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Number Systems Explained · Acedra
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Number Systems Explained
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0:00
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, and it'll help inform us when we think about other types of number systems. So we have 10 digits in our number system. So 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.
0:25
And then if I have one thing, I use the symbol 1. If I have two things, I use the symbol 2. If I have three things, I use the symbol 3. Let me scroll down a little bit, make sure 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, I'll start-- let's draw it like that-- if I have 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,
0:57
but this all has a point. If I have eight things, I use this symbol. And if I have nine things, I use this symbol. And then if I have 10 things, so 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. What symbol do I use? I've already used up my 10 digits. We only have 10 digits in a base 10 system. So we start reusing them. So what we do is we introduce this idea of number places. You say that I have 1 ten and 0 ones. So you say you have 1 ten, and then 0 ones. We call this 1, we say it's in the tens place.
1:44
This is literally saying 1, this is saying 1 tens. This is 1 tens plus 0 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 maybe we would have 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 have made the symbol star for 10. And then when you go to 11, maybe would have had another symbol for that.
2:14
Let me go to 11 just to hit the point home. So 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. So 11 in our number system, we say that this is 1 ten-- let me write it this way-- and then this is also, it's 1 ten and then 1 one.
2:37
So it's 1 ten plus 1 one. I know this is kind of strange to see it this way, but it represents this number of objects. If we had a base 11, or I guess we could say base 12 number system, maybe we would have had 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 have been? And I'll introduce higher number base systems in kind of future videos, where we see what,
3:04
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 0 and 1. And that's essentially what we're going to do, is think about how we would represent numbers in base 2. Our traditional number system is a base 10 number system. We have 10 digits, 0 through 9. How would we count in base 2?
3:36
So if you have zero things, you'd still probably say, hey, I have zero, I can use the digit 0. If I have one thing, I could still say, hey, I have one thing, because we have the digits 0 and 1. So let me make it clear. The digits here, the digits in base 2 can be 0 or 1. So if I have one thing, I can still use the number 1. 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 could I represent it?
4:09
Well, instead of having a tens place, I could create a twos place. And I know it might sound a little bit counterintuitive, but I think you'll get used to it a little bit. So over here in base 10, we said we had 1 ten and 0 ones. So in base 2, why can't we say that we have 1 two and 0 ones? Let me make that clear. So this right here is saying 1 two and 0 ones. I want to make sure you understand the analogy here. In base 10-- let me write a larger number in base 10.
4:45
So if I write the number 256 in base 10. So this is base 10 over here. What is this saying? This is saying 2 hundreds, so 2 times 100-- or maybe I should write down the word, because I don't want to confuse the symbols-- 2 hundreds plus 5 times, or maybe I should say, 2 hundreds plus 5 tens plus 6 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.
5:31
And if you know from your exponents, this is equal to 10 times 10. This right here is equal to 10 times itself only once. And this is equal to 10 times itself, I guess you could call it, zero times. Or if you know your exponents, this is 10 to the second power. This is 10 to the first power place. And this is 10 to the 0-th power place. And if you added another digit here, that would be the thousands place, which would be 10 times 10 times 10. We're going to do the exact same thing in base 2,
5:59
but instead of using 10, we're going to use 2. So now this is the twos place. This is the twos place over here. This is the ones place. If we add more digits let me make it clear. So in base 2, let me write a number here in base 2. And remember, in base 2 I can only use zeroes and ones. So in base 2 maybe I have the number 1010. So when you think about it this way, if this was base 10, you would call this the tens place, the hundreds place, and the thousands place. But this is base 2 now.
6:31
So let me be very clear. We are only using two digits. So in base 2, this right here, this is still the ones place. Now this is going to be the twos place. Remember in base 10 this was the tens place. Now this is the twos place. Now this would be, and you could take a guess here, hundreds was 10 times 10. When we go two spaces to the left in base 2, this should be the 2 times twos place, or this is the fours place.
7:00
This over here is going to be the eights place. So if you wanted to kind of think about this in terms of base 2, this is 1 eight plus 0 fours plus 1 twos plus 0 ones.
7:22
So if you wanted to represent this exact same number in base 10, it's 1 eight plus 1 two. So in base 10 this would be-- let me write it over here-- in base 10, this would be an 8 plus a 2, which is just a 10. So this is it in base 10. This is how you would represent what we know as this many things, as 10 things, this is how you would represent it in base 2, this is how we know we would represent it in base 10. Now let's continue here, just to make sure we understand things.
7:56
So, this many objects. Well, in base 2, we have 1 if you just have two objects, that's 1 two and 0 ones. Now, three objects would be one 2 plus 1 ones. So let me do it over here. So this would be 1 two plus 1 one. So this is three objects in base 2. Now when you go to this. So over here we have 1 four, 0 twos, and 0 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.
8:32
Just like we did in base 10, but now we could only use the digits 0 and 1. So now we'll have 1 four, 0 twos, 0 ones. Now when we add one more. We're going to add one more 1. So now we have 1 four, 0 twos, and 1 one. And just to be clear, this is this many things. This is this many things in base 2, this is the four place, 1 four and 1 one. If you wanted to convert this into base 10, you'd say, look, this is 1 four, 0 twos, and 1 one. So if you have a four and a one, we
9:10
would represent that with the symbol 5 in base 10. We don't have that symbol available to us in base 2. Let's go to this. So now we're going to increment one more. So, how can we represent that in base 2? This is definitely, we're going to have 1 four. And then we're going to have 1 two. And then we're going to have 0 ones. And if you keep-- it's kind of fun to practice counting in base 2. You'll start to get the hang of it. So here we have to add 1 one to this, so we get 1, 1, 1.
9:38
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 eights place. So we have 1 eight, 0 fours, 0 twos, and 0 ones. This right here, it might look like 1,000 to you, but it would be 1,000 if we were in base 10. In base 2 this is this many objects. This is eight objects in base 2. When you increment it one, we'll have this many, we'll have 1 eight and then we'll have 1 one. So it would be 1001.
10:10
And then I'll stop here at what we consider to be 10 objects. In base 10 you would say you have 1 eight, and you would need 1 two, so 0 fours, 1 two, and 0 ones. So this right here is 10 in base 2. This is 10 in base 10. Hopefully, that doesn't confuse you too much.
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