1BIN_DEC_HEX(1) rrdtool BIN_DEC_HEX(1)
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6 bin_dec_hex - How to use binary, decimal, and hexadecimal notation.
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9 Most people use the decimal numbering system. This system uses ten sym‐
10 bols to represent numbers. When those ten symbols are used up, they
11 start all over again and increment the position to the left. The digit
12 0 is only shown if it is the only symbol in the sequence, or if it is
13 not the first one.
14 If this sounds cryptic to you, this is what I've just said in numbers:
16 0
18 1
19 2
20 3
21 4
22 5
23 6
24 7
25 8
26 9
27 10
28 11
29 12
30 13
31 and so on.
33 Each time the digit nine is incremented, it is reset to 0 and the posi‐
35 tion before (to the left) is incremented (from 0 to 1). Then number 9
36 can be seen as "00009" and when we should increment 9, we reset it to
37 zero and increment the digit just before the 9 so the number becomes
38 "00010". Leading zeros we don't write except if it is the only digit
39 (number 0). And of course, we write zeros if they occur anywhere inside
40 or at the end of a number:
41 "00010" -> " 0010" -> " 010" -> " 10", but not " 1 ".
43 This was pretty basic, you already knew this. Why did I tell it? Well,
45 computers usually do not represent numbers with 10 different digits.
46 They only use two different symbols, namely "0" and "1". Apply the same
47 rules to this set of digits and you get the binary numbering system:
48 0
50 1
51 10
52 11
53 100
54 101
55 110
56 111
57 1000
58 1001
59 1010
60 1011
61 1100
62 1101
63 and so on.
65 If you count the number of rows, you'll see that these are again 14
67 different numbers. The numbers are the same and mean the same as in the
68 first list, we just used a different representation. This means that
69 you have to know the representation used, or as it is called the num‐
70 bering system or base. Normally, if we do not explicitly specify the
71 numbering system used, we implicitly use the decimal system. If we want
72 to use any other numbering system, we'll have to make that clear. There
73 are a few widely adopted methods to do so. One common form is to write
74 1010(2) which means that you wrote down a number in its binary repre‐
75 sentation. It is the number ten. If you would write 1010 without speci‐
76 fying the base, the number is interpreted as one thousand and ten using
77 base 10.
78 In books, another form is common. It uses subscripts (little charac‐
80 ters, more or less in between two rows). You can leave out the paren‐
81 theses in that case and write down the number in normal characters fol‐
82 lowed by a little two just behind it.
83 As the numbering system used is also called the base, we talk of the
85 number 1100 base 2, the number 12 base 10.
86 Within the binary system, it is common to write leading zeros. The num‐
88 bers are written down in series of four, eight or sixteen depending on
89 the context.
90 We can use the binary form when talking to computers (...program‐
92 ming...), but the numbers will have large representations. The number
93 65'535 (often in the decimal system a ' is used to separate blocks of
94 three digits for readability) would be written down as
95 1111111111111111(2) which is 16 times the digit 1. This is difficult
96 and prone to errors. Therefore, we usually would use another base,
97 called hexadecimal. It uses 16 different symbols. First the symbols
98 from the decimal system are used, thereafter we continue with alpha‐
99 betic characters. We get 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E
100 and F. This system is chosen because the hexadecimal form can be con‐
101 verted into the binary system very easily (and back).
102 There is yet another system in use, called the octal system. This was
104 more common in the old days, but is not used very often anymore. As you
105 might find it in use sometimes, you should get used to it and we'll
106 show it below. It's the same story as with the other representations,
107 but with eight different symbols.
108 Binary (2)
110 Octal (8)
111 Decimal (10)
112 Hexadecimal (16)
113 (2) (8) (10) (16)
115 00000 0 0 0
116 00001 1 1 1
117 00010 2 2 2
118 00011 3 3 3
119 00100 4 4 4
120 00101 5 5 5
121 00110 6 6 6
122 00111 7 7 7
123 01000 10 8 8
124 01001 11 9 9
125 01010 12 10 A
126 01011 13 11 B
127 01100 14 12 C
128 01101 15 13 D
129 01110 16 14 E
130 01111 17 15 F
131 10000 20 16 10
132 10001 21 17 11
133 10010 22 18 12
134 10011 23 19 13
135 10100 24 20 14
136 10101 25 21 15
137 Most computers used nowadays are using bytes of eight bits. This means
139 that they store eight bits at a time. You can see why the octal system
140 is not the most practical for that: You'd need three digits to repre‐
141 sent the eight bits and this means that you'd have to use one complete
142 digit to represent only two bits (2+3+3=8). This is a waste. For hexa‐
143 decimal digits, you need only two digits which are used completely:
144 (2) (8) (10) (16)
146 11111111 377 255 FF
147 You can see why binary and hexadecimal can be converted quickly: For
149 each hexadecimal digit there are exactly four binary digits. Take a
150 binary number: take four digits from the right and make a hexadecimal
151 digit from it (see the table above). Repeat this until there are no
152 more digits. And the other way around: Take a hexadecimal number. For
153 each digit, write down its binary equivalent.
