I'm in a computer systems course and have been struggling, in part, with two's complement. I want to understand it, but everything I've read hasn't brought the picture together for me. I've read the Wikipedia article and various other articles, including my text book.
What is two's complement, how can we use it and how can it affect numbers during operations like casts (from signed to unsigned and vice versa), bit-wise operations and bit-shift operations?
Two's complement is a clever way of storing integers so that common math problems are very simple to implement.
To understand, you have to think of the numbers in binary.
It basically says,
for zero, use all 0's.
for positive integers, start counting up, with a maximum of 2(number of bits - 1)-1.
for negative integers, do exactly the same thing, but switch the role of 0's and 1's and count down (so instead of starting with 0000, start with 1111 - that's the "complement" part).
Let's try it with a mini-byte of 4 bits (we'll call it a nibble - 1/2 a byte).
0000 - zero
0001 - one
0010 - two
0011 - three
0100 to 0111 - four to seven
That's as far as we can go in positives. 23-1 = 7.
For negatives:
1111 - negative one
1110 - negative two
1101 - negative three
1100 to 1000 - negative four to negative eight
Note that you get one extra value for negatives (1000 = -8) that you don't for positives. This is because 0000 is used for zero. This can be considered as Number Line of computers.
Distinguishing between positive and negative numbers
Doing this, the first bit gets the role of the "sign" bit, as it can be used to distinguish between nonnegative and negative decimal values. If the most significant bit is 1, then the binary can be said to be negative, where as if the most significant bit (the leftmost) is 0, you can say the decimal value is nonnegative.
"Sign-magnitude" negative numbers just have the sign bit flipped of their positive counterparts, but this approach has to deal with interpreting 1000 (one 1 followed by all 0s) as "negative zero" which is confusing.
"Ones' complement" negative numbers are just the bit-complement of their positive counterparts, which also leads to a confusing "negative zero" with 1111 (all ones).
You will likely not have to deal with Ones' Complement or Sign-Magnitude integer representations unless you are working very close to the hardware.
I wonder if it could be explained any better than the Wikipedia article.
The basic problem that you are trying to solve with two's complement representation is the problem of storing negative integers.
First, consider an unsigned integer stored in 4 bits. You can have the following
0000 = 0
0001 = 1
0010 = 2
...
1111 = 15
These are unsigned because there is no indication of whether they are negative or positive.
Sign Magnitude and Excess Notation
To store negative numbers you can try a number of things. First, you can use sign magnitude notation which assigns the first bit as a sign bit to represent +/- and the remaining bits to represent the magnitude. So using 4 bits again and assuming that 1 means - and 0 means + then you have
0000 = +0
0001 = +1
0010 = +2
...
1000 = -0
1001 = -1
1111 = -7
So, you see the problem there? We have positive and negative 0. The bigger problem is adding and subtracting binary numbers. The circuits to add and subtract using sign magnitude will be very complex.
What is
0010
1001 +
----
?
Another system is excess notation. You can store negative numbers, you get rid of the two zeros problem but addition and subtraction remains difficult.
So along comes two's complement. Now you can store positive and negative integers and perform arithmetic with relative ease. There are a number of methods to convert a number into two's complement. Here's one.
Convert Decimal to Two's Complement
Convert the number to binary (ignore the sign for now)
e.g. 5 is 0101 and -5 is 0101
If the number is a positive number then you are done.
e.g. 5 is 0101 in binary using two's complement notation.
If the number is negative then
3.1 find the complement (invert 0's and 1's)
e.g. -5 is 0101 so finding the complement is 1010
3.2 Add 1 to the complement 1010 + 1 = 1011.
Therefore, -5 in two's complement is 1011.
So, what if you wanted to do 2 + (-3) in binary? 2 + (-3) is -1.
What would you have to do if you were using sign magnitude to add these numbers? 0010 + 1101 = ?
Using two's complement consider how easy it would be.
2 = 0010
-3 = 1101 +
-------------
-1 = 1111
Converting Two's Complement to Decimal
Converting 1111 to decimal:
The number starts with 1, so it's negative, so we find the complement of 1111, which is 0000.
Add 1 to 0000, and we obtain 0001.
Convert 0001 to decimal, which is 1.
Apply the sign = -1.
