Twos Complement

Year 12 Lesson

Transcript

1. BINARY REFRESHER USING ANY METHOD YOU KNOW, CONVERT THE FOLLOWING DENARY NUMBERS INTO BINARY: 11 28 31 68 73 161
2. BINARY REFRESHER 11 28 31 68 73 161
3. TWO’S COMPLEMENT Representing Negative Numbers in Binary
4. Can’t we just put a minus sign ▪ The negative representation of a denary number in binary is called the two’s complement. ▪ Example: Take the number 2710 In binary this is represented as:
5. The sign bit ▪ To get our negative representation of 27 we need to add what is called a sign bit to represent +/- (0 (dim) for positive 1 (un) for negative (positif/negyddol), this is added to the far left of our binary number. ▪ So right now our (positive) 27 including the sign bit looks like this:
6. Next, it’s time to flip! ▪ The next step to get our two’s complement is to flip all the bits, if it’s a 0 change it to a 1 and vice versa. ▪ So now our 27 with the added sign bit and bits flipped is:
7. Finally, add 1 ▪ The final step is to add 1 to our binary number, this is then our two’s complement of 2710 ▪ Here you can see the most significant bit (farthest left) has changes to a 1 (or minus)
8. Time for some magic! ▪ -27 in two’s complement is 100101
9. One more example? ▪ Convert to binary ▪ Add sign bit ▪ Flip the bits ▪ Add one
10. Let’s get converting! Show the following denary numbers in two’s complement, show the four stages for each (convert to binary, add sign bit, flip bits, add one) ▪7 ▪15 ▪26 ▪37 ▪49
11. Your turn to do a “bit”! ▪7 ▪15 ▪26 ▪37 ▪49
12. Working backwards ▪ To convert a binary two’s complement number back to its positive counterpart we do the same thing, simply flip the bits and add one! ▪ So 100101 ▪ Becomes:
13. Let’s get positive! Show the following two’s complement binary numbers in denary, show the three stages for each (flip bits, add one, convert to denary) ▪101 ▪10010 ▪1011110 ▪1000111 ▪10100110
14. SUBTRACTING USING TWO’S COMPLEMENT
15. Subtraction example: ▪ Let’s do 8 – 16 ▪ 8 – 16 is the same as 8 + -16 ▪ 8 = 1000 ▪ 16 is 10000
16. Subtraction example: ▪ 8 = 1000 and 16 is 10000 ▪ To get -16 we add a sign bit, 010000 ▪ Flip the bits 101111 ▪ …and add one 110000 ▪ Then add the two numbers together using our binary addition rules: Binary addition rules: 0 + 0 = 0 0 + 1 =1 1 + 1 = 0 and carry 1 1+1+1 = 1 and carry 1 1000 110000 111000 This represents our negative answer (sign bit = 1)
17. Subtraction example: ▪ Now flip the bits 000111 ▪ Add one 001000 which is 8 ▪ So 8 – 16 = -8 1000 110000 111000
18. Subtraction! Do the following subtractions showing all working. ▪10 – 11 ▪15 – 21 ▪26 – 42 ▪32 – 56 ▪48 – 63
19. Subtraction worked examples ▪ 10 – 11 ▪ 15 – 21 ▪ 26 – 42 ▪ 32 – 56 ▪ 48 – 63
20. Why? ▪ You already know computers use binary for everything! ▪ This lesson has shown you how we represent negative numbers in binary. ▪ You now also know how computers can do subtraction using twos complement and binary addition. Lesson +’s Lesson –’s