154 Computers (or rather the parsers running on them) would have a hard
156 time converting a number like 1234(16). Therefore hexadecimal numbers
157 are specified with a prefix. This prefix depends on the language you're
158 writing in. Some of the prefixes are "0x" for C, "$" for Pascal, "#"
159 for HTML. It is common to assume that if a number starts with a zero,
160 it is octal. It does not matter what is used as long as you know what
161 it is. I will use "0x" for hexadecimal, "%" for binary and "0" for
162 octal. The following numbers are all the same, just their represenata‐
163 tion (base) is different: 021 0x11 17 %00010001
164 To do arithmetics and conversions you need to understand one more
166 thing. It is something you already know but perhaps you do not "see"
167 it yet:
168 If you write down 1234, (no prefix, so it is decimal) you are talking
170 about the number one thousand, two hundred and thirty four. In sort of
171 a formula:
172 1 * 1000 = 1000
174 2 * 100 = 200
175 3 * 10 = 30
176 4 * 1 = 4
177 This can also be written as:
179 1 * 10^3
181 2 * 10^2
182 3 * 10^1
183 4 * 10^0
184 where ^ means "to the power of".
186 We are using the base 10, and the positions 0,1,2 and 3. The right-
188 most position should NOT be multiplied with 10. The second from the
189 right should be multiplied one time with 10. The third from the right
190 is multiplied with 10 two times. This continues for whatever positions
191 are used.
192 It is the same in all other representations:
194 0x1234 will be
196 1 * 16^3
198 2 * 16^2
199 3 * 16^1
200 4 * 16^0
201 01234 would be
203 1 * 8^3
205 2 * 8^2
206 3 * 8^1
207 4 * 8^0
208 This example can not be done for binary as that system only uses two
210 symbols. Another example:
211 %1010 would be
213 1 * 2^3
215 0 * 2^2
216 1 * 2^1
217 0 * 2^0
218 It would have been easier to convert it to its hexadecimal form and
220 just translate %1010 into 0xA. After a while you get used to it. You
221 will not need to do any calculations anymore, but just know that 0xA
222 means 10.
223 To convert a decimal number into a hexadecimal you could use the next
225 method. It will take some time to be able to do the estimates, but it
226 will be easier when you use the system more frequently. We'll look at
227 yet another way afterwards.
228 First you need to know how many positions will be used in the other
230 system. To do so, you need to know the maximum numbers you'll be using.
231 Well, that's not as hard as it looks. In decimal, the maximum number
232 that you can form with two digits is "99". The maximum for three:
233 "999". The next number would need an extra position. Reverse this idea
234 and you will see that the number can be found by taking 10^3 (10*10*10
235 is 1000) minus 1 or 10^2 minus one.
236 This can be done for hexadecimal as well:
238 16^4 = 0x10000 = 65536
240 16^3 = 0x1000 = 4096
241 16^2 = 0x100 = 256
242 16^1 = 0x10 = 16
243 If a number is smaller than 65'536 it will fit in four positions. If
245 the number is bigger than 4'095, you must use position 4. How many
246 times you can subtract 4'096 from the number without going below zero
247 is the first digit you write down. This will always be a number from 1
248 to 15 (0x1 to 0xF). Do the same for the other positions.
249 Let's try with 41'029. It is smaller than 16^4 but bigger than 16^3-1.
251 This means that we have to use four positions. We can subtract 16^3
252 from 41'029 ten times without going below zero. The left-most digit
253 will therefore be "A", so we have 0xA????. The number is reduced to
254 41'029 - 10*4'096 = 41'029-40'960 = 69. 69 is smaller than 16^3 but
255 not bigger than 16^2-1. The second digit is therefore "0" and we now
256 have 0xA0??. 69 is smaller than 16^2 and bigger than 16^1-1. We can
257 subtract 16^1 (which is just plain 16) four times and write down "4" to
258 get 0xA04?. Subtract 64 from 69 (69 - 4*16) and the last digit is 5
259 --> 0xA045.
260 The other method builds ub the number from the right. Let's try 41'029
262 again. Divide by 16 and do not use fractions (only whole numbers).
