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23 <p>
24 </p>
25 <h1><a name="name">NAME</a></h1>
26 <p>bin_dec_hex - How to use binary, decimal, and hexadecimal notation.</p>
27 <div align="right"><a href="bin_dec_hex.pdf">PDF</a> version.</div><p>
28 </p>
29 <hr />
30 <h1><a name="description">DESCRIPTION</a></h1>
31 <p>Most people use the decimal numbering system. This system uses ten
32 symbols to represent numbers. When those ten symbols are used up, they
33 start all over again and increment the position to the left. The
34 digit 0 is only shown if it is the only symbol in the sequence, or if
35 it is not the first one.</p>
36 <p>If this sounds cryptic to you, this is what I've just said in numbers:</p>
37 <pre>
38 0
39 1
40 2
41 3
42 4
43 5
44 6
45 7
46 8
47 9
48 10
49 11
50 12
51 13</pre>
52 <p>and so on.</p>
53 <p>Each time the digit nine is incremented, it is reset to 0 and the
54 position before (to the left) is incremented (from 0 to 1). Then
55 number 9 can be seen as ``00009'' and when we should increment 9, we
56 reset it to zero and increment the digit just before the 9 so the
57 number becomes ``00010''. Leading zeros we don't write except if it is
58 the only digit (number 0). And of course, we write zeros if they occur
59 anywhere inside or at the end of a number:</p>
60 <pre>
61 "00010" -> " 0010" -> " 010" -> " 10", but not " 1 ".</pre>
62 <p>This was pretty basic, you already knew this. Why did I tell it?
63 Well, computers usually do not represent numbers with 10 different
64 digits. They only use two different symbols, namely ``0'' and ``1''. Apply
65 the same rules to this set of digits and you get the binary numbering
66 system:</p>
67 <pre>
68 0
69 1
70 10
71 11
72 100
73 101
74 110
75 111
76 1000
77 1001
78 1010
79 1011
80 1100
81 1101</pre>
82 <p>and so on.</p>
83 <p>If you count the number of rows, you'll see that these are again 14
84 different numbers. The numbers are the same and mean the same as in
85 the first list, we just used a different representation. This means
86 that you have to know the representation used, or as it is called the
87 numbering system or base. Normally, if we do not explicitly specify
88 the numbering system used, we implicitly use the decimal system. If we
89 want to use any other numbering system, we'll have to make that
90 clear. There are a few widely adopted methods to do so. One common
91 form is to write <code>1010(2)</code> which means that you wrote down a number in
92 its binary representation. It is the number ten. If you would write
93 1010 without specifying the base, the number is interpreted as one
94 thousand and ten using base 10.</p>
95 <p>In books, another form is common. It uses subscripts (little
96 characters, more or less in between two rows). You can leave out the
97 parentheses in that case and write down the number in normal
98 characters followed by a little two just behind it.</p>
99 <p>As the numbering system used is also called the base, we talk of the
100 number 1100 base 2, the number 12 base 10.</p>
101 <p>Within the binary system, it is common to write leading zeros. The
102 numbers are written down in series of four, eight or sixteen depending
103 on the context.</p>
104 <p>We can use the binary form when talking to computers
105 (...programming...), but the numbers will have large
106 representations. The number 65'535 (often in the decimal system a ' is
107 used to separate blocks of three digits for readability) would be
108 written down as <code>1111111111111111(2)</code> which is 16 times the digit 1.
109 This is difficult and prone to errors. Therefore, we usually would use
110 another base, called hexadecimal. It uses 16 different symbols. First
111 the symbols from the decimal system are used, thereafter we continue
112 with alphabetic characters. We get 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
113 A, B, C, D, E and F. This system is chosen because the hexadecimal
114 form can be converted into the binary system very easily (and back).</p>
115 <p>There is yet another system in use, called the octal system. This was
116 more common in the old days, but is not used very often anymore. As
117 you might find it in use sometimes, you should get used to it and
118 we'll show it below. It's the same story as with the other
119 representations, but with eight different symbols.</p>
120 <pre>
121 Binary (2)
122 Octal (8)
123 Decimal (10)
124 Hexadecimal (16)</pre>
125 <pre>
126 (2) (8) (10) (16)
127 00000 0 0 0
128 00001 1 1 1
129 00010 2 2 2
130 00011 3 3 3
131 00100 4 4 4
132 00101 5 5 5
133 00110 6 6 6
134 00111 7 7 7
135 01000 10 8 8
136 01001 11 9 9
137 01010 12 10 A
138 01011 13 11 B
139 01100 14 12 C
140 01101 15 13 D
141 01110 16 14 E
142 01111 17 15 F
143 10000 20 16 10
144 10001 21 17 11
145 10010 22 18 12
146 10011 23 19 13
147 10100 24 20 14
148 10101 25 21 15</pre>
149 <p>Most computers used nowadays are using bytes of eight bits. This means
150 that they store eight bits at a time. You can see why the octal system
151 is not the most practical for that: You'd need three digits to represent
152 the eight bits and this means that you'd have to use one complete digit
153 to represent only two bits (2+3+3=8). This is a waste. For hexadecimal
154 digits, you need only two digits which are used completely:</p>
155 <pre>
156 (2) (8) (10) (16)
157 11111111 377 255 FF</pre>
158 <p>You can see why binary and hexadecimal can be converted quickly: For
159 each hexadecimal digit there are exactly four binary digits. Take a
160 binary number: take four digits from the right and make a hexadecimal
161 digit from it (see the table above). Repeat this until there are no
162 more digits. And the other way around: Take a hexadecimal number. For
163 each digit, write down its binary equivalent.</p>
164 <p>Computers (or rather the parsers running on them) would have a hard
165 time converting a number like 1234(16). Therefore hexadecimal numbers
166 are specified with a prefix. This prefix depends on the language
167 you're writing in. Some of the prefixes are ``0x'' for C, ``$'' for
168 Pascal, ``#'' for HTML. It is common to assume that if a number starts
169 with a zero, it is octal. It does not matter what is used as long as
170 you know what it is. I will use ``0x'' for hexadecimal, ``%'' for binary
171 and ``0'' for octal. The following numbers are all the same, just their
172 representation (base) is different: 021 0x11 17 %00010001</p>
173 <p>To do arithmetics and conversions you need to understand one more thing.
