I’m doing computer science A-level course in a high school and currently working on two’s complement arithmetic. For some reason I don’t quite get it. I know how to convert a signed integer into its two’s complement equivalent, but here’re my confusions:
I’ve done some research and people say the op-code which is the carry-bit tells the CPU if a 2’s complement code represents a positive integer or a negative, but sometimes the carry-bit is ignored according to some people; for instance, adding 1111 (-1) to 1000 (-8) you get 10111 (-9), but if it’s a 4-bit computer, the most significant bit which is the 5th bit cannot be stored, so how does the computer deal with that ?
A somewhat trivial question is if it’s given that 00110011 represent a signed integer in two’s complement form, how do I know if the actual code is 0110011, which is a positive number (in 2’s complement form), or 110011, which is a negative number (in 2’s complement form) ?
Thanks!
Typically, what doesn't fit is lost. So, 11112 + 10002 = 01112. However, some CPUs have special flags which get set to 0 or 1 depending on whether or not there's unsigned overflow (the carry flag) and signed overflow (the overflow flag). Some CPUs may be able to automatically raise an overflow exception when an arithmetic instruction can't produce result without overflow.
Typically, the CPU does not care if 11112 is -110 or 1510. Binary addition and subtraction of two N-bit integers gives you the same N-bit sum or difference irrespective of whether you're operating on unsigned integers or 2's complement signed integers. The same is true when multiplying two N-bit signed 2's complement or unsigned integers: the N least significant bits of the product is the same in both cases.
IOW, it's your job to keep track of what types you're manipulating, to take care of overflows (if they're possible) and to select proper instructions for those types (e.g. you typically have signed and unsigned division, signed and unsigned comparison, etc).
I understand what the Two's complement is and what it is useful for. What I'd like to know is how does the computer decide that the number is in its Two's complement form?
How and when does it decide that 1111 1110 is -2 and not 254? Is it at the OS level of processing?
As far as I think it is dependable on programming language.
Lets say integer allocates 1 byte of memory (to make it simple).
If it is UNSIGNED integer (only positive numbers) you can use any number from 0 to 255 (in total of 2^8 numbers, zero included).
00000000 would be 0, and
11111111 would be 255 decimal.
But if your integer is SIGNED ( u can use both, negative and positive numbers) you can use values from -127 to 127, zero included (again 2^8 numbers).
If your compiler bumps into 11111111 SIGNED int value, it will not interpret it as 255 because signed int allows only values from 0 to 127 for positive numbers so it will take it as -1. Next one, -2 would be 11111110 (254 decimal) and so on...
The computer will already be expecting the data to be in (or not in) two's complement form (otherwise there wouldn't be a way of telling if it is - 2 or 254). And yes, that would probably be decided at the OS-level.
You can probably relate this to the same kind of idea used when setting variable types when declaring variables in a high-level programming language; you'll more than likely set the type to be "decimal", for example, or "integer" and then the compiler will expect values to stick to this type.
I need help in determining if my logic here is right or wrong.
Example Question
"Assuming I have an 8-bit signed decimal value of 200 in two's compliment form..."
My Thought Process
Now because it is 8-bits and is signed, the most significant bit must be reserved for the sign.
Thus, the maximum positive value it can have is:
2^(8-1) - 1 = 127
At first I was confused because I thought, why is the question stating that 200 is able to be 8-bits and signed? Then I thought, that's where the two's compliment statement comes into question.
Because it is two's compliment in reality, this is the case:
8-bit Signed, 2's Compliment, Decimal = 200
Convert to Binary --> 1100 1000
Because it is signed, the actual two's compliment number is ACTUALLY -56 (I would use negating methods to invert the 1's and 0's then + 1, but for the interest of time, I just found a converter online).
So my conclusion is:
8-bit Signed, 2's Compliment, Decimal value of 200 is actually -56.
Ultimate Question
Is my thought process correct with this? If so, I think the most confusing part about this is telling my brain that one number is equal to a completely different number.
Yes, I think your analysis is correct.
To expand a bit more, I think the wording of the question is awkward and would have been better stated as "What is the value of 1100 1000 in base 10, where the number is a two's complemented number?"
The trick here is to think not that 200 == -56, but that the single point of truth is the bits 11001000. These bits of numbers have no meaning by themselves. We have the computer interpret them differently based on the program. So two's complement (with 8 bit numbers) treats that as -56, an unsigned interpretation would treat that as 200, and in ASCII this would be some special character depending on the encoding.
