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.
7
9 Most people use the decimal numbering system. This system uses ten
10 symbols 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
35 position before (to the left) is incremented (from 0 to 1). Then number
36 9 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
70 numbering system or base. Normally, if we do not explicitly specify
71 the numbering system used, we implicitly use the decimal system. If we
72 want to use any other numbering system, we'll have to make that clear.
73 There are a few widely adopted methods to do so. One common form is to
74 write 1010(2) which means that you wrote down a number in its binary
75 representation. It is the number ten. If you would write 1010 without
76 specifying the base, the number is interpreted as one thousand and ten
77 using base 10.
78 In books, another form is common. It uses subscripts (little
80 characters, more or less in between two rows). You can leave out the
81 parentheses in that case and write down the number in normal characters
82 followed 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
88 numbers are written down in series of four, eight or sixteen depending
89 on the context.
90 We can use the binary form when talking to computers
92 (...programming...), but the numbers will have large representations.
93 The number 65'535 (often in the decimal system a ' is used to separate
94 blocks of 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
99 alphabetic characters. We get 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D,
100 E and F. This system is chosen because the hexadecimal form can be
101 converted 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
141 represent the eight bits and this means that you'd have to use one
142 complete digit to represent only two bits (2+3+3=8). This is a waste.
143 For hexadecimal digits, you need only two digits which are used
144 completely:
145 (2) (8) (10) (16)
147 11111111 377 255 FF
148 You can see why binary and hexadecimal can be converted quickly: For
150 each hexadecimal digit there are exactly four binary digits. Take a
151 binary number: take four digits from the right and make a hexadecimal
152 digit from it (see the table above). Repeat this until there are no
153 more digits. And the other way around: Take a hexadecimal number. For
154 each digit, write down its binary equivalent.
155 Computers (or rather the parsers running on them) would have a hard
157 time converting a number like 1234(16). Therefore hexadecimal numbers
158 are specified with a prefix. This prefix depends on the language you're
159 writing in. Some of the prefixes are "0x" for C, "$" for Pascal, "#"
160 for HTML. It is common to assume that if a number starts with a zero,
161 it is octal. It does not matter what is used as long as you know what
162 it is. I will use "0x" for hexadecimal, "%" for binary and "0" for
163 octal. The following numbers are all the same, just their
164 representation (base) is different: 021 0x11 17 %00010001
165 To do arithmetics and conversions you need to understand one more
167 thing. It is something you already know but perhaps you do not "see"
168 it yet:
169 If you write down 1234, (no prefix, so it is decimal) you are talking
171 about the number one thousand, two hundred and thirty four. In sort of
172 a formula:
173 1 * 1000 = 1000
175 2 * 100 = 200
176 3 * 10 = 30
177 4 * 1 = 4
178 This can also be written as:
180 1 * 10^3
182 2 * 10^2
183 3 * 10^1
184 4 * 10^0
185 where ^ means "to the power of".
187 We are using the base 10, and the positions 0,1,2 and 3. The right-
189 most position should NOT be multiplied with 10. The second from the
190 right should be multiplied one time with 10. The third from the right
191 is multiplied with 10 two times. This continues for whatever positions
192 are used.
193 It is the same in all other representations:
195 0x1234 will be
197 1 * 16^3
199 2 * 16^2
200 3 * 16^1
201 4 * 16^0
202 01234 would be
204 1 * 8^3
206 2 * 8^2
207 3 * 8^1
208 4 * 8^0
209 This example can not be done for binary as that system only uses two
211 symbols. Another example:
212 %1010 would be
214 1 * 2^3
216 0 * 2^2
217 1 * 2^1
218 0 * 2^0
219 It would have been easier to convert it to its hexadecimal form and
221 just translate %1010 into 0xA. After a while you get used to it. You
222 will not need to do any calculations anymore, but just know that 0xA
223 means 10.
224 To convert a decimal number into a hexadecimal you could use the next
226 method. It will take some time to be able to do the estimates, but it
227 will be easier when you use the system more frequently. We'll look at
228 yet another way afterwards.
229 First you need to know how many positions will be used in the other
231 system. To do so, you need to know the maximum numbers you'll be using.
232 Well, that's not as hard as it looks. In decimal, the maximum number
233 that you can form with two digits is "99". The maximum for three:
234 "999". The next number would need an extra position. Reverse this idea
235 and you will see that the number can be found by taking 10^3 (10*10*10
236 is 1000) minus 1 or 10^2 minus one.
237 This can be done for hexadecimal as well:
239 16^4 = 0x10000 = 65536
241 16^3 = 0x1000 = 4096
242 16^2 = 0x100 = 256
243 16^1 = 0x10 = 16
244 If a number is smaller than 65'536 it will fit in four positions. If
246 the number is bigger than 4'095, you must use position 4. How many
247 times you can subtract 4'096 from the number without going below zero
248 is the first digit you write down. This will always be a number from 1
249 to 15 (0x1 to 0xF). Do the same for the other positions.
