Negative Numbers

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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

Revision Questions

1. True or false, the biggest number we can make in twos compliment binary is 128

	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.

2. 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.

3. What is -34 in 8-bit two's complement binary?

	11011110
	01011110
	10100110
	01010111
→	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 invert the bits to get 11011110

4. What is the Number range of 8 bit two's compliment in denary?

	+128 to -128
	+128 to -127
	+127 to -128
	+127 to -127
→	The biggest number you can create in twos compliment is +127 ; 01111111
→	The smaller number you can create is -128 ; 10000000
→	This is because in twos compliment, the left-most digit counts as -128.

5.

What is the 8-bit two's compliment number 10010110 in denary?

→ 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.

6.

What is the 8-bit two's compliment number 01110101 in denary?

→ 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.

7.

What does 'msb' stand for? Please format your answer in lowercase, and check your spelling.

8. Which position does the msb hold?

	The middle
	The leftmost bit
	The rightmost bit
→	The msb is the leftmost bit in a sequence.

9.

What is the 8-bit two's compliment number 10011101 in denary?

→ When there is a 1 in the -128 column, the number will always be negative.

10.

What is the 8-bit two's compliment binary number 11111111 in denary?

→ In this question we can assume the answer to be -1 as the first digit (-128) + every other number (127) would be -1.

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