5 (decimal) in binary 00000101
-5 (two's complement) in binary 11111011
but 11111011 is also 251 (decimal)!
How does computer discern one from another??
How does it know whether it's -5 or 251??
it's THE SAME 11111011
Thanks in advance!!
Signed bytes have a maximum of 127.
Unsigned bytes cannot be negative.
The compiler knows whether the variable holding that value is of signed or unsigend type, and treats it appropriately.
If your program chooses to treat the byte as signed, the run-time system decides whether the byte is to be considered positive or negative according to the high-order bit. A 1 in that high-order bit (bit 7, counting from the low-order bit 0) means the number is negative; a 0 in that bit position means the number is positive. So, in the case of 11111011, bit 7 is set to 1 and the number is treated, accordingly, as negative.
Because the sign bit takes up one bit position, the absolute magnitude of the number can range from 0 to 127, as was said before.
If your program chooses to treat the byte as unsigned, on the other hand, what would have been the sign bit is included in the magnitude, which can then range from 0 to 255.
Two's complement is designed to allow signed numbers to be added/substracted to one another in the same way unsigned numbers are. So there are only two cases where the signed-ness of numbers affect the computer at low level.
when there are overflows
when you are performing operations on mixed: one signed, one unsigned
Different processors take different tacks for this. WRT orverflows, the MIPS RISC architecture, for example, deals with overflows using traps. See http://en.wikipedia.org/wiki/MIPS_architecture#MIPS_I_instruction_formats
To the best of my knowledge, mixing signed and unsigned needs to avoided at a program level.
If you're asking "how does the program know how to interpret the value" - in general it's because you've told the compiler the "type" of the variable you assigned the value to. The program doesn't actually care if 00000101 as "5 decimal", it just has an unsigned integer with value 00000101 that it can perform operations legal for unsigned integers upon, and will behave in a given manner if you try to compare with or cast to a different "type" of variable.
At the end of the day everything in programming comes down to binary - all data (strings, numbers, images, sounds etc etc) and the compiled code just ends up as a large binary blob.
I'm just curious if there's a reason why in order to represent -1 in binary, two's complement is used: flipping the bits and adding 1?
-1 is represented by 11111111 (two's complement) rather than (to me more intuitive) 10000001 which is binary 1 with first bit as negative flag.
Disclaimer: I don't rely on binary arithmetic for my job!
It's done so that addition doesn't need to have any special logic for dealing with negative numbers. Check out the article on Wikipedia.
Say you have two numbers, 2 and -1. In your "intuitive" way of representing numbers, they would be 0010 and 1001, respectively (I'm sticking to 4 bits for size). In the two's complement way, they are 0010 and 1111. Now, let's say I want to add them.
Two's complement addition is very simple. You add numbers normally and any carry bit at the end is discarded. So they're added as follows:
0010
+ 1111
=10001
= 0001 (discard the carry)
0001 is 1, which is the expected result of "2+(-1)".
But in your "intuitive" method, adding is more complicated:
0010
+ 1001
= 1011
Which is -3, right? Simple addition doesn't work in this case. You need to note that one of the numbers is negative and use a different algorithm if that's the case.
For this "intuitive" storage method, subtraction is a different operation than addition, requiring additional checks on the numbers before they can be added. Since you want the most basic operations (addition, subtraction, etc) to be as fast as possible, you need to store numbers in a way that lets you use the simplest algorithms possible.
Additionally, in the "intuitive" storage method, there are two zeroes:
0000 "zero"
1000 "negative zero"
Which are intuitively the same number but have two different values when stored. Every application will need to take extra steps to make sure that non-zero values are also not negative zero.
There's another bonus with storing ints this way, and that's when you need to extend the width of the register the value is being stored in. With two's complement, storing a 4-bit number in an 8-bit register is a matter of repeating its most significant bit:
0001 (one, in four bits)
00000001 (one, in eight bits)
1110 (negative two, in four bits)
11111110 (negative two, in eight bits)
It's just a matter of looking at the sign bit of the smaller word and repeating it until it pads the width of the bigger word.
With your method you would need to clear the existing bit, which is an extra operation in addition to padding:
0001 (one, in four bits)
00000001 (one, in eight bits)
1010 (negative two, in four bits)
10000010 (negative two, in eight bits)
You still need to set those extra 4 bits in both cases, but in the "intuitive" case you need to clear the 5th bit as well. It's one tiny extra step in one of the most fundamental and common operations present in every application.
