Hello in this video we're going to be talking about number systems and justifying how I guess.
Simple number systems can be.
So I'm going to be using Excel right here and I'm going to be using my pen.
I have a system that's already out here.
Decimal hex octal and binary decimal is base 10 which means that it's going to be numbers 0 1 2 3 4
5 6 7 8 and 9.
Those are the numbers allowed in base 10 and base 8.
You can only use 0 through 7 in base 2 which is binary.
This is base to that octal is base 8 hexes base 16 and decimal is base 10 hex has 16 characters in it.
Those characters are 1 2 3 4 5 6 7 8 9.
And then it starts doing a b c d e f and that's still 15 So we'll see how to justify this.
So let me give you my justification and then we can see that it can apply to any of these numbers and
we can build any of these numbers pretty quickly.
OK so I'm going to write out the numbers in a sense in this fashion.
I'm going to put 0 0 0 1 0 2 0 3 0 for that.
So you get the 0 9 that uses all of our base 10 numbers and then now I counted with a 0 in front.
Now I'm going to count with a 1 in front and those are the next numbers 10 11 12 with a 1 in front 13
14 until you get to dot dot dot 90.
After I counted to zero and I'm going to count with the two in front you see I'm just going past these
numbers the 0 1 2 is the base and then and then I go with the ones that I go with the two unfriends.
A 22 23 24 twenty nine.
Now I keep going until I get to 9 with a zero zero with a 9 in front 0 with a 1 and 9 with a 1 and a
9 and a 1 9 2 9 3 9 and afford that that 9 and 5 9 and 9.
So those are my numbers and I just play this game where I did it now if I want to go further and count
the next number all I have to do is take all of these numbers and put one more number in front of each
of these put a one in front of each of these.
That's 100 101 1 or 2 1 0 3 1 0 4 1 0 until 1 or 9.
Then you get 110 and you keep going and you just put a one in front.
So you can see that this system this idea will work for any for base 10 and that's how we count where
we have the zeros and we counted the ones that we put a 1 in front.
And if you wanted to get to the 200 you're taking this same system except instead of a 1 in front you'd
have two was in front of this whole thing.
And once you got seven times in front you you have 999 at the bottom you would have another zero and
then you'd put another one.
And that's basically the system.
So let's now just try the same thing.
But what I'm going to do is I'm going to do it in binary and hex and justified those examples for what
we would we normally do.
OK so let's count in base 16.
OK.
So well let's keep count and base.
Kept in place to first OK so base 2 is going to have a zero.
I'm going to put 0 0 0 0 and then 000 1.
That would be the zero.
And that would be the one.
OK.
Now we finish the end because we got to all the numbers in base 2.
That's all the numbers they're 0 and 1.
So now we have to put our one change that spot to or 1.
So now we have the equivalent of 1 2 3 4.
That's the equivalent of 1 2 3 4.
OK.
Now we already finished all the options with the one in front.
So let's move to the next position and I'm going to take this whole block and I'm going to change the
third zero or the second 0 2 or 1.
So I'm going to have that's going to be a one that I'm going to change that to a 1 and I'm going to
change this one to 1 and I'm going to change the next one to a 1 K so that's 0 1 1 1.
And that's our Those are our next numbers and that's our logical.
That's our logic of just applying this 5 6 0.
This is why I'm so sorry that 0 1 2 have to be careful there that's zero because that's a zero right
there.
That's one two three four five six and seven.
Now if I want to get the next thing what do I do.
Well I just apply my logic which is change this first zero to a one and let's fill it in for all of
them so I can change the first one to one filter and so it's going to be a 1 000 that's going to be
a 1 0 0 1 and a 1 0 1 0 a 1 0 1 1 and let me go down a drop and now I'm going to do the second block
1 1 0 0 1 0 1 0 1 1 1 1 0 1 1 1 1 and you can imagine if that was 7 then this will be 8.
This will be nine except for now the nice thing about this is we don't even have to say that these are
all positive numbers.
We could actually break these up and say that some of these are negative numbers.
We can just decide based on you know a very logical system to compliment whatever that means.
We can just decide to keep these as the positive numbers and these as the negative numbers.
So whatever number has a 1 in front is going to be negative.
Whatever number it has a 1 in front is going to be negative and you can see that half of them are going
to have a 1 in front.
If I were to do this again and go to the next level and add another one in front then these are all
going to have zeros in front and then they're going to be a half of them that have one in France or
half are going to be positive half to be negative.
Now this one's going to attain a zero the other it going insane.