263 41'029 / 16 is 2'564 with a remainder of 5. Write down 5.
265 2'564 / 16 is 160 with a remainder of 4. Write the 4 before the 5.
266 160 / 16 is 10 with no remainder. Prepend 45 with 0.
267 10 / 16 is below one. End here and prepend 0xA. End up with 0xA045.
268 Which method to use is up to you. Use whatever works for you. I use
270 them both without being able to tell what method I use in each case, it
271 just depends on the number, I think. Fact is, some numbers will occur
272 frequently while programming. If the number is close to one I am famil‐
273 iar with, then I will use the first method (like 32'770 which is into
274 32'768 + 2 and I just know that it is 0x8000 + 0x2 = 0x8002).
275 For binary the same approach can be used. The base is 2 and not 16, and
277 the number of positions will grow rapidly. Using the second method has
278 the advantage that you can see very easily if you should write down a
279 zero or a one: if you divide by two the remainder will be zero if it is
280 an even number and one if it is an odd number:
281 41029 / 2 = 20514 remainder 1
283 20514 / 2 = 10257 remainder 0
284 10257 / 2 = 5128 remainder 1
285 5128 / 2 = 2564 remainder 0
286 2564 / 2 = 1282 remainder 0
287 1282 / 2 = 641 remainder 0
288 641 / 2 = 320 remainder 1
289 320 / 2 = 160 remainder 0
290 160 / 2 = 80 remainder 0
291 80 / 2 = 40 remainder 0
292 40 / 2 = 20 remainder 0
293 20 / 2 = 10 remainder 0
294 10 / 2 = 5 remainder 0
295 5 / 2 = 2 remainder 1
296 2 / 2 = 1 remainder 0
297 1 / 2 below 0 remainder 1
298 Write down the results from right to left: %1010000001000101
300 Group by four:
302 %1010000001000101
304 %101000000100 0101
305 %10100000 0100 0101
306 %1010 0000 0100 0101
307 Convert into hexadecimal: 0xA045
309 Group %1010000001000101 by three and convert into octal:
311 %1010000001000101
313 %1010000001000 101
314 %1010000001 000 101
315 %1010000 001 000 101
316 %1010 000 001 000 101
317 %1 010 000 001 000 101
318 %001 010 000 001 000 101
319 1 2 0 1 0 5 --> 0120105
320 So: %1010000001000101 = 0120105 = 0xA045 = 41029
322 Or: 1010000001000101(2) = 120105(8) = A045(16) = 41029(10)
323 Or: 1010000001000101(2) = 120105(8) = A045(16) = 41029
324 At first while adding numbers, you'll convert them to their decimal
326 form and then back into their original form after doing the addition.
327 If you use the other numbering system often, you will see that you'll
328 be able to do arithmetics directly in the base that is used. In any
329 representation it is the same, add the numbers on the right, write down
330 the right-most digit from the result, remember the other digits and use
331 them in the next round. Continue with the second digit from the right
332 and so on:
333 %1010 + %0111 --> 10 + 7 --> 17 --> %00010001
335 will become
337 %1010
339 %0111 +
340 ||||
341 |||+-- add 0 + 1, result is 1, nothing to remember
342 ||+--- add 1 + 1, result is %10, write down 0 and remember 1
343 |+---- add 0 + 1 + 1(remembered), result = 0, remember 1
344 +----- add 1 + 0 + 1(remembered), result = 0, remember 1
345 nothing to add, 1 remembered, result = 1
346 --------
347 %10001 is the result, I like to write it as %00010001
348 For low values, try to do the calculations yourself, then check them
350 with a calculator. The more you do the calculations yourself, the more
351 you'll find that you didn't make mistakes. In the end, you'll do cal‐
352 culi in other bases as easily as you do them in decimal.
353 When the numbers get bigger, you'll have to realize that a computer is
355 not called a computer just to have a nice name. There are many differ‐
356 ent calculators available, use them. For Unix you could use "bc" which
357 is short for Binary Calculator. It calculates not only in decimal, but
358 in all bases you'll ever want to use (among them Binary).
359 For people on Windows: Start the calculator (start->programs->acces‐
361 sories->calculator) and if necessary click view->scientific. You now
362 have a scientific calculator and can compute in binary or hexadecimal.
363
365 I hope you enjoyed the examples and their descriptions. If you do, help
366 other people by pointing them to this document when they are asking
367 basic questions. They will not only get their answer, but at the same
368 time learn a whole lot more.
369 Alex van den Bogaerdt <alex@ergens.op.het.net>
3711.2.27 2008-02-17 BIN_DEC_HEX(1)