### Video Transcript

Three NOT gates are connected
together in series. If the input of the first NOT gate
is zero, what will the output of the final NOT gate be?

So we′re told here that we′ve got
three NOT gates and that they are connected in series. We can recall that the symbol for a
NOT gate is a triangle oriented so that one of its corners points to the right. And on this rightward-pointing
corner, there′s a small circle. We know that we′ve got three of
these NOT gates connected in series, which means that they are joined up in a line
like this.

In this arrangement of gates, the
input to the system is the input to the first left-hand NOT gate. We can then see that the output
from this first NOT gate becomes the input to the second NOT gate, and similarly the
output from the second NOT gate becomes the input for the third NOT gate. In other words, starting from the
left-hand side, the output of each NOT gate then becomes the input for the next one
along the line. When we get to the third and final
NOT gate over on the right, the output from this gate is then the overall output of
the system.

We′re told in the question that the
input to the first NOT gate is zero. Now the first NOT gate is this one
over here on the left, and we can add our input value of zero to the diagram. To work out what value this output
will have, we′re going to need to follow this initial input of zero and see what
happens to it as it goes through each NOT gate in the line. Now to do this, we need to know how
a NOT gate works.

We can recall that a NOT gate acts
to negate its input value. This means that a NOT gate turns an
input value of zero into an output value of one, while an input value of one becomes
an output of zero. This table that we have drawn here
is known as the truth table for a NOT gate. And we can use it to help us work
out what′s going on in this diagram. Let′s begin with the first NOT
gate, so that′s this one here on the left. We know that it has an input of
zero. And we can see from our truth table
that an input of zero means the NOT gate will have an output value of one.

So let′s add this output value to
the diagram. This output then becomes the input
for the second NOT gate, so that′s this one here in the middle. We′ve got an input value of
one. And our truth table tells us that
an input of one means an output of zero. Again, let′s add this value to the
diagram. This output of zero then becomes
the input to the third and final NOT gate. We know that an input of zero gives
an output of one. And so the output from this third
NOT gate must be equal to one. Since the third NOT gate is the
final one in this series arrangement, then this output value of one is the output of
the final NOT gate, which is what we were asked to find. So our answer is that for three NOT
gates connected together in series with an input to the first NOT gate of zero, the
output of the final NOT gate will be equal to one.

As an aside, it′s worth noticing
that we would get the same result for any odd number of NOT gates connected in
series. So whether that′s three NOT gates,
five NOT gates, seven NOT gates, or any other odd number, the final output value
will always be the opposite of the initial input value. So in the example we had here, we
started with an initial input of zero, and we ended up with an output of one. Likewise, if we′d started with an
input of one, we would have got a final output value of zero.

We can understand this by thinking
about how a NOT gate works. We know that a NOT gate negates its
input value; that is, zero becomes one, while a one becomes zero. So if a single NOT gate negates its
input value, then two NOT gates together in series must negate that value twice. So then after two NOT gates, the
value ends up back the same as it started. If we had a load of NOT gates
connected together in series, we could then imagine mentally pairing them off.

So for example with these seven NOT
gates here, we would pair off the first and second NOT gates, the third and fourth
gates, and the fifth and sixth ones. The first NOT gate negates its
input value, and then the second NOT gate negates this value again. So the second NOT gate undoes the
effect of the first one, and effectively they cancel each other out. The same thing then happens with
the third and fourth NOT gates. The third one negates the value,
the fourth one negates it again, and the effect is that they cancel each other
out. We can see that in the same way all
pairs of NOT gates will end up canceling each other out. So here, the fifth and sixth NOT
gates will act to cancel each other.

Whenever we′ve got an odd number of
NOT gates like this, then they′ll all pair off and cancel each other out except for
one last NOT gate. So the overall effect of any odd
number of NOT gates connected in series is just the same as that for one single NOT
gate. And we know that the effect of a
single NOT gate is to negate its input value.