SLIDE 1 Slides for Lecture 4
ENEL 353: Digital Circuits — Fall 2013 Term Steve Norman, PhD, PEng
Electrical & Computer Engineering Schulich School of Engineering University of Calgary
16 September, 2013
SLIDE 2 ENEL 353 F13 Section 02 Slides for Lecture 4
slide 2/17
Previous Lecture
◮ a little more about binary, octal, and hexadecimal
numbers
◮ introduction to signed and unsigned number systems ◮ unsigned binary integer addition ◮ sign/magnitude representation of signed integers
SLIDE 3 ENEL 353 F13 Section 02 Slides for Lecture 4
slide 3/17
Today’s Lecture
◮ drawbacks of sign/magnitude representation of signed
integers
◮ two’s complement representation of signed integers
Related reading in Harris & Harris: Section 1.4.6.
SLIDE 4
ENEL 353 F13 Section 02 Slides for Lecture 4
slide 4/17
Some important ideas from the previous lecture
Number systems in digital circuits and computer systems usually have a fixed quantity of bits available to represent a number; this quantity is called the width of the number system. Number systems that allow negative numbers, zero, and positive numbers are called signed number systems. sign/magnitude is one way to set up a system of signed integers.
SLIDE 5 ENEL 353 F13 Section 02 Slides for Lecture 4
slide 5/17
Sign/magnitude numbers (last slide from previous lecture)
We use sign/magnitude in daily life with decimal numbers, such as +5, −37, 44. (If the sign is left out, we assume that it’s +.) For binary numbers, we can use 1 to stand for − and 0 to stand for +. Let’s look at some examples of binary sign/magnitude
- representation. (Done in previous lecture.)
A new question for today: What is the range of an n-bit sign/magnitude system?
SLIDE 6 ENEL 353 F13 Section 02 Slides for Lecture 4
slide 6/17
Drawbacks of sign/magnitude systems
It’s annoying that zero has two different representations: 000 · · · 00 and 100 · · · 00. Ideally, two numbers should be equal only if they have identical bit patterns. Complicated logic is needed to add two numbers that have
SLIDE 7
ENEL 353 F13 Section 02 Slides for Lecture 4
slide 7/17
What makes a number format good?
In digital systems design it is important that arithmetic circuits are small and efficient. It is not important that numbers be easy for humans to read! By these criteria, sign/magnitude is not a winning format—circuits for the common operations of comparison and addition are more complicated than necessary.
SLIDE 8
ENEL 353 F13 Section 02 Slides for Lecture 4
slide 8/17
Bit numbering, the MSB, and the LSB
Each bit in an n-bit pattern can be given a number to identify the position of the bit within the pattern. The range of bit numbers runs from 0 up to and including n − 1. Bit n − 1, on the far left, is called the MSB (most significant bit). Bit 0, on the far right, is called the LSB (least significant bit). Let’s draw a diagram. For the eight-bit pattern 01110101, what are the bit numbers and values of the MSB and LSB? What is the value of bit 4?
SLIDE 9 ENEL 353 F13 Section 02 Slides for Lecture 4
slide 9/17
Two’s complement systems for signed integers
Two’s complement is way, way more widely used than sign/magnitude or any other method for representing signed integers. Examples:
◮ With current processor chips, operating systems, and C
and C++ compilers, the int type is almost certain to be 32-bit two’s-complement.
◮ In Java programming, the int type is guaranteed to be
32-bit two’s-complement, and the long type is guaranteed to be 64-bit two’s-complement.
SLIDE 10
ENEL 353 F13 Section 02 Slides for Lecture 4
slide 10/17
Two’s complement systems: Zero and positive numbers
In two’s complement, zero and positive numbers are just like sign/magnitude: the MSB is 0, and all the other bits are used for the magnitude. What are the 5-bit two’s-complement representations of 0 and 910?
SLIDE 11 ENEL 353 F13 Section 02 Slides for Lecture 4
slide 11/17
Two’s complement systems: Negative numbers
Suppose that a is a positive n-bit number. Then the bit pattern for the n-bit two’s-complement representation of −a is the same as the bit pattern for the unsigned number 2n − a. Example: What is the 5-bit two’s complement representation
Let’s compare 5-bit two’s-complement and sign/magnitude rep’s of +1010 and −1010.
SLIDE 12 ENEL 353 F13 Section 02 Slides for Lecture 4
slide 12/17
Two’s complement systems: Finding the n-bit pattern for a negative number
The algorithm is:
- 1. Get the n-bit binary representation for the magnitude of
the number.
- 2. Invert (in other words, complement) each bit. Each 0
changes to 1, and each 1 changes to 0.
- 3. Add 1 to the result using n-bit unsigned addition. (That
implies ignoring the carry out of the MSB.) Let’s demonstrate for the 5-bit rep of −1010. What happens if we try to find the 5-bit rep of −0 (“negative zero”)?
SLIDE 13 ENEL 353 F13 Section 02 Slides for Lecture 4
slide 13/17
6-bit two’s complement examples
What are the 6-bit two’s-complement representations of −1710 and 2310? Attention: A width must be specified. It does not make sense to ask, “What are the two’s-complement representations
- f −1710 and 2310?” without any particular width in mind.
SLIDE 14 ENEL 353 F13 Section 02 Slides for Lecture 4
slide 14/17
Two’s-complement negation
Review from two slides back . . .
- 1. Get the n-bit binary representation for the magnitude of
the number.
- 2. Invert (in other words, complement) each bit. Each 0
changes to 1, and each 1 changes to 0.
- 3. Add 1 to the result using n-bit unsigned addition. (That
implies ignoring the carry out of the MSB.) Steps 2 and 3 together are called two’s-complement negation. Let’s try two’s-complement negation starting with a negative number.
SLIDE 15
ENEL 353 F13 Section 02 Slides for Lecture 4
slide 15/17
Two good things (among many) about two’s complement
(1) There is only one way to represent zero, and it’s obvious: all bits are 0. (2) Perhaps surprising, but true and extremely useful: An unsigned binary adder circuit will correctly perform two’s-complement addition! Let’s demonstrate this with our 6-bit two’s-complement bit patterns for −1710 and 2310.
SLIDE 16
ENEL 353 F13 Section 02 Slides for Lecture 4
slide 16/17
Why does a circuit for unsigned addition also work for two’s-complement addition?
It might seem like we’ve picked a system more or less at random for representing negative numbers. Therefore it might seem like a fluky accident that an adder designed for unsigned numbers will handle negative numbers correctly! In fact, straightforward but time-consuming mathematics proves why this is true. Your textbook and lectures both skip the proofs!
SLIDE 17 ENEL 353 F13 Section 02 Slides for Lecture 4
slide 17/17
Next Lecture
◮ ranges for two’s-complement systems ◮ overflow in two’s-complement addition ◮ BCD (binary coded decimal) codes for decimal digits ◮ Gray codes
Related reading in Harris & Harris: Section 1.4.6, for the two’s-complement stuff. Harris & Harris do not really cover BCD or Gray codes—there are very brief descriptions on pages 258 (BCD) and 76 (Gray codes).
