## Sunday, March 12, 2017

### Find the number of bits needed to store an integer, and its binary representation (Python)

By Vasudev Ram

I wrote this post yesterday:

Analyse this Python code snippet

I was initially going to give the solution (to the question asked in that post) today, but then realized that I posted it at the weekend. So, to give a bit of time for anyone to attempt it, including some of my programming students, I decided to post the solution early next week.

But in the meantime, inspired some Q&A in a class I taught, I had the idea of creating this simple program to find the the number of bits needed to represent integers of various sizes (0 to 256, specifically, though the code can easily be modified to do it for any size of int). Note that this is the calculation of the minimum number of bits needed to represent some integers per se, not necessarily the number of bits that Python or any other language actually uses to store those same integers, which can be more than the minimum. This is because, at least in the case of Python, being a dynamic language, most data types have more capabilities than just being data - e.g. ints in Python are objects, so they incur some overhead for being objects (instances of classes, such as having a dictionary of attributes, and so on). The other reason is that data objects in dynamic languages often take up extra pre-allocated space, to store some metadata or to allow for future expansion in the size of the value being stored - e.g. that may or not apply in the case of ints, but it can in the case of lists.

(See this earlier post by me: Exploring sizes of data types in Python for more on this topic.)

Note: the level of this post is targeted towards relative beginners to programming, who might not be too familiar with computer representation of numbers, or even towards somewhat experienced programmers, who are not familiar with that topic either (I've come across some). Believe it or not, I've come across people (ex-colleagues in some cases, as well as others, and developers as well as system administrators) who did not know that a compiled binary for one processor will usually not run on another type of processor [1], even though they may be running the same OS (such as Unix), because their (processor) instructions sets are different. (What's an instruction set? - some of them might ask.) This is like expecting a person who only knows language A to understand something spoken in language B - impossible, at least without some source of help, be it another person or a dictionary.

Having some knowledge in these areas (i.e. system internals or under-the-hood stuff, even at a basic level) is quite useful to have, and almost needed, for many real-life situations, ranging from things like choosing an appropriate data representation for your data, finding and fixing bugs quicker, system integration, data reading / writing / transformation / conversion, to bigger-picture issues like system performance and portability.

[1] Though there are exceptions to that these days, such as fat binaries.

Anyway, end of rant :)

I chose 256 as the upper limit because it is the size (+1) of the highest unsigned integer that can be stored in a single byte, and because values in the range 0 to 255 or 256 are very commonly used in low-level code, such as bit manipulation, assembly or machine language, C, some kinds of data processing (e.g. of many binary file formats), and so on. Of course values of word size (2 bytes / 16 bits) or double-word (4 bytes / 32 bits) are also often used in those areas, but the program can be modified to handle them too.

If you want to get a preview (and a clue) about what is coming up, check this snippet and its output first:
```>>> for item in (0, 1, 2, 4, 8, 16):
...     print item.bit_length()
...
0
1
2
3
4
5```
Hint: Notice that those are all powers of 2 in the tuple above, and correlate that fact with the output values.

Here is the program to find the number of bits needed to store an integer, and its binary representation (Python):
```# int_bit_length_and_binary_repr.py
# Purpose: For integers from 0 to 256, print the number of
# bits needed to represent them, and their values in binary.
# Author: Vasudev Ram
# Website: https://vasudevram.github.io

