Difference between revisions of "Negative Numbers"
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Two's Complement can be used to convert binary numbers from positive to negative, to do this we need to: | Two's Complement can be used to convert binary numbers from positive to negative, to do this we need to: | ||
− | 1) Write the number | + | 1) Write the number as its equivalent positive binary form |
2) Add 0's to the number to make it 8 bit | 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 | 3) Invert each bit, changing 0's to 1's and 1's to 0's | ||
Line 27: | Line 27: | ||
For example, represent -41 in two's complement form: | For example, represent -41 in two's complement form: | ||
− | First calculate +41 in binary using your preferred method | + | 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 | + | Then Invert the bits: |
− | + | 11010110 | |
− | Then Add 1 to the number | + | Then Add 1 to the number: |
− | + | 11010110 | |
+ | + 1 | ||
+ | -------- | ||
+ | 11010111 | ||
+ | -------- | ||
− | 11010111 = -128+64+16+4+2+1 = -41 | + | 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= | =Method 2= | ||
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For example, represent -46 in twos complement: | For example, represent -46 in twos complement: | ||
− | First calculate +46 in binary using your preferred method | + | 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: | ||
+ | |||
+ | 00101110 | ||
− | + | Then find the first one and keep it: | |
− | + | 001011'''1'''0 | |
+ | ^ | ||
− | Then | + | 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 | + | To check our answer we can convert the number to denary, remembering that the msb represent -128: |
− | -128+64+16+2= -46 | + | 11010010 = -128+64+16+2= -46 |
=Revision Questions= | =Revision Questions= |
Latest revision as of 08:25, 25 September 2020
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Video from 0:00 until 6.25
https://www.youtube.com/watch?v=CglODZZm_Z4&list=PLCiOXwirraUDGCeSoEPSN-e2o9exXdOka&index=3 (0:00 - 6:25)
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 as 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:
00101110
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:
11010010 = -128+64+16+2= -46