Wikipedia says it all:
The two's-complement system has the advantage of not requiring that the addition and subtraction circuitry examine the signs of the operands to determine whether to add or subtract. This property makes the system both simpler to implement and capable of easily handling higher precision arithmetic. Also, zero has only a single representation, obviating the subtleties associated with negative zero, which exists in ones'-complement systems.
In other words, adding is the same, wether or not the number is negative.
Even though this question is old , let me put in my 2 cents.
Before I explain this ,lets get back to basics. 2' complement is 1's complement + 1 .
Now what is 1's complement and what is its significance in addition.
Sum of any n-bit number and its 1's complement gives you the highest possible number that can be represented by those n-bits.
Example:
0010 (2 in 4 bit system)
+1101 (1's complement of 2)
___________________________
1111 (the highest number that we can represent by 4 bits)
Now what will happen if we try to add 1 more to the result. It will results in an overflow.
The result will be 1 0000 which is 0 ( as we are working with 4 bit numbers , (the 1 on left is an overflow )
So ,
Any n-bit number + its 1's complement = max n-bit number
Any n-bit number + its 1'complement + 1 = 0 ( as explained above, overflow will occur as we are adding 1 to max n-bit number)
Someone then decided to call 1's complement + 1 as 2'complement. So the above statement becomes:
Any n'bit number + its 2's complement = 0
which means 2's complement of a number = - (of that number)
All this yields one more question , why can we use only the (n-1) of the n bits to represent positive number and why does the left most nth bit represent sign (0 on the leftmost bit means +ve number , and 1 means -ve number ) . eg why do we use only the first 31 bits of an int in java to represent positive number if the 32nd bit is 1 , its a -ve number.
1100 (lets assume 12 in 4 bit system)
+0100(2's complement of 12)
___________________________
1 0000 (result is zero , with the carry 1 overflowing)
Thus the system of (n + 2'complement of n) = 0 , still works. The only ambiguity here is 2's complement of 12 is 0100 which ambiguously also represents +8 , other than representing -12 in 2s complement system.
This problem will be solved if positive numbers always have a 0 in their left most bit. In that case their 2's complement will always have a 1 in their left most bit , and we wont have the ambiguity of the same set of bits representing a 2's complement number as well as a +ve number.
Two's complement allows addition and subtraction to be done in the normal way (like you wound for unsigned numbers). It also prevents -0 (a separate way to represent 0 that would not be equal to 0 with the normal bit-by-bit method of comparing numbers).
Two's complement allows negative and positive numbers to be added together without any special logic.
If you tried to add 1 and -1 using your method
10000001 (-1)
+00000001 (1)
you get
10000010 (-2)
Instead, by using two's complement, we can add
11111111 (-1)
+00000001 (1)
you get
00000000 (0)
The same is true for subtraction.
Also, if you try to subtract 4 from 6 (two positive numbers) you can 2's complement 4 and add the two together 6 + (-4) = 6 - 4 = 2
This means that subtraction and addition of both positive and negative numbers can all be done by the same circuit in the cpu.
this is to simplify sums and differences of numbers. a sum of a negative number and a positive one codified in 2's complements is the same as summing them up in the normal way.
The usual implementation of the operation is "flip the bits and add 1", but there's another way of defining it that probably makes the rationale clearer. 2's complement is the form you get if you take the usual unsigned representation where each bit controls the next power of 2, and just make the most significant term negative.
Taking an 8-bit value a7 a6 a5 a4 a3 a2 a1 a0
The usual unsigned binary interpretation is:
27*a7 + 26*a6 + 25*a5 + 24*a4 + 23*a3 + 22*a2 + 21*a1 + 20*a0
11111111 = 128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = 255
The two's complement interpretation is:
-27*a7 + 26*a6 + 25*a5 + 24*a4 + 23*a3 + 22*a2 + 21*a1 + 20*a0
11111111 = -128 + 64 + 32 + 16 + 8 + 4 + 2 + 1 = -1
None of the other bits change meaning at all, and carrying into a7 is "overflow" and not expected to work, so pretty much all of the arithmetic operations work without modification (as others have noted). Sign-magnitude generally inspect the sign bit and use different logic.
To expand on others answers:
In two's complement
Adding is the same mechanism as plain positive integers adding.
Subtracting doesn't change too
Multiplication too!
Division does require a different mechanism.
