1. COMP 1321 Digital Infrastructure
Richard Henson
University of Worcester
October 2018

2. Week 2: Binary Numbers and CPUs
Objectives of session:
Explain the CPU as the fundamental part of the computer
Explain why binary numbers are fundamental to computing
Convert between binary and hexadecimal numbers
Explain the structure of typical CPU

3. Origins of the CPU All about processing data Needs a mechanism for
adding data (input)
removing processed data (output)
storing data outside the CPU
In early days (e.g. Bletchley Park) data input, output, movement, storage were all paper-based… very slow!

4. Binary Numbers Base 2 number system… based on symbols 0 and 1
perfect fit with Boole & logic states (1850s)
surprising no-one before Shannon (1930s) grasped this idea!!!
To represent binary in real world… need “on/off” switches
mechanical… slow
but gotta start somewhere… (!)

5. Electrical Storage of Binary?
Possible using:
Relays
on – magnet has current/off – magnet no current
Transistors
Semiconductor creates
current on/off states
binary numbers
electrical off = 0; on = 1

6. More from Shannon (1948)
Demonstrated the essential unity of all information media:
text
telephone signals
radio waves
pictures, film, etc…
ALL could be represented as 0/1
needed Boolean logic…
for transistor inputs/outputs

7. Binary Logic and Truth Tables
Predicting and calculating outputs for transistor combinations
could use Boolean Algebra…
could use truth tables with results already calculated
Overview:

8. Binary Logic and CPU Design
If memory and data are binary (0/1)…
processing must be binary as well!
Transistors need to be wired together to make a binary data processor (or CPU)
Overview: 8

9. Number Systems (binary and others)
Number of different symbols…
“base” of number system
many tried over the years
Examples:
base 2: binary (2 symbols)
base 10: decimal (10 symbols)
base 12: (inches in a foot, etc. flawed… only 10 symbols)
base 16: hexadecimal (16, and 16 symbols)
Nystrom (1859!)

10. Logic and Digital Logic
“Logical”… through mental activity (human CPU?)
(Mr Spock)
“Emotional”… following “gut feeling”
Now add “digital”
specific type of logic
required to program a CPU!

11. Some Definitions… binary digit bit 0 or 1 byte: a group of 8 bits
(nibble: a group of 4 bits)
“word”: a group of bits of a fixed length
actual length of a word depends on the register size in a CPU 11

12. Number Theory: what does decimal 2314 mean?
Thousands
Hundreds
Tens
units
10x10x10
10x10 10
103
102
101
100
bracket form:
(2 x 103) + (3 x 102) + (1 x 101) + (4 x100 )
most significant digit 2 
least significant digit 4 12

13. Counting in Binary… Same logic as decimal (denary) system Two symbols
When both symbols used start another column…

14. Counting in Hexadecimal
16 symbols
0-9 plus A-F
Same logic as decimal or binary
Why bother?
represents a nibble as a single symbol!

15. What does binary 1101 mean? 1 1 0 1 2x2x2 2x2 2 1 23 22 21 20 
2x2x2 2x  
bracket form – for conversion...  
(1 x 23) + (1 x 22) + (0 x 21) + (1 x20 )  
= 13 in denary (decimal) 15

16. Binary representation of the 8 bits: 1011 0101     
= 181
Q. How many different binary numbers can an 8 bit word hold?
A (= 28) ranging from to 16

17. “Significance” of a bit
All bits aren’t equal!
As with decimal, the greater the value…
the more significant it becomes
e.g. decimal 13821
1 in 10000s column…
has much greater value than 1 in the 1s column

18. What about a 16 bit number
(a) To what decimal number is it equal?
(b) What is the value of the most significant bit?
(c) How many different 16 bit binary numbers can be represented? 18

19. The 16 bit number
(a) To what decimal number is it equal?
Answer: = 53
(b) What is the value of the most significant bit?
Answer: 0
(c) How many different 16 bit binary numbers can be represented?
Answer: 216 = (which is 64 k) 19

20. Shorthand Rows of many 1s and 0s can be very confusing
easy to make mistakes
Solution: divide into blocks of 4 digits (nibbles)
use hexadecimal equivalent for each nibble
very useful in hard disk analysis, when searching for data 20

21. Hexadecimal notation Decimal Binary Hexadecimal 0 0000 0 1 0001 1 21

22. Hexadecimal notation Decimal Binary Hexadecimal 10 1010 A 11 1011 BF 22

23. Notation
Useful to know what type of number we are dealing with e.g. “110”
use subscript at the end
11010 = 110 (denary)
1102 = 110 (binary) = 6 (denary)
110H or = 110 (hexadecimal) = 272 (denary)
So now you know! 23

24. Binary Logic & CPU Binary logic + Boolean algebra
logic gates
mathematically predictable effects of combining them
documented as truth tables
Problem: valves wired together produce very large, energy hungry logic gates:
CPU large, use a lot of energy!

25. Transistors Same effect as valves Wonderful discovery much smaller
use much less energy
Wonderful discovery
very exciting for Shannon?
computers could be much smaller than he expected…

26. Effects of Miniaturisation
Transistors first used in radios
very popular with 1950s “teenagers”
Took some time to become part of more sophisticated devices
breakthrough: integrated circuits (ICs)
many transistors on a single component
first IC with enough transistors to be a CPU with programmed instructions… Intel (1972)

27. Now, let’s make a Computer
… or at least the CPU
(millions of transistors)
Ultra Sparc 1
Pentium 4
Opteron
21364
Itanium 2 McKinley

28. Computer Program (Code)
CPU with INPUT & OUTPUT
Computer Program (Code)
1 do this…
2 do that
3 now this
4 goto 1
CPU
Memory
VDU
Plus Data…
Keyboard

29. So THAT’S how they work!?  Next week: Structure of a CPU and programming it!