16</U>):<BR><BR><IMG SRC="num8.gif" WIDTH=420 HEIGHT=185><BR><BR>As you see we got this hexadecimal number: <B>27h</B>.<BR>All remainders were below <B>10</B> in the above example, sowe do not use any letters.<BR><BR>Here is another more complex example:<BR>let's convert decimal number <B>43868</B> to hexadecimal form:<BR><BR><IMG SRC="num9.gif" WIDTH=419 HEIGHT=310><BR><BR>The result is <B>0AB5Ch</B>, we are using <A HREF="#hextable">the above table</A>to convert remainders over <B>9</B> to corresponding letters.<BR><BR>Using the same principle we can convert to binary form (using <B>2</B> as the divider),or convert to hexadecimal number, and then convert it to binary number using<A HREF="#hextable">the above table</A>:<BR><BR><IMG SRC="num10.gif" WIDTH=233 HEIGHT=88><BR><BR>As you see we got this binary number: <B>1010101101011100b</B><BR><BR><HR><BR><FONT SIZE=+2><B>Signed Numbers</B></FONT><BR><BR>There is no way to say for sure whether the hexadecimal byte <B>0FFh</B> ispositive or negative, it can represent both decimal value "<B>255</B>" and "<B>- 1</B>".<BR><BR>8 bits can be used to create <B>256</B> combinations (including zero), so we simplypresume that first <B>128</B> combinations (<B>0..127</B>) will represent positive numbersand next <B>128</B> combinations (<B>128..256</B>) will represent negative numbers.<BR><BR>In order to get "<B>- 5</B>", we should subtract <B>5</B> from the number ofcombinations (<B>256</B>), so it we'll get: <NOBR><B>256 - 5 = 251</B>.</NOBR><BR><BR>Using this complex way to represent negative numbers has some meaning, in mathwhen you add "<B>- 5</B>" to "<B>5</B>" you should get zero.<BR>This is what happens when processor adds two bytes <B>5</B> and <B>251</B>,the result gets over <B>255</B>, because of the overflow processor gets zero!<BR><BR><IMG SRC="num11.gif" WIDTH=271 HEIGHT=191><BR><BR>When combinations <B>128..256</B> are used the high bit is always <B>1</B>, sothis maybe used to determine the sign of a number.<BR><BR>The same principle is used for <B>words</B> (16 bit values),16 bits create <B>65536</B> combinations, first 32768 combinations (<B>0..32767</B>)are used to represent positive numbers, and next 32768 combinations (<B>32767..65535</B>)represent negative numbers.<BR><BR><HR><BR>There are some handy tools in <I>Emu8086</I> to convert numbers, and makecalculations of any numerical expressions, all you need is aclick on <B>Math</B> menu:<BR><BR><IMG SRC="num12.gif" WIDTH=555 HEIGHT=382><BR><BR><B>Number Convertor</B> allows you to convert numbers from any system andto any system. Just type a value in any text-box, and the valuewill be automatically converted to all other systems. You can work bothwith <B>8 bit</B> and <B>16 bit</B> values.<BR><BR><B>Expression Evaluator</B> can be used to make calculations betweennumbers in different systems and convert numbers from one system to another.Type an expression and press enter, result will appear in chosen numbering system.You can work with values up to <B>32 bits</B>. When <B>Signed</B> is checkedevaluator assumes that all values (except decimal and double words) should be treated as<B>signed</B>. Double words are always treated as signed values, so<B>0FFFFFFFFh</B> is converted to <B>-1</B>.<BR>For example you want to calculate: <FONT FACE="Fixedsys"><NOBR>0FFFFh * 10h + 0FFFFh</NOBR></FONT>(maximum memory location that can be accessed by 8086 CPU).If you check <B>Signed</B> and <B>Word</B> you will get<FONT FACE="Fixedsys">-17</FONT> (because it is evaluated as<FONT FACE="Fixedsys"><NOBR>(-1) * 16 + (-1) </NOBR></FONT>. To makecalculation with unsigned values uncheck <B>Signed</B>so that the evaluation will be <FONT FACE="Fixedsys"><NOBR>65535 * 16 + 65535</NOBR></FONT>and you should get <FONT FACE="Fixedsys">1114095</FONT>.<BR>You can also use the <B>Number Convertor</B> to convertnon-decimal digits to <B>signed decimal</B> values, anddo the calculation with decimal values (if it's easier for you).<BR><BR>These operation are supported:<BR><PRE><FONT FACE="Fixedsys">~ not (inverts all bits).* multiply./ divide.% modulus.+ sum.- subtract (and unary -).<< shift left.>> shift right.& bitwise AND.^ bitwise XOR.| bitwise OR.</FONT></PRE><BR>Binary numbers must have "<B>b</B>" suffix, example:<BR><FONT FACE="Fixedsys"> 00011011b</FONT><BR><BR>Hexadecimal numbers must have "<B>h</B>" suffix, and start with a zero<BR>when first digit is a letter (A..F), example:<BR><FONT FACE="Fixedsys"> 0ABCDh</FONT><BR><BR>Octal (base 8) numbers must have "<B>o</B>" suffix, example:<BR><FONT FACE="Fixedsys"> 77o</FONT><BR><BR><BR><BR><BR><HR><CENTER><A HREF="asm_tutorial_01.html"><B> >>> Next Tutorial >>> </B></A></CENTER><HR></TD></TR></TABLE><BR></FONT></BODY></HTML>⌨️ 快捷键说明
复制代码
Ctrl + C
搜索代码
Ctrl + F
全屏模式
F11
切换主题
Ctrl + Shift + D
显示快捷键
?
增大字号
Ctrl + =
减小字号
Ctrl + -