Binary Numbers Understanding how numbers are represented. Introduction to binary numbers
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- What I want to do in this video is revisit some ideas that you've probably taken for granted
- since the time that you were, like, three or four years old, but hopefully you'll kind of view it in a new light
- that will help inform us when we think about other types of number systems. So, we have ten digits in our number systems.
- Let me just start counting. So if I have nothing I use the symbol 0. Then if I have one object
- I use the symbol 1. Actually, let me draw this out. So nothing, then if I have one thing, I use the symbol 1.
- If I have two things, I use the the symbol 2. If I have three things, I use the symbol 3. Let me scroll
- down a little bit so you can see that. If I have four things, I use this symbol right over here. If I
- have five things, I use this symbol. If I have six things... let's draw it like that... if I have I six things, I use that symbol.
- If I have seven things, I use that symbol. I know this might be getting a little bit tedious, but this
- all has a point. If I have eight things, eight things, I use this symbol. And if I have nine things I
- use this symbol. And then if I have ten things... what symbol do I use? I've already used up my ten digits, we only have ten digits in a base ten
- system, so we start reusing them. So what we do is we reintroduce this idea of number places. You said
- that I have one ten and zero ones. So you say you have one ten and zero ones.
- ...and then zero ones. We call this one, we say it's in the ten's place. This is literally saying one,
- this is saying one tens, this is one tens plus zero ones. So that's what this is saying.
- But we didn't have to reuse it. We could have had maybe more symbols.
- Maybe this was a symbol, or instead of, or maybe we would've created a new symbol.
- Instead of, you know, all of these had their own symbol, so instead of having to reuse the old ones
- maybe we could've made... the symbol star for ten. And then when you go to eleven we would
- have had another symbol for that... let me go to eleven, just to hit the point home.
- So... two, three, four, five, six, seven, eight, nine, ten, eleven.
- So, eleven in our number system, we say that this is one ten... we say this one ten
- ... let me write it this way... one ten. And then this is also, it's one ten, and then one one.
- ... and then one one. So, it's one ten, plus one one. I know this is kind of strange to see
- this way, but it represents this number of objects. If we had a base eleven, or I guess we could
- say base twelve number system, maybe we would have a symbol for this
- instead of reusing our old digits. Maybe a symbol could have been something wacky
- ... maybe it would've been a smiley face. Who knows what it would've been. And I'll introduce
- higher number base systems in, kind of, future videos where we see, kind of, the symbols
- that are actually used. But, what I want to do in this video is think about
- how would we count, or what symbols would we use
- if we had fewer digits, and in particular, how would we
- count things if we only had two digits - if we only had
- zero and one. Essentially, what we're going to do is think about
- how we would represent numbers in base two.
- Our traditional number system is a base ten number system.
- We have ten digits - zero through nine.
- How would we count in base two?
- So, if you have zero things, you'd still probably say
- "hey, I have zero. I can use the digit zero."
- If I have one thing, I can still say
- "hey, I have one thing"... because, we
- have the digits of zero and one. So, let me make it clear.
- The digits here, the digits in base two, can be zero or one.
- So, if I have one thing, I can still use the number one.
- But, now all of a sudden I have these two objects here,
- and I'm saying that I'm limited... to only these two digits over here.
- So, how can I represent it. Well, instead of
- having a ten's place, I could create a two's place.
- ...and I know it might sound a little bit counter-intuitive but I think you'll
- get used to it a little bit. So, over here in base ten, we said we had one ten and zero ones.
- So in base two, why can't we that we have
- one two - one two - and zero ones.
- Let me make that clear. So, this right here is saying
- one two and zero ones.
- I want to make sure you understand the analogy here.
- In base ten... let me write a larger number in base ten...
- ...and so if I write the number 256 in base ten...
- so, this is base ten over here, what does this say?
- This is saying two hundreds, so, two times a hundred...
- or, maybe I should write down the word so I don't confuse the symbols...
- two hundreds plus five times... or maybe I should I say two hundreds
- plus five tens... two hundreds, plus five tens, plus six ones.