Tada!
Like most explanations I've seen, the ones above are clear about how to work with 2's complement, but don't really explain what they are mathematically. I'll try to do that, for integers at least, and I'll cover some background that's probably familiar first.
Recall how it works for decimal: 2345 is a way of writing 2 × 103 + 3 × 102 + 4 × 101 + 5 × 100.
In the same way, binary is a way of writing numbers using just 0 and 1 following the same general idea, but replacing those 10s above with 2s. Then in binary, 1111is a way of writing 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20and if you work it out, that turns out to equal 15 (base 10). That's because it is 8+4+2+1 = 15.
This is all well and good for positive numbers. It even works for negative numbers if you're willing to just stick a minus sign in front of them, as humans do with decimal numbers. That can even be done in computers, sort of, but I haven't seen such a computer since the early 1970's. I'll leave the reasons for a different discussion.
For computers it turns out to be more efficient to use a complement representation for negative numbers. And here's something that is often overlooked. Complement notations involve some kind of reversal of the digits of the number, even the implied zeroes that come before a normal positive number. That's awkward, because the question arises: all of them? That could be an infinite number of digits to be considered.
Fortunately, computers don't represent infinities. Numbers are constrained to a particular length (or width, if you prefer). So let's return to positive binary numbers, but with a particular size. I'll use 8 digits ("bits") for these examples. So our binary number would really be 00001111or 0 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20
To form the 2's complement negative, we first complement all the (binary) digits to form 11110000and add 1 to form 11110001but how are we to understand that to mean -15?
The answer is that we change the meaning of the high-order bit (the leftmost one). This bit will be a 1 for all negative numbers. The change will be to change the sign of its contribution to the value of the number it appears in. So now our 11110001 is understood to represent -1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 1 × 20Notice that "-" in front of that expression? It means that the sign bit carries the weight -27, that is -128 (base 10). All the other positions retain the same weight they had in unsigned binary numbers.
Working out our -15, it is -128 + 64 + 32 + 16 + 1 Try it on your calculator. it's -15.
Of the three main ways that I've seen negative numbers represented in computers, 2's complement wins hands down for convenience in general use. It has an oddity, though. Since it's binary, there have to be an even number of possible bit combinations. Each positive number can be paired with its negative, but there's only one zero. Negating a zero gets you zero. So there's one more combination, the number with 1 in the sign bit and 0 everywhere else. The corresponding positive number would not fit in the number of bits being used.
What's even more odd about this number is that if you try to form its positive by complementing and adding one, you get the same negative number back. It seems natural that zero would do this, but this is unexpected and not at all the behavior we're used to because computers aside, we generally think of an unlimited supply of digits, not this fixed-length arithmetic.
This is like the tip of an iceberg of oddities. There's more lying in wait below the surface, but that's enough for this discussion. You could probably find more if you research "overflow" for fixed-point arithmetic. If you really want to get into it, you might also research "modular arithmetic".
2's complement is very useful for finding the value of a binary, however I thought of a much more concise way of solving such a problem(never seen anyone else publish it):
take a binary, for example: 1101 which is [assuming that space "1" is the sign] equal to -3.
using 2's complement we would do this...flip 1101 to 0010...add 0001 + 0010 ===> gives us 0011. 0011 in positive binary = 3. therefore 1101 = -3!
What I realized:
instead of all the flipping and adding, you can just do the basic method for solving for a positive binary(lets say 0101) is (23 * 0) + (22 * 1) + (21 * 0) + (20 * 1) = 5.
Do exactly the same concept with a negative!(with a small twist)
take 1101, for example:
for the first number instead of 23 * 1 = 8 , do -(23 * 1) = -8.
then continue as usual, doing -8 + (22 * 1) + (21 * 0) + (20 * 1) = -3
Imagine that you have a finite number of bits/trits/digits/whatever. You define 0 as all digits being 0, and count upwards naturally:
00
01
02
..
Eventually you will overflow.
98
99
00
We have two digits and can represent all numbers from 0 to 100. All those numbers are positive! Suppose we want to represent negative numbers too?
What we really have is a cycle. The number before 2 is 1. The number before 1 is 0. The number before 0 is... 99.