174 It is something you already know but perhaps you do not ``see'' it yet:</p>
175 <p>If you write down 1234, (no prefix, so it is decimal) you are talking
176 about the number one thousand, two hundred and thirty four. In sort of
177 a formula:</p>
178 <pre>
179 1 * 1000 = 1000
180 2 * 100 = 200
181 3 * 10 = 30
182 4 * 1 = 4</pre>
183 <p>This can also be written as:</p>
184 <pre>
185 1 * 10^3
186 2 * 10^2
187 3 * 10^1
188 4 * 10^0</pre>
189 <p>where ^ means ``to the power of''.</p>
190 <p>We are using the base 10, and the positions 0,1,2 and 3.
191 The right-most position should NOT be multiplied with 10. The second
192 from the right should be multiplied one time with 10. The third from
193 the right is multiplied with 10 two times. This continues for whatever
194 positions are used.</p>
195 <p>It is the same in all other representations:</p>
196 <p>0x1234 will be</p>
197 <pre>
198 1 * 16^3
199 2 * 16^2
200 3 * 16^1
201 4 * 16^0</pre>
202 <p>01234 would be</p>
203 <pre>
204 1 * 8^3
205 2 * 8^2
206 3 * 8^1
207 4 * 8^0</pre>
208 <p>This example can not be done for binary as that system only uses two
209 symbols. Another example:</p>
210 <p>%1010 would be</p>
211 <pre>
212 1 * 2^3
213 0 * 2^2
214 1 * 2^1
215 0 * 2^0</pre>
216 <p>It would have been easier to convert it to its hexadecimal form and
217 just translate %1010 into 0xA. After a while you get used to it. You will
218 not need to do any calculations anymore, but just know that 0xA means 10.</p>
219 <p>To convert a decimal number into a hexadecimal you could use the next
220 method. It will take some time to be able to do the estimates, but it
221 will be easier when you use the system more frequently. We'll look at
222 yet another way afterwards.</p>
223 <p>First you need to know how many positions will be used in the other
224 system. To do so, you need to know the maximum numbers you'll be
225 using. Well, that's not as hard as it looks. In decimal, the maximum
226 number that you can form with two digits is ``99''. The maximum for
227 three: ``999''. The next number would need an extra position. Reverse
228 this idea and you will see that the number can be found by taking 10^3
229 (10*10*10 is 1000) minus 1 or 10^2 minus one.</p>
230 <p>This can be done for hexadecimal as well:</p>
231 <pre>
232 16^4 = 0x10000 = 65536
233 16^3 = 0x1000 = 4096
234 16^2 = 0x100 = 256
235 16^1 = 0x10 = 16</pre>
236 <p>If a number is smaller than 65'536 it will fit in four positions.
237 If the number is bigger than 4'095, you must use position 4.
238 How many times you can subtract 4'096 from the number without going below
239 zero is the first digit you write down. This will always be a number
240 from 1 to 15 (0x1 to 0xF). Do the same for the other positions.</p>
241 <p>Let's try with 41'029. It is smaller than 16^4 but bigger than 16^3-1. This
242 means that we have to use four positions.
243 We can subtract 16^3 from 41'029 ten times without going below zero.
244 The left-most digit will therefore be ``A'', so we have 0xA????.
245 The number is reduced to 41'029 - 10*4'096 = 41'029-40'960 = 69.
246 69 is smaller than 16^3 but not bigger than 16^2-1. The second digit
247 is therefore ``0'' and we now have 0xA0??.
248 69 is smaller than 16^2 and bigger than 16^1-1. We can subtract 16^1
249 (which is just plain 16) four times and write down ``4'' to get 0xA04?.
250 Subtract 64 from 69 (69 - 4*16) and the last digit is 5 --> 0xA045.</p>
251 <p>The other method builds up the number from the right. Let's try 41'029
252 again. Divide by 16 and do not use fractions (only whole numbers).</p>
253 <pre>
254 41'029 / 16 is 2'564 with a remainder of 5. Write down 5.