- That's what I represent here, and the way we know that
- is that we know that if we go two places to the left, this is the hundreds
- place, this is the tens place, and this is the ones place.
- And if you know from your exponents, this is equal to ten times ten.
- This right here is equal to ten times itself only once
- and this is equal to ten times itself, I guess
- you could call it, zero times.
- Or, if you know your exponents, this is
- ten to the second power, this is ten to the first power place,
- and this is ten to the zeroth power place.
- And if you added another digit here, that
- would be the thousandths place, which would be
- ten times ten times ten.
- We're going to do the exact same thing in base two
- but, instead of using ten, we're going to
- use two. So, now this is the two's place.
- This is the two's place over here. This is the one's place.
- If we add more digits... let me make it clear...
- So in base two... let me write a number in base two...
- remember, in base two I can only use zeros and ones.
- So, in base two, maybe I have the number 1010.
- So, when you think about it this way, if this was base ten
- you would call this the ten's place, the hundred's place, and the thousandth's place.
- But, this is base two now. So let me be very clear.
- We are only using two digits. So, in base two
- this right here is still the one's place
- now this is going to be the two's place
- remember, in base ten this was the ten's place, now
- this is the two's place.
- Now this would be, and you can take a guess here
- hundreds was ten times ten.
- When we go two spaces to the left in base two
- this should be the two times two's place.
- Or this is the four's place. This over here is going to be the eight's place.
- So, if you wanted to kind of think about this in terms
- of base two, this is one, one eight, plus zero fours,
- plus one twos, plus zero ones. Plus zero ones.
- So, if you wanted to represent this exact same number
- in base ten, it's one eight, plus one two.
- So, in base ten this would be... let me write it over here...
- in base ten this would be an eight plus a two, which is just a ten
- So, this is in base ten. This is how you'd represent
- what we know as this many things - as ten things.
- This is how you'd represent it in base two.
- This is how we know we would represent it in base ten.
- Now let's continue here, just to make sure we understand things.
- So, this many objects, well, in base two we have one...
- if you just have two objects - that's one two and zero ones...
- now three objects would be one two plus one ones.
- So, let me do it over here, so this would be one two
- plus one ones.
- So, this is three objects in base two.
- Now when you go to this, so over here we have one four...
- zero twos and zero ones.
- So, now we're going to go to the four place.
- Because we've essentially maxed out everything.
- If we increment more, we have to go to one more place
- just like we did in base ten, but now we can only use
- the digits zero and one.
- So, now we'll have one four, zero twos, zero ones.
- Now when we add one more, we're going to add one more one
- so, now we have one four, zero twos, and one one.
- and just to be clear, this is this many things.
- This is this many things in base two, this is the four place
- one four and one one. If you wanted to convert
- this into base ten, you'd say look
- "this is one four, zero twos, and one one."
- So, if you have a four and a one, we would represent
- that with a symbol 5 in base ten, but
- we don't have that symbol to us in base two.
- Let's go to this. So, now we're going to increment one more.
- So, how can we represent that in base two?
- This is definitely, we're going to have one four...
- and then we're going to have one two... and then
- we're going to have zero ones.
- And if you keep it... it's kind of fun counting
- in base two, you'll start to get the hang of it.
- So, here we'll have to add one one to this so we
- get one, one, one.
- And now when we get to eight, there's no
- way to kind of increment any of these any
- higher, so we have to get a new place... we have to go to
- the eight's place. So, we have one eight...
- zero fours, zero twos, and zero ones.
- This right here, it might look like a thousand to you
- but it would be a thousand if we were in base ten.
- In base two, this is this many objects. This is eight objects in base two.
- When you go... when you increment at one, we'll
- have this many, we'll have one eight, and then we'll have one one.
- So, it'll be 1001.
- And then, I'll stop here, at what we consider to be ten objects....
- in base two, you would say you have one eight, and you would need one two...
- so zero fours, one two, and zero ones.
- So, this right here is ten in base two.
- This is ten in base ten.
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