So, for simplicity, let's say that any number over 50 is negative. "0" through "49" represent 0 through 49. "99" is -1, "98" is -2, ... "50" is -50.
This representation is ten's complement. Computers typically use two's complement, which is the same except using bits instead of digits.
The nice thing about ten's complement is that addition just works. You do not need to do anything special to add positive and negative numbers!
I read a fantastic explanation on Reddit by jng, using the odometer as an analogy.
It is a useful convention. The same circuits and logic operations that
add / subtract positive numbers in binary still work on both positive
and negative numbers if using the convention, that's why it's so
useful and omnipresent.
Imagine the odometer of a car, it rolls around at (say) 99999. If you
increment 00000 you get 00001. If you decrement 00000, you get 99999
(due to the roll-around). If you add one back to 99999 it goes back to
00000. So it's useful to decide that 99999 represents -1. Likewise, it is very useful to decide that 99998 represents -2, and so on. You have
to stop somewhere, and also by convention, the top half of the numbers
are deemed to be negative (50000-99999), and the bottom half positive
just stand for themselves (00000-49999). As a result, the top digit
being 5-9 means the represented number is negative, and it being 0-4
means the represented is positive - exactly the same as the top bit
representing sign in a two's complement binary number.
Understanding this was hard for me too. Once I got it and went back to
re-read the books articles and explanations (there was no internet
back then), it turned out a lot of those describing it didn't really
understand it. I did write a book teaching assembly language after
that (which did sell quite well for 10 years).
Two complement is found out by adding one to 1'st complement of the given number.
Lets say we have to find out twos complement of 10101 then find its ones complement, that is, 01010 add 1 to this result, that is, 01010+1=01011, which is the final answer.
Lets get the answer 10 – 12 in binary form using 8 bits:
What we will really do is 10 + (-12)
We need to get the compliment part of 12 to subtract it from 10.
12 in binary is 00001100.
10 in binary is 00001010.
To get the compliment part of 12 we just reverse all the bits then add 1.
12 in binary reversed is 11110011. This is also the Inverse code (one's complement).
Now we need to add one, which is now 11110100.
So 11110100 is the compliment of 12! Easy when you think of it this way.
Now you can solve the above question of 10 - 12 in binary form.
00001010
11110100
-----------------
11111110
Looking at the two's complement system from a math point of view it really makes sense. In ten's complement, the idea is to essentially 'isolate' the difference.
Example: 63 - 24 = x
We add the complement of 24 which is really just (100 - 24). So really, all we are doing is adding 100 on both sides of the equation.
Now the equation is: 100 + 63 - 24 = x + 100, that is why we remove the 100 (or 10 or 1000 or whatever).
Due to the inconvenient situation of having to subtract one number from a long chain of zeroes, we use a 'diminished radix complement' system, in the decimal system, nine's complement.
When we are presented with a number subtracted from a big chain of nines, we just need to reverse the numbers.
Example: 99999 - 03275 = 96724
That is the reason, after nine's complement, we add 1. As you probably know from childhood math, 9 becomes 10 by 'stealing' 1. So basically it's just ten's complement that takes 1 from the difference.
In Binary, two's complement is equatable to ten's complement, while one's complement to nine's complement. The primary difference is that instead of trying to isolate the difference with powers of ten (adding 10, 100, etc. into the equation) we are trying to isolate the difference with powers of two.
It is for this reason that we invert the bits. Just like how our minuend is a chain of nines in decimal, our minuend is a chain of ones in binary.
Example: 111111 - 101001 = 010110
Because chains of ones are 1 below a nice power of two, they 'steal' 1 from the difference like nine's do in decimal.
When we are using negative binary number's, we are really just saying:
0000 - 0101 = x
1111 - 0101 = 1010
1111 + 0000 - 0101 = x + 1111
In order to 'isolate' x, we need to add 1 because 1111 is one away from 10000 and we remove the leading 1 because we just added it to the original difference.
1111 + 1 + 0000 - 0101 = x + 1111 + 1
10000 + 0000 - 0101 = x + 10000
Just remove 10000 from both sides to get x, it's basic algebra.