for an_int in range(0, 256 + 1):
print an_int, "takes", an_int.bit_length(), "bits to represent,",
print "and equals", bin(an_int), "in binary"
```

Before showing the output (because it is long, since I've shown all 257 rows of it:

If you found this post informative, you may also be interested in this earlier one on a related topic:

Converting numeric strings to integers with handrolled code

(I didn't remember to say it in that earlier post, but the image at the top of it is of a roti being rolled out with a rolling pin:)

And here is the output when I run the program:
```\$ python int_bit_length_and_binary_repr.py
0 takes 0 bits to represent, and equals 0b0 in binary
1 takes 1 bits to represent, and equals 0b1 in binary
2 takes 2 bits to represent, and equals 0b10 in binary
3 takes 2 bits to represent, and equals 0b11 in binary
4 takes 3 bits to represent, and equals 0b100 in binary
5 takes 3 bits to represent, and equals 0b101 in binary
6 takes 3 bits to represent, and equals 0b110 in binary
7 takes 3 bits to represent, and equals 0b111 in binary
8 takes 4 bits to represent, and equals 0b1000 in binary
9 takes 4 bits to represent, and equals 0b1001 in binary
10 takes 4 bits to represent, and equals 0b1010 in binary
11 takes 4 bits to represent, and equals 0b1011 in binary
12 takes 4 bits to represent, and equals 0b1100 in binary
13 takes 4 bits to represent, and equals 0b1101 in binary
14 takes 4 bits to represent, and equals 0b1110 in binary
15 takes 4 bits to represent, and equals 0b1111 in binary
16 takes 5 bits to represent, and equals 0b10000 in binary
17 takes 5 bits to represent, and equals 0b10001 in binary
18 takes 5 bits to represent, and equals 0b10010 in binary
19 takes 5 bits to represent, and equals 0b10011 in binary
20 takes 5 bits to represent, and equals 0b10100 in binary
21 takes 5 bits to represent, and equals 0b10101 in binary
22 takes 5 bits to represent, and equals 0b10110 in binary
23 takes 5 bits to represent, and equals 0b10111 in binary
24 takes 5 bits to represent, and equals 0b11000 in binary
25 takes 5 bits to represent, and equals 0b11001 in binary
26 takes 5 bits to represent, and equals 0b11010 in binary
27 takes 5 bits to represent, and equals 0b11011 in binary
28 takes 5 bits to represent, and equals 0b11100 in binary
29 takes 5 bits to represent, and equals 0b11101 in binary
30 takes 5 bits to represent, and equals 0b11110 in binary
31 takes 5 bits to represent, and equals 0b11111 in binary
32 takes 6 bits to represent, and equals 0b100000 in binary
33 takes 6 bits to represent, and equals 0b100001 in binary
34 takes 6 bits to represent, and equals 0b100010 in binary
35 takes 6 bits to represent, and equals 0b100011 in binary
36 takes 6 bits to represent, and equals 0b100100 in binary
37 takes 6 bits to represent, and equals 0b100101 in binary
38 takes 6 bits to represent, and equals 0b100110 in binary
39 takes 6 bits to represent, and equals 0b100111 in binary
40 takes 6 bits to represent, and equals 0b101000 in binary
41 takes 6 bits to represent, and equals 0b101001 in binary
42 takes 6 bits to represent, and equals 0b101010 in binary
43 takes 6 bits to represent, and equals 0b101011 in binary
44 takes 6 bits to represent, and equals 0b101100 in binary
45 takes 6 bits to represent, and equals 0b101101 in binary
46 takes 6 bits to represent, and equals 0b101110 in binary
47 takes 6 bits to represent, and equals 0b101111 in binary
48 takes 6 bits to represent, and equals 0b110000 in binary
49 takes 6 bits to represent, and equals 0b110001 in binary
50 takes 6 bits to represent, and equals 0b110010 in binary
51 takes 6 bits to represent, and equals 0b110011 in binary
52 takes 6 bits to represent, and equals 0b110100 in binary
53 takes 6 bits to represent, and equals 0b110101 in binary
54 takes 6 bits to represent, and equals 0b110110 in binary
55 takes 6 bits to represent, and equals 0b110111 in binary
56 takes 6 bits to represent, and equals 0b111000 in binary
57 takes 6 bits to represent, and equals 0b111001 in binary
58 takes 6 bits to represent, and equals 0b111010 in binary
59 takes 6 bits to represent, and equals 0b111011 in binary
60 takes 6 bits to represent, and equals 0b111100 in binary
61 takes 6 bits to represent, and equals 0b111101 in binary
62 takes 6 bits to represent, and equals 0b111110 in binary
63 takes 6 bits to represent, and equals 0b111111 in binary
64 takes 7 bits to represent, and equals 0b1000000 in binary