All these are true because two's complement is just normal modular arithmetic, where we choose to look at some numbers as negative by subtracting the modulo.
Reading the answers to this question, I came across this comment [edited].
2's complement of 0100(4) will be 1100. Now 1100 is 12 if I say normally. So,
when I say normal 1100 then it is 12, but when I say 2's complement 1100 then
it is -4? Also, in Java when 1100 (lets assume 4 bits for now) is stored then
how it is determined if it is +12 or -4 ?? – hagrawal Jul 2 at 16:53
In my opinion, the question asked in this comment is quite interesting and so I'd like first of all to rephrase it and then to provide an answer and an example.
QUESTION – How can the system establish how one or more adjacent bytes have to be interpreted? In particular, how can the system establish whether a given sequence of bytes is a plain binary number or a 2's complement number?
ANSWER – The system establishes how to interpret a sequence of bytes through types.
Types define
how many bytes have to be considered
how those bytes have to be interpreted
EXAMPLE – Below we assume that
char's are 1 byte long
short's are 2 bytes long
int's and float's are 4 bytes long
Please note that these sizes are specific to my system. Although pretty common, they can be different from system to system. If you're curious of what they are on your system, use the sizeof operator.
First of all we define an array containing 4 bytes and initialize all of them to the binary number 10111101, corresponding to the hexadecimal number BD.
// BD(hexadecimal) = 10111101 (binary)
unsigned char l_Just4Bytes[ 4 ] = { 0xBD, 0xBD, 0xBD, 0xBD };
Then we read the array content using different types.
unsigned char and signed char
// 10111101 as a PLAIN BINARY number equals 189
printf( "l_Just4Bytes as unsigned char -> %hi\n", *( ( unsigned char* )l_Just4Bytes ) );
// 10111101 as a 2'S COMPLEMENT number equals -67
printf( "l_Just4Bytes as signed char -> %i\n", *( ( signed char* )l_Just4Bytes ) );
unsigned short and short
// 1011110110111101 as a PLAIN BINARY number equals 48573
printf( "l_Just4Bytes as unsigned short -> %hu\n", *( ( unsigned short* )l_Just4Bytes ) );
// 1011110110111101 as a 2'S COMPLEMENT number equals -16963
printf( "l_Just4Bytes as short -> %hi\n", *( ( short* )l_Just4Bytes ) );
unsigned int, int and float
// 10111101101111011011110110111101 as a PLAIN BINARY number equals 3183328701
printf( "l_Just4Bytes as unsigned int -> %u\n", *( ( unsigned int* )l_Just4Bytes ) );
// 10111101101111011011110110111101 as a 2'S COMPLEMENT number equals -1111638595
printf( "l_Just4Bytes as int -> %i\n", *( ( int* )l_Just4Bytes ) );
// 10111101101111011011110110111101 as a IEEE 754 SINGLE-PRECISION number equals -0.092647
printf( "l_Just4Bytes as float -> %f\n", *( ( float* )l_Just4Bytes ) );
The 4 bytes in RAM (l_Just4Bytes[ 0..3 ]) always remain exactly the same. The only thing that changes is how we interpret them.
Again, we tell the system how to interpret them through types.
For instance, above we have used the following types to interpret the contents of the l_Just4Bytes array
unsigned char: 1 byte in plain binary
signed char: 1 byte in 2's complement
unsigned short: 2 bytes in plain binary notation
short: 2 bytes in 2's complement
unsigned int: 4 bytes in plain binary notation
int: 4 bytes in 2's complement
float: 4 bytes in IEEE 754 single-precision notation
[EDIT] This post has been edited after the comment by user4581301. Thank you for taking the time to drop those few helpful lines!
Two's complement is used because it is simpler to implement in circuitry and also does not allow a negative zero.
If there are x bits, two's complement will range from +(2^x/2+1) to -(2^x/2). One's complement will run from +(2^x/2) to -(2^x/2), but will permit a negative zero (0000 is equal to 1000 in a 4 bit 1's complement system).
It's worthwhile to note that on some early adding machines, before the days of digital computers, subtraction would be performed by having the operator enter values using a different colored set of legends on each key (so each key would enter nine minus the number to be subtracted), and press a special button would would assume a carry into a calculation. Thus, on a six-digit machine, to subtract 1234 from a value, the operator would hit keys that would normally indicate "998,765" and hit a button to add that value plus one to the calculation in progress. Two's complement arithmetic is simply the binary equivalent of that earlier "ten's-complement" arithmetic.
