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1. Math 260scarbnick@gmail.com Two’s Complement Arithmetic

2. Two’s Complement Arithmetic • Our goal in these slides is showing how positive and negative numbers are stored in a byte, which is the standard unit for computer memory.

3. Two’s Complement Arithmetic • Topics • Binary Numbers • Two’s Complement Basics • Using Two’s Complement to Store Numbers on a Computer • Adding Numbers on a Computer with Two’s Complement

4. Two’s Complement ArithmeticBinary Numbers • Binary Numbers - A numerical system that represents integers in the following format: dn2n + dn -12 n-1 + … d1 21 + d0 20 • Each dI is either 0 or 1. • Example: 110012 = 1 x24 + 1 x 2 3 + 0 x 22 + 0 x21 + 1 x20 = 2510

5. Two’s Complement ArithmeticBinary Numbers • 1 2 + 0 2 = 1 2 • 0 2 + 1 2 = 1 2 • 1 2 + 1 2 = 1 0 2 , since we must carry into the 2’s place • 11011 2 + 01101 2 = 101000 2

6. Two’s Complement ArithmeticBinary Numbers • Example of Subtraction: 100 2 - 001 2

7. Two’s Complement ArithmeticTwo’s Complement Basics • Definition of Two’s Complement: Let the term, a, represent a positive integer. The two’s complement of a relative to a fixed bit length n is the n-bit binary representation of 2n – a. • For simplicity, we will let n = 8, though on modern computers n is typically set to 32.

8. Two’s Complement ArithmeticTwo’s Complement Basics • Computing the two’s complement of a number: • Note that 28 - a = (28 - 1) - a + 1 (b) 28 = 1 0000 0000 2 ( c ) 28 - 1 = 1 0000 0000 2 - 1 = 1111 1111 2 (continued on the next slide)

9. Two’s Complement ArithmeticTwo’s Complement Basics • Computing the two’s complement of a number: (d) Assume a is in binary notation. To compute 1111 1111 2 – a, we just toggle the digits of a. Example: 1111 11112 ─ 0101 10112 1010 01002 (Continued on the next slide)

10. Two’s Complement ArithmeticTwo’s Complement Basics • Computing the two’s complement of a number: • To compute 28 - a = (28 - 1) - a + 1 , we just toggle the digits of a and then add 1.

11. Two’s Complement ArithmeticTwo’s Complement Basics • Example: Calculate the two complement of 1012 relative to a fixed bit length of 8 • Make sure to write 1012 with 8-bits: 0000 01012 • Toggle the bits to get 1111 10102 • Add 1 to arrive at 1111 10112

12. Two’s Complement ArithmeticTwo’s Complement Basics • Suppose we have the two’s complement of a number. How do we find the original number? • Note that 2n – (2n – a) = a • Therefore, to find the original number we just calculate the two’s complement of the two’s complement!

13. Two’s Complement ArithmeticUsing Two’s Complement to Store Numbers on a Computer • Based on the previous slides, the 8-bit two’s complement of the numbers, 1 through 128, have a leading bit of 1. • Verify the two’s complement of 12810 = 1000 00002 is 1000 00002

14. Two’s Complement ArithmeticUsing Two’s Complement to Store Numbers on a Computer • This means 8-bits can represent all numbers between -12810 and 12710 by using the following formulas (a) if 0 ≤ a ≤ 127, store a in its binary representation. (b) if -128 ≤ a < 0, store a in the binary representation of 28 - |a | (In other words, the number, a, is represented by the two’s complement of the absolute value of a.)

15. Two’s Complement ArithmeticUsing Two’s Complement to Store Numbers on a Computer • If the leading digit in an 8-bit representation is 1, the number is negative. To find the original number, calculate the two’s complement and then place a negative sign in front of this number. • Example: 1111 00002 is stored in a computer byte. The two’s complement of this number is 0001 00002. Therefore, we stored -1610 in the computer byte.

16. Two’s Complement ArithmeticUsing Two’s Complement to Store Numbers on a Computer • If the leading digit in an 8-bit representation is 0, the number is positive. This means the largest number that can be stored in one-byte is 0111 11112 = 12710.

17. Two’s Complement ArithmeticAdding Numbers on a Computer with Two’s Complement • Directly add bits • Ignore any overflow into the (n+1)st bit

18. Two’s Complement ArithmeticAdding Numbers on a Computer with Two’s Complement Example 1: Add -610 + -10110 The two’s complement of 610 = 1111 10102 The two’s complement of 10110= 1001 10112 1111 10102 +1001 10112 = 1 1001 01012 Ignore the bit in 28 position to get 1001 01012. The leading bit is 1, so we have a negative number. The two’s complement of 1001 01012 = 0110 10112 = 10710. Therefore, the sum is equal to -10710

19. Two’s Complement ArithmeticAdding Numbers on a Computer with Two’s Complement • Example 2: Add 5010 + ( - 3810 ) Since 5010 is positive, we use its binary representation = 0011 00102 Since -3810 is negative, we use the two’s complement of 38 = 1101 10102 = 0011 00102 +1101 10102 = 1 0000 11002 Ignore the bit in 28 position to get 0000 11002. The leading bit is 0, so we have a positive number. Therefore, the sum is equal to 0000 11002 = 1210

20. New Topic – Hexadecimal Notation • Base 16 is referred to as Hexadecimal • Hexadecimal means we represent numbers as powers of 16. • In hexadecimal A = 10, B = 11, C = 12, D = 13, E = 14, and F = 15 • A hexadecimal number is written as dn 16n + dn -116 n-1 + … d1 161 + d0 160,where each di ranges from 0 to F.

21. Hexadecimal Notation • Examples C16 = 1210 1AB16 = 1 x 162 + 10 x 161 + 11 x 160 = 42710 D00016 = 13 x 163 = 53,24810

22. Hexadecimal Notation • To convert from binary to hexadecimal, break the binary number into groups with four digits and then convert each group to a hexadecimal number • Example: Convert 101 1101 00102 to hexadecimal • Answer is 5C216