255 2'564 / 16 is 160 with a remainder of 4. Write the 4 before the 5.
256 160 / 16 is 10 with no remainder. Prepend 45 with 0.
257 10 / 16 is below one. End here and prepend 0xA. End up with 0xA045.</pre>
258 <p>Which method to use is up to you. Use whatever works for you. I use
259 them both without being able to tell what method I use in each case,
260 it just depends on the number, I think. Fact is, some numbers will
261 occur frequently while programming. If the number is close to one I am
262 familiar with, then I will use the first method (like 32'770 which is
263 into 32'768 + 2 and I just know that it is 0x8000 + 0x2 = 0x8002).</p>
264 <p>For binary the same approach can be used. The base is 2 and not 16,
265 and the number of positions will grow rapidly. Using the second method
266 has the advantage that you can see very easily if you should write down
267 a zero or a one: if you divide by two the remainder will be zero if it
268 is an even number and one if it is an odd number:</p>
269 <pre>
270 41029 / 2 = 20514 remainder 1
271 20514 / 2 = 10257 remainder 0
272 10257 / 2 = 5128 remainder 1
273 5128 / 2 = 2564 remainder 0
274 2564 / 2 = 1282 remainder 0
275 1282 / 2 = 641 remainder 0
276 641 / 2 = 320 remainder 1
277 320 / 2 = 160 remainder 0
278 160 / 2 = 80 remainder 0
279 80 / 2 = 40 remainder 0
280 40 / 2 = 20 remainder 0
281 20 / 2 = 10 remainder 0
282 10 / 2 = 5 remainder 0
283 5 / 2 = 2 remainder 1
284 2 / 2 = 1 remainder 0
285 1 / 2 below 0 remainder 1</pre>
286 <p>Write down the results from right to left: %1010000001000101</p>
287 <p>Group by four:</p>
288 <pre>
289 %1010000001000101
290 %101000000100 0101
291 %10100000 0100 0101
292 %1010 0000 0100 0101</pre>
293 <p>Convert into hexadecimal: 0xA045</p>
294 <p>Group %1010000001000101 by three and convert into octal:</p>
295 <pre>
296 %1010000001000101
297 %1010000001000 101
298 %1010000001 000 101
299 %1010000 001 000 101
300 %1010 000 001 000 101
301 %1 010 000 001 000 101
302 %001 010 000 001 000 101
303 1 2 0 1 0 5 --> 0120105</pre>
304 <pre>
305 So: %1010000001000101 = 0120105 = 0xA045 = 41029
306 Or: 1010000001000101(2) = 120105(8) = A045(16) = 41029(10)
307 Or: 1010000001000101(2) = 120105(8) = A045(16) = 41029</pre>
308 <p>At first while adding numbers, you'll convert them to their decimal
309 form and then back into their original form after doing the addition.
310 If you use the other numbering system often, you will see that you'll
311 be able to do arithmetics directly in the base that is used.
312 In any representation it is the same, add the numbers on the right,
313 write down the right-most digit from the result, remember the other
314 digits and use them in the next round. Continue with the second digit
315 from the right and so on:</p>
316 <pre>
317 %1010 + %0111 --> 10 + 7 --> 17 --> %00010001</pre>
318 <p>will become</p>
319 <pre>
320 %1010
321 %0111 +
322 ||||
323 |||+-- add 0 + 1, result is 1, nothing to remember
324 ||+--- add 1 + 1, result is %10, write down 0 and remember 1
325 |+---- add 0 + 1 + 1(remembered), result = 0, remember 1
326 +----- add 1 + 0 + 1(remembered), result = 0, remember 1
327 nothing to add, 1 remembered, result = 1
328 --------
329 %10001 is the result, I like to write it as %00010001</pre>
330 <p>For low values, try to do the calculations yourself, then check them with
331 a calculator. The more you do the calculations yourself, the more you'll
332 find that you didn't make mistakes. In the end, you'll do calculi in
333 other bases as easily as you do them in decimal.</p>
334 <p>When the numbers get bigger, you'll have to realize that a computer is
335 not called a computer just to have a nice name. There are many
336 different calculators available, use them. For Unix you could use ``bc''
337 which is short for Binary Calculator. It calculates not only in
338 decimal, but in all bases you'll ever want to use (among them Binary).</p>
339 <p>For people on Windows:
340 Start the calculator (start->programs->accessories->calculator)
341 and if necessary click view->scientific. You now have a scientific
342 calculator and can compute in binary or hexadecimal.</p>
343 <p>
344 </p>
345 <hr />
346 <h1><a name="author">AUTHOR</a></h1>
347 <p>I hope you enjoyed the examples and their descriptions. If you do, help
348 other people by pointing them to this document when they are asking
349 basic questions. They will not only get their answer, but at the same
350 time learn a whole lot more.</p>
351 <p>Alex van den Bogaerdt <<a href="mailto:alex@vandenbogaerdt.nl">alex@vandenbogaerdt.nl</a>></p>
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