Computers in the real world
Objectives
•
•
•
•
Understand what is meant by binary
Why do computers use binary?
Understand what is meant by a truth table.
Be able to read simple logic gate circuit
diagrams
• Be able to create and append to circuit
diagrams.
Computers in the real world
Binary 1 or 0
Computer only understand binary. A binary digit can only be either 1 or
0.
By arranging 1’s and 0’s together in special ways we can create anything!
•Numbers
•Letters
•Pictures
•Music
•Software!
When you read text on a screen the computer sees
a series of 1’s and 0’s!
Computers in the real world
Why?
High Voltage
Low Voltage
0 Voltage
Computers communicate binary by setting a voltage on a wire. A high voltage
represents a 1 and a low voltage represents a 0. 0 voltage DOES NOT means a 0
binary digit! It means the line is broke!
Computers in the real world
Why?
The voltage is measured at fixed time intervals. That way a binary value can be
“read”. The value here is 10011!
Computers in the real world
Why?
What is wrong with this? Voltages sometimes fluctuate. When the values are
close together (like in the diagram below) a fluctuation could mean the difference
between a 7 being seen as a 9! As this is very common it would mean the
computer would crash MUCH more often!
9
7
5
3
1
0 Voltage
1
Computers in the real world
Why?
In order to represent two values we need two separate voltages. In a perfect
world we could have more than two voltages. The diagram below shows what a
signal along a 10 (or normal number) line may look like.
7
5
3
1
0 Voltage
1
Computers in the real world
Why?
High Voltage
Middle Voltage
Low Voltage
0 Voltage
Lets see what happens if a voltage changes in this example.
Computers in the real world
Why?
High Voltage
Middle Voltage
Low Voltage
0 Voltage
Even though it has moved it is still clearly in the low voltage area. The computer
would have no problem recognising this as a 0. Computers and binary go
together like bread and butter. Or cake and chocolate... Mmm chocolate cake 
Computers in the real world
Implications of binary
Ever wondered why computer memory has odd sizes? Like 512 Mb or
1024 Mb?
We already know computers use binary...
This means we use binary or “base 2” for sizes.
ALL sizes on computers are 2x
So the pattern will be 2,4,8,16,32,64,128,256,512,1024,4048 and so on.
Computers in the real world
Activity
Add a new section to your revision notes called binary logic. Explain
•What is binary?
•Why do computers use binary?
You are not expected to be able to convert number
or text into binary!!
Computers in the real world
Computers and logic
You may not believe this but...
Computers are logical!
Honest!
Computers in the real world
What is binary logic
Computers use binary logic to do many great things. It is one of the key
skills needed in order to be able to program a computer. It also helps to
understand why computers can be a “bit thick” sometimes. More on that
later 
Binary logic is where we say if something is either true or false. There is
no maybes or kind of. Just true or false.
For example “Is that a car?” would result in true or false. Or “Mr Hamflett
has a beard today”. Statements like “Is Mr Hamflett awesome?” would
NOT be binary logic as the answer will not be answered the same by
different people.
Computers in the real world
Predictable
Binary logic must be predictable. That means that if you asked anyone
you would always get the same true or false answer. We must be able to
predict the outcome.
It has a posh name. Binary logic is deterministic. You do not need to know
that word but it may be good way to confuse your mates!
Computers are deterministic. That means that their behaviour must be
predictable given the same input and circumstances. As I have already
said you may thing computers have a mind of their own and are NOT
predictable. I can assure you they are and there are reasons why they act
the way they do. More on that later!
Computers in the real world
Activity
Some students like to travel on trains. Not all students like cheese.
Discuss in groups which of the following statements are true based on the
statements above.
a)
b)
c)
d)
Students who like cheese do not like to travel on a train
Student who travel by train like eating cheese
Some students who travel by train will like cheese
Students who do not like cheese may still like the train.
Computers in the real world
Answers
Some students like to travel on trains. Not all students like cheese.
Discuss in groups which of the following statements are true based on the
statements above.
a)
b)
c)
d)
Students who like cheese do not like to travel on a train
Student who travel by train like eating cheese
Some students who travel by train will like cheese
Students who do not like cheese may still like the train.
Computers in the real world
Activity
Open the “understanding logic” work sheet and answer all of the
questions.
Computers in the real world
Binary operators
We can connect truth statements together by using special words. We call
these binary (or Boolean) operators.
