Difference between revisions of "Negative Numbers"
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<quiz display=simple> | <quiz display=simple> | ||
− | { What is -34 in 8-bit twos | + | |
+ | { True or false, the biggest number we can make in twos compliment binary is 128 | ||
+ | | type="()" } | ||
+ | -true | ||
+ | +false | ||
+ | ||We cannot make 128 in twos compliment, as the first bit is equal to -128. If we add up every bit after that, we would only be able to reach 127. | ||
+ | |||
+ | { What is the difference between the two methods for calculating negative binary values? | ||
+ | - One involves inverting the bits, one does not | ||
+ | - In one of the methods, you have to make it a 8 bit value. | ||
+ | + For one you have to add 1 to the end of the number, for the other you must invert all the bits aside from the right-most 1. | ||
+ | - One allows you to skip inverting the bits | ||
+ | || In both methods you must invert the bits, which you cannot skip, and it must be a 8bit number. | ||
+ | || In method 1, you must invert all the bits and add 1 to the rightmost digit ; in method 2, you keep the rightmost one, and then invert the rest. | ||
+ | |||
+ | { What is -34 in 8-bit twos complement binary? | ||
| type="()" } | | type="()" } | ||
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- 01010111 | - 01010111 | ||
|| We know the answer must be 11011110 or 10100110, because if the first bit is a 0, the denary would be positive. | || We know the answer must be 11011110 or 10100110, because if the first bit is a 0, the denary would be positive. | ||
− | || Next, we should calculate what +38 is in binary. This would be 00100010. We can | + | || Next, we should calculate what +38 is in binary. This would be 00100010. We can invert the bits to get 11011110 |
{ What is the Number range of 8 bit twos compliment in denary? | { What is the Number range of 8 bit twos compliment in denary? | ||
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What is the 8-bit twos compliment number 10010110 in denary? | What is the 8-bit twos compliment number 10010110 in denary? | ||
{ -106 } | { -106 } | ||
− | + | || Firstly, we can invert the numbers to get the positive denary value. The inverted binary would be 01101010. | |
+ | || 01101010 would be 64+32+8+2. This is 106, which we can flip directly to a negative number to get the answer. | ||
{ | { | ||
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What is the 8-bit twos compliment number 01110101 in denary? | What is the 8-bit twos compliment number 01110101 in denary? | ||
{ 117 } | { 117 } | ||
+ | || We can identify here that the answer will be a positive number, as the first bit is a 0 | ||
+ | || The 1s in this fall on 64, 32, 16, 2, and 8. Adding these up gives us 117. | ||
+ | |||
+ | { | ||
+ | | type="{}"} | ||
+ | What does 'msb' stand for? | ||
+ | { most significant bit, msb } | ||
+ | |||
</quiz> | </quiz> |
Revision as of 10:25, 20 September 2017
Two's Complement
Two's Complement uses a similar number system to binary except the msb or left hand bit is a negative value, meaning for 8 bit two's complement it would be -128 instead of 128 like it is in regular 8bit binary.
-128 64 32 16 8 4 2 1
We can see that a 1 in the msb position, or the position of -128 would result in the binary number being negative as the other bits 64-1 only total 127. This means that even if there was a 1 in every position a two's complement number of 11111111 in binary would equal -1.
this means that in two's complement if the msb is a 0 the number is positive and if it is a 1 the number is negative.
Therefore we know that the smallest possible value in 8bit two's complement binary is 10000000 = -128 and the largest value is 01111111 = +127.
Method 1
Two's Complement can be used to convert binary numbers from positive to negative, to do this we need to:
1) Write the number is its equivalent positive binary form 2) Add 0's to the number to make it 8 bit 3) Invert each bit, changing 0's to 1's and 1's to 0's 4) Add 1 to the number to make it a two's complement number
For example, represent -41 in two's complement form:
First calculate +41 in binary using your preferred method
41= 32+0+8+0+0+1 = 101001
Then add 0's to make it 8 bit. 00101001
Then Invert the bits. 11010110.
Then Add 1 to the number. 11010110+1= 11010111.
To check our answer we can convert the number to denary, remembering that the msb represent -128.
11010111 = -128+64+16+4+2+1 = -41
Method 2
There is one other method for representing numbers in two's complement form, without using calculations. To do this we need to:
1) Write the number is its equivalent positive binary form 2) Add 0's to the number to make it 8 bit 3) Starting from the right and going left find the first 1 and keep it 4) Invert each bit after, changing 0's to 1's and 1's to 0's, but don't invert the 1 you kept or any 0's to the right of it
For example, represent -46 in twos complement:
First calculate +46 in binary using your preferred method
46= 32+0+8+4+2+0 = 101110
Then add 0's to make it up to 8bit
Then find the first one and keep it 00101110
Then invert the bits excluding the 1 you kept and all 0's to the right of it. 11010010
To check our answer we can convert the number to denary, remembering that the msb represent -128.
-128+64+16+2= -46