- [Voiceover] For as long as human
beings have been around we've
been counting things, and we've been
looking for ways to keep track and
represent those things that we counted.
So, for example if you were
an early human and you were
trying to keep track of the days
since it last rained you might say
okay let's see it didn't rain today so
one day has gone by, and we now use
the word one, but they might have
not used it back then.
Now another day goes by.
Then another day goes by.
Another day goes by.
Another day goes by, then it rained.
And so when his friend comes
he says, "Well, how long has it been
since we last rained."
Well you would say, "Well, this is how
many days it's been."
And your friend would say, "Okay,
I think I have a general sense of that."
And at some point they probably
realized that it's useful to have names
for these.
So they would call this one, two, three,
four, five, six, seven.
Obviously every language in the world
has different names for these.
I'm sure there are lost languages
that had other names for them.
But very quickly you start to
realize that this is a pretty bulky
way of representing numbers.
One it takes a long time to write down.
It takes up a lot of space,
and then later if someone wants to
read the number they have to sit here
and count.
It's hard enough with seven,
but you could imagine if there were
what we call 27 of it, or 1000 of it.
Then it would take up, possibly, a whole
page and even when you counted
you might make a mistake.
And to solve this human beings
have invented number systems.
And it's something that we take for granted.
You might say, "Oh, isn't that just the
way you've always counted?
But hopefully over the course
of this video you'll start to appreciate
the beauty of a number system
and to realize our number system isn't
the only number system that is around.
The number system that most
of us are familiar with is the base 10
number system.
Often called the decimal, the decimal
number system.
And why 10?
Well probably because we have 10 fingers.
Or most of us have 10 fingers.
So, it was very natural to think
in terms of bundles of 10 or to have 10
symbols.
So however many bundles you have
you can use your fingers and eventually
your symbols to think about how many there are.
And since we needed 10 symbols
we came up with zero, one, two, three, four,
five, six, seven, eight, nine.
These 10 digits, these are our 10 symbols
that we use in the base 10 system.
To just give us a little bit of a reminder
how we use them imagine the number 231.
So, 231. 231.
What does this represent?
Well, what's neat about number systems
is we have place value.
This place all the way to the right,
this is the ones place.
This is the ones place.
This literally means one, one.
One bundle of one.
So, this is one, one right over here.
This right over here, this is in the 10s place.
This is in the 10s place.
This three here, literally means three 10s.
So this literally means three 10s.
And this two here, this two here is in
the 100s place.
It's in the 100s place.
So, this represent two 100s.
You add them together and once again
I'm still thinking in base 10, you'd
get 231.
This is two 100s plus three 10s plus one.
In our base 10 system notice every
time we move to the left we're thinking
in bundles of 10 of the space
to the right.
So, this is the ones place.
You multiply by 10, you go to the 10s place.
You want to go to the next place
you multiply by 10 again.
You get the 100s place.
If you're familiar with exponents,
one is the same thing as 10 to the zero power.
10 is the same thing as 10 to the first power.
So this is the 10s place. Three tens.
And 100 is the same thing as 10
to the second power.
Obviously we could keep going on and on
and on and on and on.
That is the power of the base 10 system.
So, you might be curious now.
"Well, what if this wasn't 10 here?
What if we did, let's just go as simple
as we can. You can almost view this
as a base one system.
You only have one symbol right over here.
But what if we went to something slightly
more complex, a base two system.
You'd be happy to know that not only
can we do this, but the base two system
often called the binary system.
This is called the decimal system.
The base two system often called
the binary system is the basis of all
modern computing.
It's the underlying mathematics
and operations that computers perform
are based on binary.
And in binary you have two symbols.
You have zero and you have one.
The reason why this is useful for computation
is because all the hardware that we use
to make our modern computers, all
of the transistors and the logic gates
they either result in an on or an off state.
On or an off state.
And so what we do is when you use
your calculator or whatever you might
be operating in base 10, but underlying
everything it is doing the operations
in binary.
But you might say well how do we
actually think in terms of binary?
Well, we can construct similar places here,
but instead of them being powers of 10
they're going to be powers of two.
So, let's set up some places here.
So, all the way on the right
two to the zero power is still one.
So we can still call that the ones place.
Then we can move to the left of that.
We can move to the left of that.
That would be two to the first power.
So we could call that the twos place,
and I can even write it out if I want.
Twos place instead of the 10s place.
Then I could keep going.
Instead of this space being the 10 to
the second or the 100s place, it will be
the two to the second, or the fours place.
And I can keep going.
I encourage you actually to pause
the video and try to build this out
for yourself.
What would this be?
Well this would be two to the third,
or the eights place.
Notice every time we're doing this
we're multiplying by two.
Everytime we go to the left,
just like we multiplied by 10 here.
So notice everywhere you see this 10s
we're now dealing with twos.
Let's keep going.
Let's keep going and then we can actually
represent this number using binary.
So, let's do that.
So, this right over here I've already
used that color.
This right over here, this is two
to the fourth.
We could call that the 16s place.
Then we could have --
I'll reuse some of these colors.
This is two to the fifth.
We could call this the 32s place.
Then we can go two to the sixth.
We can call that, multiply by two again,
or two to the six is 64.
So this is the 64s place.
Tells us how many 64s we have. Zero or one 64s.
We'll see that in a second.
Then we can go over here.
This would be two to the seventh.
That would be the 128s place.
And we can obviously keep going on
and on and on, but this should be enough for
me to represent this number.
In future videos I will show you how
to do that, but let's actually represent
the number.
It turns out that this number
in decimal can be represented
as 11100111 in binary.
What does this mean?
This means you have one 128 plus one 64,
plus one 32, plus no 16s, plus no eights,
plus one four, plus one two, plus one one.
So you can see that these are going
to be the same thing.
Notice, this is one 128.
So it's 128, plus 64, plus 32.
We have zero 16s, zero eights.
So we're not going to add those.
Plus four, one four.
Plus one two.
Plus one one.
And add these together,
and once again when we're doing this,
when I'm writing it this way I'm
kind of using the number system
that we're most familiar with.
We're most used to doing the operations in,
but when you do it you will see that
this is the exact same number as 231.
This is just another representation.
One isn't better than the other.
The only reason why I converted this
is this is what I'm used to thinking in.
It's what I'm used to doing operations in.
So, hopefully you find that pretty interesting.
To me, this kind of opened my mind
to the power of even our decimal system.
In future videos we'll explore other
number systems.
The most used ones, base 10 is
used very heavily, binary and there's
also hexadecimal where you don't have
two digits or not 10 digits, but you have 16 digits.
And we'll explore those in future videos
and how to convert between or rewrite the
the different representations and different bases.