The words we can use are –
AND
OR
NOT
By doing this we can derive something else which is true. For example 1. It is raining
2. I have an umbrella
It is raining AND I have an umbrella means I will not get wet
Computers in the real world
Truth tables
We can consider all of the possibilities of the last statement in a little
table. This is known as a truth table.
It is raining
I have an umbrella
I wont get wet
True
True
True
True
False
False
False
True
False
False
False
False
The above table shows the truth for AND. Both things MUST be true for
the result to be true. But hang on there is a mistake! If it is not raining i
will not get wet either! Duh! We need to use our boolean operators to try
and fix this. Let us look at what OR and NOT do!
Computers in the real world
AND
AND – If A and B are both true then A AND B is true.
A
B
A AND B
True
True
True
True
False
False
False
True
False
False
False
False
Key point – Both MUST be true for AND to be true!
Computers in the real world
OR
OR– If either A or B are true then A OR B is true
A
B
A OR B
True
True
True
True
False
True
False
True
True
False
False
False
Key point – At least one of A or B must be true!
Computers in the real world
NOT
NOT– reverses truth. True becomes false and false becomes true!
A
NOT A
True
False
False
True
Key point – This flips truth!
Computers in the real world
Activity
For all of these questions assume that A = True and B = false. Write down
if the result will be true or false.
1)
2)
3)
4)
5)
A AND B
NOT B
NOT A
A OR B
NOT (A AND B)
Computers in the real world
Coming back to our problem
We already know there are issues with the statement below. We want the
truth below but AND / OR / NOT on their own will create it! We will need
to combine them in order to create the correct table.
It is raining
I have an umbrella
I wont get wet
True
True
True
True
False
False
False
True
True
False
False
True
We need to get creative to create the correct truth table!
Computers in the real world
Coming back to our problem
If we flip the raining we get the column in yellow. So rather that say true if
it is raining, we say false.
It is raining
NOT (It is
raining)
I have an
umbrella
I wont get
wet
True
False
True
True
True
False
False
False
False
True
True
True
False
True
False
True
Now look at what we have. We can now use OR to get the required
answers on the left. So our final truth statement is
NOT (it is raining) OR I have an umbrella
Computers in the real world
Activity
For all of these questions assume that A = True and B = false. Write down
if the result will be true or false.
1)
2)
3)
4)
(NOT A) AND (NOT B)
NOT (A OR B)
(A OR B) AND (B OR A)
NOT (A OR B) OR (A AND B)
Once we have gone over the answers you should do the “truth table”
worksheet
Computers in the real world
Truth is a matter of on or off!
Now that we have an idea of what we mean by AND / OR / NOT we can
start looking at how truth is represented on a computer. If you do
electronics then you will already know this!
True = 1 (on)
False = 0 (off)
So rather than writing True or false we can write 1 or 0.
Let us see how this changes the truth tables.
Exam tip – You will be using 1 / 0 in the exam so get used to doing it this
way!
Computers in the real world
AND
AND – If A and B are both 1 then A AND B is 1.
A
B
A AND B
1
1
1
1
0
0
0
1
0
0
0
0
Key point – Both MUST be 1 for AND to be 1!
Computers in the real world
OR
OR– If either A or B are 1 then A OR B is 1
A
B
A OR B
1
1
1
1
0
1
0
1
1
0
0
0
Key point – At least one of A or B must be 1!
Computers in the real world
NOT
NOT– reverses truth. 1 becomes 0 and 0 becomes 1!
A
NOT A
1
0
0
1
Key point – This flips truth!
Computers in the real world
Activity
Copy the truth tables into your revision notes. Make sure you copy the
ones with 1’s and 0’s!
Yes i did say copy! It does not happen very often I know! Bask in the lazy
glow!
Computers in the real world
Logic gates
Now that we have an idea of truth and understand this on / off / 1 / 0
malarkey we can move onto logic gates.
You will need to be able to read them, append to them and create them.
You will also be expected to create truth tables from them.
Below are what AND / OR and NOT look like
Computers in the real world
Remembering them!
You need to remember these symbols. Let us use some memory
techniques to help.
The OR looks a bit like a badge. There is lots
of badges to choose from but your only
allowed to wear one. So could it be that one
OR that one OR that one ....
NOT has a circle at the end which is a bit like
a full stop. You are NOT having that, full
stop!!!
AND looks a bit like a pac man ghost.