The word complement derives from completeness. In the decimal world the numerals 0 through 9 provide a complement (complete set) of numerals or numeric symbols to express all decimal numbers. In the binary world the numerals 0 and 1 provide a complement of numerals to express all binary numbers. In fact The symbols 0 and 1 must be used to represent everything (text, images, etc) as well as positive (0) and negative (1).
In our world the blank space to the left of number is considered as zero:
35=035=000000035.
In a computer storage location there is no blank space. All bits (binary digits) must be either 0 or 1. To efficiently use memory numbers may be stored as 8 bit, 16 bit, 32 bit, 64 bit, 128 bit representations. When a number that is stored as an 8 bit number is transferred to a 16 bit location the sign and magnitude (absolute value) must remain the same. Both 1's complement and 2's complement representations facilitate this.
As a noun:
Both 1's complement and 2's complement are binary representations of signed quantities where the most significant bit (the one on the left) is the sign bit. 0 is for positive and 1 is for negative.
2s complement does not mean negative. It means a signed quantity. As in decimal the magnitude is represented as the positive quantity. The structure uses sign extension to preserve the quantity when promoting to a register [] with more bits:
[0101]=[00101]=[00000000000101]=5 (base 10)
[1011]=[11011]=[11111111111011]=-5(base 10)
As a verb:
2's complement means to negate. It does not mean make negative. It means if negative make positive; if positive make negative. The magnitude is the absolute value:
if a >= 0 then |a| = a
if a < 0 then |a| = -a = 2scomplement of a
This ability allows efficient binary subtraction using negate then add.
a - b = a + (-b)
The official way to take the 1's complement is for each digit subtract its value from 1.
1'scomp(0101) = 1010.
This is the same as flipping or inverting each bit individually. This results in a negative zero which is not well loved so adding one to te 1's complement gets rid of the problem.
To negate or take the 2s complement first take the 1s complement then add 1.
Example 1 Example 2
0101 --original number 1101
1's comp 1010 0010
add 1 0001 0001
2's comp 1011 --negated number 0011
In the examples the negation works as well with sign extended numbers.
Adding:
1110 Carry 111110 Carry
0110 is the same as 000110
1111 111111
sum 0101 sum 000101
SUbtracting:
1110 Carry 00000 Carry
0110 is the same as 00110
-0111 +11001
---------- ----------
sum 0101 sum 11111
Notice that when working with 2's complement, blank space to the left of the number is filled with zeros for positive numbers butis filled with ones for negative numbers. The carry is always added and must be either a 1 or 0.
Cheers
2's complement is essentially a way of coming up with the additive inverse of a binary number. Ask yourself this: Given a number in binary form (present at a fixed length memory location), what bit pattern, when added to the original number (at the fixed length memory location), would make the result all zeros ? (at the same fixed length memory location). If we could come up with this bit pattern then that bit pattern would be the -ve representation (additive inverse) of the original number; as by definition adding a number to its additive inverse always results in zero. Example: take 5 which is 101 present inside a single 8 bit byte. Now the task is to come up with a bit pattern which when added to the given bit pattern (00000101) would result in all zeros at the memory location which is used to hold this 5 i.e. all 8 bits of the byte should be zero. To do that, start from the right most bit of 101 and for each individual bit, again ask the same question: What bit should I add to the current bit to make the result zero ? continue doing that taking in account the usual carry over. After we are done with the 3 right most places (the digits that define the original number without regard to the leading zeros) the last carry goes in the bit pattern of the additive inverse. Furthermore, since we are holding in the original number in a single 8 bit byte, all other leading bits in the additive inverse should also be 1's so that (and this is important) when the computer adds "the number" (represented using the 8 bit pattern) and its additive inverse using "that" storage type (a byte) the result in that byte would be all zeros.
1 1 1
----------
1 0 1
1 0 1 1 ---> additive inverse
---------
0 0 0
Many of the answers so far nicely explain why two's complement is used to represent negative numbers, but do not tell us what two's complement number is, particularly not why a '1' is added, and in fact often added in a wrong way.
The confusion comes from a poor understanding of the definition of a complement number. A complement is the missing part that would make something complete.
The radix complement of an n digit number x in radix b is, by definition, b^n-x.
In binary 4 is represented by 100, which has 3 digits (n=3) and a radix of 2 (b=2). So its radix complement is b^n-x = 2^3-4=8-4=4 (or 100 in binary).
