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