65 takes 7 bits to represent, and equals 0b1000001 in binary
66 takes 7 bits to represent, and equals 0b1000010 in binary
67 takes 7 bits to represent, and equals 0b1000011 in binary
68 takes 7 bits to represent, and equals 0b1000100 in binary
69 takes 7 bits to represent, and equals 0b1000101 in binary
70 takes 7 bits to represent, and equals 0b1000110 in binary
71 takes 7 bits to represent, and equals 0b1000111 in binary
72 takes 7 bits to represent, and equals 0b1001000 in binary
73 takes 7 bits to represent, and equals 0b1001001 in binary
74 takes 7 bits to represent, and equals 0b1001010 in binary
75 takes 7 bits to represent, and equals 0b1001011 in binary
76 takes 7 bits to represent, and equals 0b1001100 in binary
77 takes 7 bits to represent, and equals 0b1001101 in binary
78 takes 7 bits to represent, and equals 0b1001110 in binary
79 takes 7 bits to represent, and equals 0b1001111 in binary
80 takes 7 bits to represent, and equals 0b1010000 in binary
81 takes 7 bits to represent, and equals 0b1010001 in binary
82 takes 7 bits to represent, and equals 0b1010010 in binary
83 takes 7 bits to represent, and equals 0b1010011 in binary
84 takes 7 bits to represent, and equals 0b1010100 in binary
85 takes 7 bits to represent, and equals 0b1010101 in binary
86 takes 7 bits to represent, and equals 0b1010110 in binary
87 takes 7 bits to represent, and equals 0b1010111 in binary
88 takes 7 bits to represent, and equals 0b1011000 in binary
89 takes 7 bits to represent, and equals 0b1011001 in binary
90 takes 7 bits to represent, and equals 0b1011010 in binary
91 takes 7 bits to represent, and equals 0b1011011 in binary
92 takes 7 bits to represent, and equals 0b1011100 in binary
93 takes 7 bits to represent, and equals 0b1011101 in binary
94 takes 7 bits to represent, and equals 0b1011110 in binary
95 takes 7 bits to represent, and equals 0b1011111 in binary
96 takes 7 bits to represent, and equals 0b1100000 in binary
97 takes 7 bits to represent, and equals 0b1100001 in binary
98 takes 7 bits to represent, and equals 0b1100010 in binary
99 takes 7 bits to represent, and equals 0b1100011 in binary
100 takes 7 bits to represent, and equals 0b1100100 in binary
101 takes 7 bits to represent, and equals 0b1100101 in binary
102 takes 7 bits to represent, and equals 0b1100110 in binary
103 takes 7 bits to represent, and equals 0b1100111 in binary
104 takes 7 bits to represent, and equals 0b1101000 in binary
105 takes 7 bits to represent, and equals 0b1101001 in binary
106 takes 7 bits to represent, and equals 0b1101010 in binary
107 takes 7 bits to represent, and equals 0b1101011 in binary
108 takes 7 bits to represent, and equals 0b1101100 in binary
109 takes 7 bits to represent, and equals 0b1101101 in binary
110 takes 7 bits to represent, and equals 0b1101110 in binary
111 takes 7 bits to represent, and equals 0b1101111 in binary
112 takes 7 bits to represent, and equals 0b1110000 in binary
113 takes 7 bits to represent, and equals 0b1110001 in binary
114 takes 7 bits to represent, and equals 0b1110010 in binary
115 takes 7 bits to represent, and equals 0b1110011 in binary
116 takes 7 bits to represent, and equals 0b1110100 in binary
117 takes 7 bits to represent, and equals 0b1110101 in binary
118 takes 7 bits to represent, and equals 0b1110110 in binary
119 takes 7 bits to represent, and equals 0b1110111 in binary
120 takes 7 bits to represent, and equals 0b1111000 in binary
121 takes 7 bits to represent, and equals 0b1111001 in binary
122 takes 7 bits to represent, and equals 0b1111010 in binary
123 takes 7 bits to represent, and equals 0b1111011 in binary
124 takes 7 bits to represent, and equals 0b1111100 in binary
125 takes 7 bits to represent, and equals 0b1111101 in binary
126 takes 7 bits to represent, and equals 0b1111110 in binary
127 takes 7 bits to represent, and equals 0b1111111 in binary
128 takes 8 bits to represent, and equals 0b10000000 in binary
129 takes 8 bits to represent, and equals 0b10000001 in binary
130 takes 8 bits to represent, and equals 0b10000010 in binary
131 takes 8 bits to represent, and equals 0b10000011 in binary
132 takes 8 bits to represent, and equals 0b10000100 in binary
133 takes 8 bits to represent, and equals 0b10000101 in binary
134 takes 8 bits to represent, and equals 0b10000110 in binary
135 takes 8 bits to represent, and equals 0b10000111 in binary
136 takes 8 bits to represent, and equals 0b10001000 in binary
137 takes 8 bits to represent, and equals 0b10001001 in binary