However, in binary obtaining a radix's complement is not as easy as getting its diminished radix complement, which is defined as (b^n-1)-y, just 1 less than that of radix complement. To get a diminished radix complement, you simply flip all the digits.
100 -> 011 (diminished (one's) radix complement)
to obtain the radix (two's) complement, we simply add 1, as the definition defined.
011 +1 ->100 (two's complement).
Now with this new understanding, let's take a look of the example given by Vincent Ramdhanie (see above second response):
Converting 1111 to decimal:
The number starts with 1, so it's negative, so we find the complement of 1111, which is 0000.
Add 1 to 0000, and we obtain 0001.
Convert 0001 to decimal, which is 1.
Apply the sign = -1.
Tada!
Should be understood as:
The number starts with 1, so it's negative. So we know it is a two's complement of some value x. To find the x represented by its two's complement, we first need find its 1's complement.
two's complement of x: 1111
one's complement of x: 1111-1 ->1110;
x = 0001, (flip all digits)
Apply the sign -, and the answer =-x =-1.
I liked lavinio's answer, but shifting bits adds some complexity. Often there's a choice of moving bits while respecting the sign bit or while not respecting the sign bit. This is the choice between treating the numbers as signed (-8 to 7 for a nibble, -128 to 127 for bytes) or full-range unsigned numbers (0 to 15 for nibbles, 0 to 255 for bytes).
It is a clever means of encoding negative integers in such a way that approximately half of the combination of bits of a data type are reserved for negative integers, and the addition of most of the negative integers with their corresponding positive integers results in a carry overflow that leaves the result to be binary zero.
So, in 2's complement if one is 0x0001 then -1 is 0x1111, because that will result in a combined sum of 0x0000 (with an overflow of 1).
2’s Complements: When we add an extra one with the 1’s complements of a number we will get the 2’s complements. For example: 100101 it’s 1’s complement is 011010 and 2’s complement is 011010+1 = 011011 (By adding one with 1's complement) For more information
this article explain it graphically.
Two's complement is mainly used for the following reasons:
To avoid multiple representations of 0
To avoid keeping track of carry bit (as in one's complement) in case of an overflow.
Carrying out simple operations like addition and subtraction becomes easy.
Two's complement is one of the ways of expressing a negative number and most of the controllers and processors store a negative number in two's complement form.
In simple terms, two's complement is a way to store negative numbers in computer memory. Whereas positive numbers are stored as a normal binary number.
Let's consider this example,
The computer uses the binary number system to represent any number.
x = 5;
This is represented as 0101.
x = -5;
When the computer encounters the - sign, it computes its two's complement and stores it.
That is, 5 = 0101 and its two's complement is 1011.
The important rules the computer uses to process numbers are,
If the first bit is 1 then it must be a negative number.
If all the bits except first bit are 0 then it is a positive number, because there is no -0 in number system (1000 is not -0 instead it is positive 8).
If all the bits are 0 then it is 0.
Else it is a positive number.
To bitwise complement a number is to flip all the bits in it. To two’s complement it, we flip all the bits and add one.
Using 2’s complement representation for signed integers, we apply the 2’s complement operation to convert a positive number to its negative equivalent and vice versa. So using nibbles for an example, 0001 (1) becomes 1111 (-1) and applying the op again, returns to 0001.
The behaviour of the operation at zero is advantageous in giving a single representation for zero without special handling of positive and negative zeroes. 0000 complements to 1111, which when 1 is added. overflows to 0000, giving us one zero, rather than a positive and a negative one.
A key advantage of this representation is that the standard addition circuits for unsigned integers produce correct results when applied to them. For example adding 1 and -1 in nibbles: 0001 + 1111, the bits overflow out of the register, leaving behind 0000.
For a gentle introduction, the wonderful Computerphile have produced a video on the subject.
The question is 'What is “two's complement”?'
The simple answer for those wanting to understand it theoretically (and me seeking to complement the other more practical answers): 2's complement is the representation for negative integers in the dual system that does not require additional characters, such as + and -.
Two's complement of a given number is the number got by adding 1 with the ones' complement of the number.