The advantage of performing subtraction by the complement method is reduction in the hardware
complexity.The are no need of the different digital circuit for addition and subtraction.both
addition and subtraction are performed by adder only.
I have a slight addendum that is important in some situations: two's compliment is the only representation that is possible given these constraints:
Unsigned numbers and two's compliment are commutative rings with identity. There is a homomorphism between them.
They share the same representation, with a different branch cut for negative numbers, (hence, why addition and multiplication are the same between them.)
The high bit determines the sign.
To see why, it helps to reduce the cardinality; for example, Z_4.
Sign and magnitude and ones' compliment both do not form a ring with the same number of elements; a symptom is the double zero. It is therefore difficult to work with on the edges; to be mathematically consistent, they require checking for overflow or trap representations.
Well, your intent is not really to reverse all bits of your binary number. It is actually to subtract each its digit from 1. It's just a fortunate coincidence that subtracting 1 from 1 results in 0 and subtracting 0 from 1 results in 1. So flipping bits is effectively carrying out this subtraction.
But why are you finding each digit's difference from 1? Well, you're not. Your actual intent is to compute the given binary number's difference from another binary number which has the same number of digits but contains only 1's. For example if your number is 10110001, when you flip all those bits, you're effectively computing (11111111 - 10110001).
This explains the first step in the computation of Two's Complement. Now let's include the second step -- adding 1 -- also in the picture.
Add 1 to the above binary equation:
11111111 - 10110001 + 1
What do you get? This:
100000000 - 10110001
This is the final equation. And by carrying out those two steps you're trying to find this, final difference: the binary number subtracted from another binary number with one extra digit and containing zeros except at the most signification bit position.
But why are we hankerin' after this difference really? Well, from here on, I guess it would be better if you read the Wikipedia article.
We perform only addition operation for both addition and subtraction. We add the second operand to the first operand for addition. For subtraction we add the 2's complement of the second operand to the first operand.
With a 2's complement representation we do not need separate digital components for subtraction—only adders and complementers are used.
A major advantage of two's-complement representation which hasn't yet been mentioned here is that the lower bits of a two's-complement sum, difference, or product are dependent only upon the corresponding bits of the operands. The reason that the 8 bit signed value for -1 is 11111111 is that subtracting any integer whose lowest 8 bits are 00000001 from any other integer whose lowest 8 bits are 0000000 will yield an integer whose lowest 8 bits are 11111111. Mathematically, the value -1 would be an infinite string of 1's, but all values within the range of a particular integer type will either be all 1's or all 0's past a certain point, so it's convenient for computers to "sign-extend" the most significant bit of a number as though it represented an infinite number of 1's or 0's.
Two's-complement is just about the only signed-number representation that works well when dealing with types larger than a binary machine's natural word size, since when performing addition or subtraction, code can fetch the lowest chunk of each operand, compute the lowest chunk of the result, and store that, then load the next chunk of each operand, compute the next chunk of the result, and store that, etc. Thus, even a processor which requires all additions and subtractions to go through a single 8-bit register can handle 32-bit signed numbers reasonably efficiently (slower than with a 32-bit register, of course, but still workable).
When using of the any other signed representations allowed by the C Standard, every bit of the result could potentially be affected by any bit of the operands, making it necessary to either hold an entire value in registers at once or else follow computations with an extra step that would, in at least some cases, require reading, modifying, and rewriting each chunk of the result.
There are different types of representations those are:
unsigned number representation
signed number representation
one's complement representation
Two's complement representation
-Unsigned number representation used to represent only positive numbers
-Signed number representation used to represent positive as well as a negative number. In Signed number representation MSB bit represents sign bit and rest bits represents the number. When MSB is 0 means number is positive and When MSB is 1 means number is negative.
Problem with Signed number representation is that there are two values for 0.
Problem with one's complement representation is that there are two values for 0.
But if we use Two's complement representation then there will only one value for 0 that's why we represent negative numbers in two's complement form.
Source:Why negative numbers are stored in two's complement form bytesofgigabytes
One satisfactory answer of why Two2's Complement is used to represent negative numbers rather than One's Complement system is that
Two's Complement system solves the problem of multiple representations of 0 and the need for end-around-carry which exist in the One's complement system of representing negative numbers.
For more information Visit https://en.wikipedia.org/wiki/Signed_number_representations
For End-around-carry Visit
https://en.wikipedia.org/wiki/End-around_carry