Number Systems: Signed numbers

Number representation

• Computers are digital systems:
• data stored in the form of 0s and 1s
• Computers commonly perform arithmetic operations on numbers:
• integers (signed and unsigned) and real numbers
• How we represent numbers with bits profoundly impacts how:
• Bits are used
• arithmetic operations are implemented.

Signed numbers

• So far we’ve been concentrating on usigned (positive only) numbers
• How can we represent signed numbers in a computer?
• There are a number of schemes for representing signed numbers in binary format.
• sign-magnitude representation
• twos-complement representation.

Sign Magnitude

• Use the most significant bit to represent the sign:
• 0 is positive
• 1 is negative
• Eg. sign magnitude in 8 bits (Big Endian):

0 0 0 0 0 1 1 0 610

1 0 0 0 0 1 1 0 -610

SIGN-MAGNITUDE STEPS

Find the sign magnitude representation of 7010 Step 1: find binary representation using 8 bits

7010 = 010001102

Step 2: if the number is a negative number flip left most bit 01000110 (no flipping, since it is +ve)

So: 7010 = 010001102 (in 8-bit sign/magnitude form)

SIGN-MAGNITUDE TRADEOFFS

• Trade-offs: + Simple and intuitive + easy to implement
• Reduced value range (we lose a bit!)
• How to represent 0?

2’S COMPLEMENT

• We can solve these issues using 2’s complement representation
• We can think of 2’s complement as shifting the range of possible values so that “0” is in the middle of the range.
• Value range: [-2N-1, 2N-1 – 1], where N is the number of bits
• To do this requires a series of steps which you will need to remember

2’S COMPLEMENT REPRESENTATION

Find the 2’s complement representation of –610

Step1: find binary representation in 8 bits
610 = 000001102

Step 2: Complement the entire positive number, and then add one 00000110 (complemented) -> 11111001 11111010

So: -610 = 111110102 (in 2's complement form, using any of above

methods)

Find the Two’s Complement of 7210 Step 1: Find the 8 bit binary representation of the positive value.

7210 = 010010002

Step 2: Since number is positive do nothing.

So: 7210 = 010010002 (in 2's complement form, using

any of above methods)

2’S COMPLEMENT TRADEOFFS

+ M e m o r y e f f i c i e n t (i.e, i t r e t a i n s t h e f u l l representational capacity of the bits) + Retains property that most significant bit still indicates sign + zero represented unambiguously But:

• Slightly more complex transformation (but easily achieved using ALU)

SIGN-EXTENSION

• Sometimes we want to represent a value within a larger word size (eg., an 8, 16 or 32 bit word)
• We can extend any signed binary number by repeating the sign bit up to the size needed

EXAMPLE: SIGN EXTENSION (8 TO 16 BIT) signext.c

• e.g. -16 (stored as 2’s compliment): 27 26 25 24 23 22 21 20 128 64 32 16 8 4 2 1 1 1 1 1 0 0 0 0 becomes: 215 214 213 212 211 210 29 28 27 26 25 24 23 22 21 20

3276 163 819 409 204 102 512 256 128 64 32 16 8 4 2 1
8 84 2 6 8 4
1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 0

+16

0 0 0 1 0 0 0 0

becomes:

0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0

SUMMARY

• Number representation a fundamental design choice of computer systems
• Signed numbers can be represented in different ways:
• Sign magnitude
• 2’s complement
• We can extend either to larger register sizes using sign extension
• Next Lecture: representing real numbers!