Suppose, we have a binary number: 10111001101
Its 1's complement is: 01000110010
And its two's complement will be: 01000110011
Reference: Two's Complement (Thomas Finley)
I invert all the bits and add 1. Programmatically:
// In C++11
int _powers[] = {
1,
2,
4,
8,
16,
32,
64,
128
};
int value = 3;
int n_bits = 4;
int twos_complement = (value ^ ( _powers[n_bits]-1)) + 1;
You can also use an online calculator to calculate the two's complement binary representation of a decimal number: http://www.convertforfree.com/twos-complement-calculator/
The simplest answer:
1111 + 1 = (1)0000. So 1111 must be -1. Then -1 + 1 = 0.
It's perfect to understand these all for me.
Is there an existing subset of the alphanumerics that is easier to read? In particular, is there a subset that has fewer characters that are visually ambiguous, and by removing (or equating) certain characters we reduce human error?
I know "visually ambiguous" is somewhat waffly of an expression, but it is fairly evident that D, O and 0 are all similar, and 1 and I are also similar. I would like to maximize the size of the set of alpha-numerics, but minimize the number of characters that are likely to be misinterpreted.
The only precedent I am aware of for such a set is the Canada Postal code system that removes the letters D, F, I, O, Q, and U, and that subset was created to aid the postal system's OCR process.
My initial thought is to use only capital letters and numbers as follows:
A
B = 8
C = G
D = 0 = O = Q
E = F
H
I = J = L = T = 1 = 7
K = X
M
N
P
R
S = 5
U = V = Y
W
Z = 2
3
4
6
9
This problem may be difficult to separate from the given type face. The distinctiveness of the characters in the chosen typeface could significantly affect the potential visual ambiguity of any two characters, but I expect that in most modern typefaces the above characters that are equated will have a similar enough appearance to warrant equating them.
I would be grateful for thoughts on the above – are the above equations suitable, or perhaps are there more characters that should be equated? Would lowercase characters be more suitable?
I needed a replacement for hexadecimal (base 16) for similar reasons (e.g. for encoding a key, etc.), the best I could come up with is the following set of 16 characters, which can be used as a replacement for hexadecimal:
0 1 2 3 4 5 6 7 8 9 A B C D E F Hexadecimal
H M N 3 4 P 6 7 R 9 T W C X Y F Replacement
In the replacement set, we consider the following:
All characters used have major distinguishing features that would only be omitted in a truly awful font.
Vowels A E I O U omitted to avoid accidentally spelling words.
Sets of characters that could potentially be very similar or identical in some fonts are avoided completely (none of the characters in any set are used at all):
0 O D Q
1 I L J
8 B
5 S
2 Z
By avoiding these characters completely, the hope is that the user will enter the correct characters, rather than trying to correct mis-entered characters.
For sets of less similar but potentially confusing characters, we only use one character in each set, hopefully the most distinctive:
Y U V
Here Y is used, since it always has the lower vertical section, and a serif in serif fonts
C G
Here C is used, since it seems less likely that a C would be entered as G, than vice versa
X K
Here X is used, since it is more consistent in most fonts
F E
Here F is used, since it is not a vowel
In the case of these similar sets, entry of any character in the set could be automatically converted to the one that is actually used (the first one listed in each set). Note that E must not be automatically converted to F if hexadecimal input might be used (see below).
Note that there are still similar-sounding letters in the replacement set, this is pretty much unavoidable. When reading aloud, a phonetic alphabet should be used.
Where characters that are also present in standard hexadecimal are used in the replacement set, they are used for the same base-16 value. In theory mixed input of hexadecimal and replacement characters could be supported, provided E is not automatically converted to F.
Since this is just a character replacement, it should be easy to convert to/from hexadecimal.
Upper case seems best for the "canonical" form for output, although lower case also looks reasonable, except for "h" and "n", which should still be relatively clear in most fonts:
h m n 3 4 p 6 7 r 9 t w c x y f
Input can of course be case-insensitive.
There are several similar systems for base 32, see http://en.wikipedia.org/wiki/Base32 However these obviously need to introduce more similar-looking characters, in return for an additional 25% more information per character.