138 takes 8 bits to represent, and equals 0b10001010 in binary
139 takes 8 bits to represent, and equals 0b10001011 in binary
140 takes 8 bits to represent, and equals 0b10001100 in binary
141 takes 8 bits to represent, and equals 0b10001101 in binary
142 takes 8 bits to represent, and equals 0b10001110 in binary
143 takes 8 bits to represent, and equals 0b10001111 in binary
144 takes 8 bits to represent, and equals 0b10010000 in binary
145 takes 8 bits to represent, and equals 0b10010001 in binary
146 takes 8 bits to represent, and equals 0b10010010 in binary
147 takes 8 bits to represent, and equals 0b10010011 in binary
148 takes 8 bits to represent, and equals 0b10010100 in binary
149 takes 8 bits to represent, and equals 0b10010101 in binary
150 takes 8 bits to represent, and equals 0b10010110 in binary
151 takes 8 bits to represent, and equals 0b10010111 in binary
152 takes 8 bits to represent, and equals 0b10011000 in binary
153 takes 8 bits to represent, and equals 0b10011001 in binary
154 takes 8 bits to represent, and equals 0b10011010 in binary
155 takes 8 bits to represent, and equals 0b10011011 in binary
156 takes 8 bits to represent, and equals 0b10011100 in binary
157 takes 8 bits to represent, and equals 0b10011101 in binary
158 takes 8 bits to represent, and equals 0b10011110 in binary
159 takes 8 bits to represent, and equals 0b10011111 in binary
160 takes 8 bits to represent, and equals 0b10100000 in binary
161 takes 8 bits to represent, and equals 0b10100001 in binary
162 takes 8 bits to represent, and equals 0b10100010 in binary
163 takes 8 bits to represent, and equals 0b10100011 in binary
164 takes 8 bits to represent, and equals 0b10100100 in binary
165 takes 8 bits to represent, and equals 0b10100101 in binary
166 takes 8 bits to represent, and equals 0b10100110 in binary
167 takes 8 bits to represent, and equals 0b10100111 in binary
168 takes 8 bits to represent, and equals 0b10101000 in binary
169 takes 8 bits to represent, and equals 0b10101001 in binary
170 takes 8 bits to represent, and equals 0b10101010 in binary
171 takes 8 bits to represent, and equals 0b10101011 in binary
172 takes 8 bits to represent, and equals 0b10101100 in binary
173 takes 8 bits to represent, and equals 0b10101101 in binary
174 takes 8 bits to represent, and equals 0b10101110 in binary
175 takes 8 bits to represent, and equals 0b10101111 in binary
176 takes 8 bits to represent, and equals 0b10110000 in binary
177 takes 8 bits to represent, and equals 0b10110001 in binary
178 takes 8 bits to represent, and equals 0b10110010 in binary
179 takes 8 bits to represent, and equals 0b10110011 in binary
180 takes 8 bits to represent, and equals 0b10110100 in binary
181 takes 8 bits to represent, and equals 0b10110101 in binary
182 takes 8 bits to represent, and equals 0b10110110 in binary
183 takes 8 bits to represent, and equals 0b10110111 in binary
184 takes 8 bits to represent, and equals 0b10111000 in binary
185 takes 8 bits to represent, and equals 0b10111001 in binary
186 takes 8 bits to represent, and equals 0b10111010 in binary
187 takes 8 bits to represent, and equals 0b10111011 in binary
188 takes 8 bits to represent, and equals 0b10111100 in binary
189 takes 8 bits to represent, and equals 0b10111101 in binary
190 takes 8 bits to represent, and equals 0b10111110 in binary
191 takes 8 bits to represent, and equals 0b10111111 in binary
192 takes 8 bits to represent, and equals 0b11000000 in binary
193 takes 8 bits to represent, and equals 0b11000001 in binary
194 takes 8 bits to represent, and equals 0b11000010 in binary
195 takes 8 bits to represent, and equals 0b11000011 in binary
196 takes 8 bits to represent, and equals 0b11000100 in binary
197 takes 8 bits to represent, and equals 0b11000101 in binary
198 takes 8 bits to represent, and equals 0b11000110 in binary
199 takes 8 bits to represent, and equals 0b11000111 in binary
200 takes 8 bits to represent, and equals 0b11001000 in binary
201 takes 8 bits to represent, and equals 0b11001001 in binary
202 takes 8 bits to represent, and equals 0b11001010 in binary
203 takes 8 bits to represent, and equals 0b11001011 in binary
204 takes 8 bits to represent, and equals 0b11001100 in binary
205 takes 8 bits to represent, and equals 0b11001101 in binary
206 takes 8 bits to represent, and equals 0b11001110 in binary
207 takes 8 bits to represent, and equals 0b11001111 in binary
208 takes 8 bits to represent, and equals 0b11010000 in binary