250 Let's try with 41'029. It is smaller than 16^4 but bigger than 16^3-1.
252 This means that we have to use four positions. We can subtract 16^3
253 from 41'029 ten times without going below zero. The left-most digit
254 will therefore be "A", so we have 0xA????. The number is reduced to
255 41'029 - 10*4'096 = 41'029-40'960 = 69. 69 is smaller than 16^3 but
256 not bigger than 16^2-1. The second digit is therefore "0" and we now
257 have 0xA0??. 69 is smaller than 16^2 and bigger than 16^1-1. We can
258 subtract 16^1 (which is just plain 16) four times and write down "4" to
259 get 0xA04?. Subtract 64 from 69 (69 - 4*16) and the last digit is 5
260 --> 0xA045.
261 The other method builds up the number from the right. Let's try 41'029
263 again. Divide by 16 and do not use fractions (only whole numbers).
264 41'029 / 16 is 2'564 with a remainder of 5. Write down 5.
266 2'564 / 16 is 160 with a remainder of 4. Write the 4 before the 5.
267 160 / 16 is 10 with no remainder. Prepend 45 with 0.
268 10 / 16 is below one. End here and prepend 0xA. End up with 0xA045.
269 Which method to use is up to you. Use whatever works for you. I use
271 them both without being able to tell what method I use in each case, it
272 just depends on the number, I think. Fact is, some numbers will occur
273 frequently while programming. If the number is close to one I am
274 familiar with, then I will use the first method (like 32'770 which is
275 into 32'768 + 2 and I just know that it is 0x8000 + 0x2 = 0x8002).
276 For binary the same approach can be used. The base is 2 and not 16, and
278 the number of positions will grow rapidly. Using the second method has
279 the advantage that you can see very easily if you should write down a
280 zero or a one: if you divide by two the remainder will be zero if it is
281 an even number and one if it is an odd number:
282 41029 / 2 = 20514 remainder 1
284 20514 / 2 = 10257 remainder 0
285 10257 / 2 = 5128 remainder 1
286 5128 / 2 = 2564 remainder 0
287 2564 / 2 = 1282 remainder 0
288 1282 / 2 = 641 remainder 0
289 641 / 2 = 320 remainder 1
290 320 / 2 = 160 remainder 0
291 160 / 2 = 80 remainder 0
292 80 / 2 = 40 remainder 0
293 40 / 2 = 20 remainder 0
294 20 / 2 = 10 remainder 0
295 10 / 2 = 5 remainder 0
296 5 / 2 = 2 remainder 1
297 2 / 2 = 1 remainder 0
298 1 / 2 below 0 remainder 1
299 Write down the results from right to left: %1010000001000101
301 Group by four:
303 %1010000001000101
305 %101000000100 0101
306 %10100000 0100 0101
307 %1010 0000 0100 0101
308 Convert into hexadecimal: 0xA045
310 Group %1010000001000101 by three and convert into octal:
312 %1010000001000101
314 %1010000001000 101
315 %1010000001 000 101
316 %1010000 001 000 101
317 %1010 000 001 000 101
318 %1 010 000 001 000 101
319 %001 010 000 001 000 101
320 1 2 0 1 0 5 --> 0120105
321 So: %1010000001000101 = 0120105 = 0xA045 = 41029
323 Or: 1010000001000101(2) = 120105(8) = A045(16) = 41029(10)
324 Or: 1010000001000101(2) = 120105(8) = A045(16) = 41029
325 At first while adding numbers, you'll convert them to their decimal
327 form and then back into their original form after doing the addition.
328 If you use the other numbering system often, you will see that you'll
329 be able to do arithmetics directly in the base that is used. In any
330 representation it is the same, add the numbers on the right, write down
331 the right-most digit from the result, remember the other digits and use
332 them in the next round. Continue with the second digit from the right
333 and so on:
334 %1010 + %0111 --> 10 + 7 --> 17 --> %00010001
336 will become
338 %1010
340 %0111 +
341 ||||
342 |||+-- add 0 + 1, result is 1, nothing to remember
343 ||+--- add 1 + 1, result is %10, write down 0 and remember 1
344 |+---- add 0 + 1 + 1(remembered), result = 0, remember 1
345 +----- add 1 + 0 + 1(remembered), result = 0, remember 1
346 nothing to add, 1 remembered, result = 1
347 --------
348 %10001 is the result, I like to write it as %00010001
349 For low values, try to do the calculations yourself, then check them
351 with a calculator. The more you do the calculations yourself, the more
352 you'll find that you didn't make mistakes. In the end, you'll do
353 calculi in other bases as easily as you do them in decimal.
354 When the numbers get bigger, you'll have to realize that a computer is
356 not called a computer just to have a nice name. There are many
357 different calculators available, use them. For Unix you could use "bc"
358 which is short for Binary Calculator. It calculates not only in
359 decimal, but in all bases you'll ever want to use (among them Binary).
360 For people on Windows: Start the calculator
362 (start->programs->accessories->calculator) and if necessary click
363 view->scientific. You now have a scientific calculator and can compute
364 in binary or hexadecimal.
365
367 I hope you enjoyed the examples and their descriptions. If you do, help
368 other people by pointing them to this document when they are asking
369 basic questions. They will not only get their answer, but at the same
370 time learn a whole lot more.
371 Alex van den Bogaerdt <alex@vandenbogaerdt.nl>
3731.8.0 2022-03-14 BIN_DEC_HEX(1)