Apparently the following set was also used for Windows product keys in base 24, but again has more similar-looking characters:
B C D F G H J K M P Q R T V W X Y 2 3 4 6 7 8 9
My set of 23 unambiguous characters is:
c,d,e,f,h,j,k,m,n,p,r,t,v,w,x,y,2,3,4,5,6,8,9
I needed a set of unambiguous characters for user input, and I couldn't find anywhere that others have already produced a character set and set of rules that fit my criteria.
My requirements:
No capitals: this supposed to be used in URIs, and typed by people who might not have a lot of typing experience, for whom even the shift key can slow them down and cause uncertainty. I also want someone to be able to say "all lowercase" so as to reduce uncertainty, so I want to avoid capital letters.
Few or no vowels: an easy way to avoid creating foul language or surprising words is to simply omit most vowels. I think keeping "e" and "y" is ok.
Resolve ambiguity consistently: I'm open to using some ambiguous characters, so long as I only use one character from each group (e.g., out of lowercase s, uppercase S, and five, I might only use five); that way, on the backend, I can just replace any of these ambiguous characters with the one correct character from their group. So, the input string "3Sh" would be replaced with "35h" before I look up its match in my database.
Only needed to create tokens: I don't need to encode information like base64 or base32 do, so the exact number of characters in my set doesn't really matter, besides my wanting to to be as large as possible. It only needs to be useful for producing random UUID-type id tokens.
Strongly prefer non-ambiguity: I think it's much more costly for someone to enter a token and have something go wrong than it is for someone to have to type out a longer token. There's a tradeoff, of course, but I want to strongly prefer non-ambiguity over brevity.
The confusable groups of characters I identified:
A/4
b/6/G
8/B
c/C
f/F
9/g/q
i/I/1/l/7 - just too ambiguous to use; note that european "1" can look a lot like many people's "7"
k/K
o/O/0 - just too ambiguous to use
p/P
s/S/5
v/V
w/W
x/X
y/Y
z/Z/2
Unambiguous characters:
I think this leaves only 9 totally unambiguous lowercase/numeric chars, with no vowels:
d,e,h,j,m,n,r,t,3
Adding back in one character from each of those ambiguous groups (and trying to prefer the character that looks most distinct, while avoiding uppercase), there are 23 characters:
c,d,e,f,h,j,k,m,n,p,r,t,v,w,x,y,2,3,4,5,6,8,9
Analysis:
Using the rule of thumb that a UUID with a numerical equivalent range of N possibilities is sufficient to avoid collisions for sqrt(N) instances:
an 8-digit UUID using this character set should be sufficient to avoid collisions for about 300,000 instances
a 16-digit UUID using this character set should be sufficient to avoid collisions for about 80 billion instances.
Mainly drawing inspiration from this ux thread, mentioned by #rwb,
Several programs use similar things. The list in your post seems to be very similar to those used in these programs, and I think it should be enough for most purposes. You can add always add redundancy (error-correction) to "forgive" minor mistakes; this will require you to space-out your codes (see Hamming distance), though.
No references as to particular method used in deriving the lists, except trial and error
with humans (which is great for non-ocr: your users are humans)
It may make sense to use character grouping (say, groups of 5) to increase context ("first character in the second of 5 groups")
Ambiguity can be eliminated by using complete nouns (from a dictionary with few look-alikes; word-edit-distance may be useful here) instead of characters. People may confuse "1" with "i", but few will confuse "one" with "ice".
Another option is to make your code into a (fake) word that can be read out loud. A markov model may help you there.
If you have the option to use only capitals, I created this set based on characters which users commonly mistyped, however this wholly depends on the font they read the text in.
Characters to use: A C D E F G H J K L M N P Q R T U V W X Y 3 4 6 7 9
Characters to avoid:
B similar to 8
I similar to 1
O similar to 0
S similar to 5
Z similar to 2
What you seek is an unambiguous, efficient Human-Computer code. What I recommend is to encode the entire data with literal(meaningful) words, nouns in particular.
I have been developing a software to do just that - and most efficiently. I call it WCode. Technically its just Base-1024 Encoding - wherein you use words instead of symbols.
Here are the links:
Presentation: https://docs.google.com/presentation/d/1sYiXCWIYAWpKAahrGFZ2p5zJX8uMxPccu-oaGOajrGA/edit
Documentation: https://docs.google.com/folder/d/0B0pxLafSqCjKOWhYSFFGOHd1a2c/edit
Project: https://github.com/San13/WCode (Please wait while I get around uploading...)