209 takes 8 bits to represent, and equals 0b11010001 in binary
210 takes 8 bits to represent, and equals 0b11010010 in binary
211 takes 8 bits to represent, and equals 0b11010011 in binary
212 takes 8 bits to represent, and equals 0b11010100 in binary
213 takes 8 bits to represent, and equals 0b11010101 in binary
214 takes 8 bits to represent, and equals 0b11010110 in binary
215 takes 8 bits to represent, and equals 0b11010111 in binary
216 takes 8 bits to represent, and equals 0b11011000 in binary
217 takes 8 bits to represent, and equals 0b11011001 in binary
218 takes 8 bits to represent, and equals 0b11011010 in binary
219 takes 8 bits to represent, and equals 0b11011011 in binary
220 takes 8 bits to represent, and equals 0b11011100 in binary
221 takes 8 bits to represent, and equals 0b11011101 in binary
222 takes 8 bits to represent, and equals 0b11011110 in binary
223 takes 8 bits to represent, and equals 0b11011111 in binary
224 takes 8 bits to represent, and equals 0b11100000 in binary
225 takes 8 bits to represent, and equals 0b11100001 in binary
226 takes 8 bits to represent, and equals 0b11100010 in binary
227 takes 8 bits to represent, and equals 0b11100011 in binary
228 takes 8 bits to represent, and equals 0b11100100 in binary
229 takes 8 bits to represent, and equals 0b11100101 in binary
230 takes 8 bits to represent, and equals 0b11100110 in binary
231 takes 8 bits to represent, and equals 0b11100111 in binary
232 takes 8 bits to represent, and equals 0b11101000 in binary
233 takes 8 bits to represent, and equals 0b11101001 in binary
234 takes 8 bits to represent, and equals 0b11101010 in binary
235 takes 8 bits to represent, and equals 0b11101011 in binary
236 takes 8 bits to represent, and equals 0b11101100 in binary
237 takes 8 bits to represent, and equals 0b11101101 in binary
238 takes 8 bits to represent, and equals 0b11101110 in binary
239 takes 8 bits to represent, and equals 0b11101111 in binary
240 takes 8 bits to represent, and equals 0b11110000 in binary
241 takes 8 bits to represent, and equals 0b11110001 in binary
242 takes 8 bits to represent, and equals 0b11110010 in binary
243 takes 8 bits to represent, and equals 0b11110011 in binary
244 takes 8 bits to represent, and equals 0b11110100 in binary
245 takes 8 bits to represent, and equals 0b11110101 in binary
246 takes 8 bits to represent, and equals 0b11110110 in binary
247 takes 8 bits to represent, and equals 0b11110111 in binary
248 takes 8 bits to represent, and equals 0b11111000 in binary
249 takes 8 bits to represent, and equals 0b11111001 in binary
250 takes 8 bits to represent, and equals 0b11111010 in binary
251 takes 8 bits to represent, and equals 0b11111011 in binary
252 takes 8 bits to represent, and equals 0b11111100 in binary
253 takes 8 bits to represent, and equals 0b11111101 in binary
254 takes 8 bits to represent, and equals 0b11111110 in binary
255 takes 8 bits to represent, and equals 0b11111111 in binary
256 takes 9 bits to represent, and equals 0b100000000 in binary
```
If you look carefully at the values in the output, you can notice some interesting bit patterns, e.g.:

1. Look at the bit patterns for the values of (2 ** n) - 1, i.e. values one less than each power of 2.
2. The same for the values halfway between any two adjacent powers of 2.

Notice any patterns or regularities?

The number columns in the output should really be right-justified, and the repeated (and hence redundant) text in between numbers in the rows should be replaced by a header line at the top, but this time, I've leaving this as an elementary exercise for the reader :)

Enjoy.

- Vasudev Ram - Online Python training and consulting

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Vasudev Ram said...

Notice any possible error in the output?

Vasudev Ram said...

Another observation:

In this Python program, I used the bit_length() method and the bin() function to calculate the number of bits needed for the integer, and the binary value of the int, respectively.

In some other languages, we might have to do at least the former manually, through some bit-manipulation code, and maybe the latter too via some additional code, unless the language has the equivalent of C's printf but for binary format.

Vasudev Ram said...

TIL about the unit shannon, named after Claude Shannon, father of information theory:

https://en.wikipedia.org/wiki/Shannon_(unit)

https://en.wikipedia.org/wiki/Claude_Shannon

And check out his inventions section at the above page :)