1. COMP 1321 Digital Infrastructure
Richard Henson
University of Worcester
October 2017

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… 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
binary numbers
relay off = 0
relay on = 1

6. Shannon & Relays
1938: first scientist to use Boolean Logic with relays: on/off 1/0
realised binary numbers could replace decimal numbers & represent real data values
Figured that relays could be wired together to create logic gates and do Boolean Algebra (!)
problem… relays slow

7. A Fast Switch Electronic… But once working, it did work, and quickly…
called valves, but actually glass tubes
air taken out; near vacuum
didn’t work until “warmed up”
needed a lot of energy to work
took up lots of space
But once working, it did work, and quickly…
Head of IBM (1950s) said that the world would only need four of these computers anyway (!)

8. More from Shannon (1948)
Demonstrated the essential unity of all information media:
text
telephone signals
radio waves
pictures, film, etc…
ALL could be encoded in the universal language of binary digits
used term “bit” for a binary digit

9. Binary Logic and CPU Design
Putting theory into practice…
Data stored as blocks of ‘cells’ (effectively switches)
voltage off/on
in binary either “0” or “1”
If memory and data binary…
processing must be binary as well! 9

10. 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!)

11. 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!

12. 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 12

13. 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 13

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

15. 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!

16. 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) 16

17. 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 17

18. “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

19. 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? 19

20. 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) 20

21. 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 21

22. Hexadecimal notation Decimal Binary Hexadecimal 0 0000 0 1 0001 1 22

23. Hexadecimal notation Decimal Binary Hexadecimal 10 1010 A 11 1011 BF 23

24. 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! 24

25. 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!

26. 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…

27. 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)

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

30. 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

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