This would be a general problem in OCR. Thus for end to end solution where in OCR encoding is controlled - specialised fonts have been developed to solve the "visual ambiguity" issue you mention of.
See: http://en.wikipedia.org/wiki/OCR-A_font
as additional information : you may want to know about Base32 Encoding - wherein symbol for digit '1' is not used as it may 'confuse' the users with the symbol for alphabet 'l'.
Unambiguous looking letters for humans are also unambiguous for optical character recognition (OCR). By removing all pairs of letters that are confusing for OCR, one obtains:
!+2345679:BCDEGHKLQSUZadehiopqstu
See https://www.monperrus.net/martin/store-data-paper
It depends how large you want your set to be. For example, just the set {0, 1} will probably work well. Similarly the set of digits only. But probably you want a set that's roughly half the size of the original set of characters.
I have not done this, but here's a suggestion. Pick a font, pick an initial set of characters, and write some code to do the following. Draw each character to fit into an n-by-n square of black and white pixels, for n = 1 through (say) 10. Cut away any all-white rows and columns from the edge, since we're only interested in the black area. That gives you a list of 10 codes for each character. Measure the distance between any two characters by how many of these codes differ. Estimate what distance is acceptable for your application. Then do a brute-force search for a set of characters which are that far apart.
Basically, use a script to simulate squinting at the characters and see which ones you can still tell apart.
Here's some python I wrote to encode and decode integers using the system of characters described above.
def base20encode(i):
"""Convert integer into base20 string of unambiguous characters."""
if not isinstance(i, int):
raise TypeError('This function must be called on an integer.')
chars, s = '012345689ACEHKMNPRUW', ''
while i > 0:
i, remainder = divmod(i, 20)
s = chars[remainder] + s
return s
def base20decode(s):
"""Convert string to unambiguous chars and then return integer from resultant base20"""
if not isinstance(s, str):
raise TypeError('This function must be called on a string.')
s = s.translate(bytes.maketrans(b'BGDOQFIJLT7KSVYZ', b'8C000E11111X5UU2'))
chars, i, exponent = '012345689ACEHKMNPRUW', 0, 1
for number in s[::-1]:
i += chars.index(number) * exponent
exponent *= 20
return i
base20decode(base20encode(10))
base58:123456789ABCDEFGHJKLMNPQRSTUVWXYZabcdefghijkmnopqrstuvwxyz
I had always been taught 0–9 to represent values zero to nine, and A, B, C, D, E, F for 10-15.
I see this format 0x00000000 and it doesn't fit into the pattern of hexadecimal. Is there a guide or a tutor somewhere that can explain it?
I googled for hexadecimal but I can't find any explanation of it.
So my 2nd question is, is there a name for the 0x00000000 format?
0x simply tells you the number after it will be in hex
so 0x00 is 0, 0x10 is 16, 0x11 is 17 etc
The 0x is just a prefix (used in C and many other programming languages) to mean that the following number is in base 16.
Other notations that have been used for hex include:
$ABCD
ABCDh
X'ABCD'
"ABCD"X
Yes, it is hexadecimal.
Otherwise, you can't represent A, for example. The compiler for C and Java will treat it as variable identifier. The added prefix 0x tells the compiler it's hexadecimal number, so:
int ten_i = 10;
int ten_h = 0xA;
ten_i == ten_h; // this boolean expression is true
The leading zeroes indicate the size: 0x0080 hints the number will be stored in two bytes; and 0x00000080 represents four bytes. Such notation is often used for flags: if a certain bit is set, that feature is enabled.
P.S. As an off-topic note: if the number starts with 0, then it's interpreted as octal number, for example 010 == 8. Here 0 is also a prefix.
Everything after the x are hex digits (the 0x is just a prefix to designate hex), representing 32 bits (if you were to put 0xFFFFFFFF in binary, it would be 1111 1111 1111 1111 1111 1111 1111 1111).
hexadecimal digits are often prefaced with 0x to indicate they are hexadecimal digits.
In this case, there are 8 digits, each representing 4 bits, so that is 32 bits or a word. I"m guessing you saw this in an error, and it is a memory address. this value means null, as the hex value is 0.