You know the negative eight that are paid.
That's still fine.
But we can see that we have a reasonable counting system.
We can name something cause if we were in and we can name something negative.
OK.
And zero not being negative obviously.
Now what we can do is we could.
So we basically are justifying hopefully in a reasonable way a counting system based on any base in
base 16 we can do the same thing.
We're going to have to have units in our units are going to be 0 1 2 3 4 5 6 7 8 9.
Now 10 is not a unit.
So instead we're going to say a b c d he s that A is going to represent for us 10.
Now that's only represented in terms of binary and decimal right 10 11 12 13 14 and 15.
So this is base 16 because we have the zero there so there's 16 units there.
So that's how we can know what would come after death.
Well if this is the system that we're using We're going to employ the same logic we can just put a one
in the next number in front of all of these and we would get that that that that until we got a one
a one B that at that and one f what would the next numbers be 2 0 to 1 that dot to a to b that to F
X cetra and we would keep going until we got that and that's our logical system.
Now what does 10 really represent that represents 16.
What does 11 represent 17 12 is 18 etc. and you can see that I did this in Excel actually just to show
you that you can see down here this column which I will color in so that way we can identify which is
the decimal.
OK so this is the decimal column and I'm going to give it a border OK in the decimal column.
We could see that 10 11 k I'm going to put in a freeze panes.
It's a nice useful thing there.
OK so that Hex has 10 11 12 13 16 is that.
Now let's just justify and of course and actually could do that.
I used formulas deaths to hex that changed the decimal to hex deaths to Auct. that change to octo and
d'Este to bind every change in the binary so you can see all the binary and hex numbers in Excel has
it for you.
You can go the other way as well.
Now what I'd like to do is I would like to justify that 10 should be 16.
Now here's the justification in any of these numbers systems that we've been talking about.
Let's just look at base 2.
What we do is we have each spot in here is going to be 2 2 0 2 2 1 2 2 2 2 2 3 2 2 to 4 2 to 5 etc..
If I want to count it that I include it and I put one next to the two if not I put a zero and to add
these all up.
So let's just pick this number right here 1 1 0 1 so I'm going to include this include this.
Don't include that include this.
So that's going to be 1 times 2 to 3 which is a that's 1 times to do the 2 which is 4 0 and that's 1
times 2 to 1 which is 1 because the what is 1 and so that gives me that give me 12.
That gives me 8 10 plus for K and 2 2 2 2 do in decimal that's 32 which is a plus for plus 1 which is
13.
OK so why don't we try this with some other numbers.
For instance let's try it in terms of hex that was with base 2.
We did that.
I'm going to try it in terms of hex.
That's going to be 16 because it's base 16 to the zero 16 to the widest 16 to 16 to 3 etc..
And of course all of these keep going 16.
Let's see what 1 0 is.
Well one in zero means I'm going to include the 16.
What does that mean include the 16 that's 16 plus 0.
So that is perfect because in decimal that's base 10 over there in decimal.
I get 1 0.
If I'm talking about base 16.
So base 10.
Is there some of that 16 in in hex that's 1 0.
If I wanted to check out what hex is for other numbers like let's just go a little bit further and find
another number that might be nice.
OK one f..
Let's see what one f is.
OK so I'm going to see one f so I'm going to really just need two spots.
I have my 16 to the zero and 16 to the 1.
Now I have one time 16 to the 1 plus f time 16 to 0 Well that's 16 plus in base 10.
This is all in base 10 right here.
This is base 10 right there.
I'm just converting it.
Plus f times 1.
Well what's f f is really 15.
OK.
So let's see.
One should give me 31 which it does in base 10 which is perfect.
So those are the numbers that I'm counting now.
You'll notice that if I go further and further down till I get to about 100 or so in the binary There
are a lot of numbers.
In octal there's three spots taken up and in hex there are only two spots taken up and you could imagine
that if I were to continue to drag this down it would just it would continue on in this fashion.
Hex is still only two numbers.
Decimal is already three numbers octal actually three numbers and binary is a bunch and you can imagine
if you keep going.
This is at B B this is at 187.
OK well what's b b.
Let's just double check make sure we're good with that baby.
That's 16 to 1 and 16 to 0.
That's b times 16 plus B times 1.
Because 7:44 as hero is 1 What was B.
Well a was 10 so B is 11 so that's 11 times 16 plus 11.
OK.
Which baby is 187 in base 10.
So that's what you get.
So this is the conversion between them.
We can look this up but this is the logic behind how these things work.