Pacman needs to eat power kills AND avoid
ghosts.
Computers in the real world
Activity
In your revision notes copy the symbols. Now think of a fun / interesting
way to remember what they look like. Note it down next to the icon.
This may seem silly but lets face it what does AND look like??? It will be
hard to remember in a years time!
Computers in the real world
Reading results off a logic gate
The diagram below has been labelled with letters. It has two Inputs and
one out put. The inputs are labelled X and Y while the output is Labelled P.
X
P
Y
What would be the output if the inputs were –
X=1
Y=0
P = ???
Computers in the real world
Reading results off a logic gate
X
P
Here is the truth table for this diagram.
Do you recognise it? You should because
this is....
Y
X
Y
P
1
1
1
1
0
0
0
1
0
0
0
0
Computers in the real world
Activity
Complete the questions below
X
P
Y
What would be the output if the inputs were –
X=1
Y=0
P = ???
Create the truth table for this.
Computers in the real world
Combining gates
The diagram below is P = NOT (X OR B)
X
P
Y
How? Well the output of the OR is fed into the NOT. So we do X OR Y and
then NOT it. When you want to NOT the whole of a truth statement you
need to put the NOT at the end. Even though we write it down at the
front.
Bit confusing but it will become clearer over time.
Computers in the real world
Truth table of P = NOT( X AND Y)
X
P
Y
X
Y
P
1
1
0
1
0
0
0
1
0
0
0
1
Computers in the real world
Need help? TRUTH TABLE IT!
X
P
Y
X
Y
X OR Y
P
1
1
1
0
1
0
1
0
0
1
1
0
0
0
0
1
If you add a column for each stage then it will help you work out the
answer! With all these 1’s and 0’s floating around it is easy to get
confuzzled.
Computers in the real world
Activity
Create the truth table (including any help) for the logic gate diagram
below.
L
K
D
You may wish you create a extra column for the first symbol!
Computers in the real world
More that 2 inputs
It is possible to have more than 2 inputs. The diagram below shows three
inputs (A, B and C). The equation is Z = (A OR B) AND (NOT C)
A
B
Z
C
You can create truth tables for these beasts as well!
Computers in the real world
More that 2 inputs
It is possible to have more than 2 inputs. The diagram below shows three
inputs (A, B and C). The equation is Z = (A OR B) AND (NOT C)
A
B
C
Z
1
1
0
1
1
0
0
1
0
1
0
1
0
0
0
0
1
1
1
0
1
0
1
0
0
1
1
0
0
0
1
0
So how did I make this? I will
take you through the steps
Computers in the real world
Step 1 – Write out all the permutations
The first step is write down all of the possibilities. As there are now 3
inputs there are extra rows.
A
B
C
1
1
0
1
0
0
0
1
0
0
0
0
1
1
1
1
0
1
0
1
1
0
0
1
Every possibility must be
written down. As a rule of
thumb
1 input = 2 rows
2 inputs = 4 rows
3 inputs = 8 rows
4 inputs = 16 rows..
They double each time!
As a tip. When you add a extra
output copy all of the rows and
then add 0’s in the new input
for the first half and 1’s for the
second half.
Computers in the real world
Step 2 – Perform the first logic gate
Now we create a new column for the first logic gate. That will help us
work out the output of that gate. This will then be the input to the next!
A
B
C
A OR B
1
1
0
1
1
0
0
1
0
1
0
1
0
0
0
0
1
1
1
1
1
0
1
1
0
1
1
1
0
0
1
0
Computers in the real world
Step 3 – Do the same for the next logic gate
We just add a new column for each logic gate. Things are starting to take
shape!
A
B
C
A OR B
NOT C
1
1
0
1
1
1
0
0
1
1
0
1
0
1
1
0
0
0
0
1
1
1
1
1
0
1
0
1
1
0
0
1
1
1
0
0
0
1
0
0
Computers in the real world
Step 4 – The result!
Z = (NOT C) AND (A OR B)
A
B
C
A OR
B
NOT C Z
1
1
0
1
1
1
1
0
0
1
1
1
0
1
0
1
1
1
0
0
0
0
1
0
1
1
1
1
0
0
1
0
1
1
0
0
0
1
1
1
0
0
0
0
1
0
0
0
Computers in the real world
Activity
Write down the equation related to this diagram and then create a truth
table.
L
D
K
A
The key to success here is HOW you do it, not the answer!