WEBVTT Kind: captions Language: en 00:00:00.750 --> 00:00:03.320 What I want to do in this video is revisit some ideas 00:00:03.320 --> 00:00:05.630 that you've probably taken for granted since the time 00:00:05.630 --> 00:00:08.010 that you were like three or four years old. 00:00:08.010 --> 00:00:10.380 But hopefully, you'll kind of view it in a new light, 00:00:10.380 --> 00:00:12.130 and it'll help inform us when we think 00:00:12.130 --> 00:00:15.520 about other types of number systems. 00:00:15.520 --> 00:00:17.920 So we have 10 digits in our number system. 00:00:17.920 --> 00:00:19.170 So let me just start counting. 00:00:19.170 --> 00:00:21.610 So if I have nothing, I use the symbol 0. 00:00:21.610 --> 00:00:23.730 Then if I have one object, I use the symbol 1. 00:00:23.730 --> 00:00:25.110 Actually, let me draw this out. 00:00:25.110 --> 00:00:25.769 So nothing. 00:00:25.769 --> 00:00:27.810 And then if I have one thing, I use the symbol 1. 00:00:27.810 --> 00:00:31.000 If I have two things, I use the symbol 2. 00:00:31.000 --> 00:00:33.730 If I have three things, I use the symbol 3. 00:00:33.730 --> 00:00:36.230 Let me scroll down a little bit, make sure you can see that. 00:00:36.230 --> 00:00:40.760 If I have four things, I use this symbol right over here. 00:00:40.760 --> 00:00:45.900 If I have five things, I use this symbol. 00:00:45.900 --> 00:00:47.780 If I have six things, I'll start-- 00:00:47.780 --> 00:00:49.980 let's draw it like that-- if I have six things, 00:00:49.980 --> 00:00:51.670 I use that symbol. 00:00:51.670 --> 00:00:55.567 If I have seven things, I use that symbol. 00:00:55.567 --> 00:00:57.650 I know this might be getting a little bit tedious, 00:00:57.650 --> 00:00:58.920 but this all has a point. 00:00:58.920 --> 00:01:03.650 If I have eight things, I use this symbol. 00:01:03.650 --> 00:01:09.940 And if I have nine things, I use this symbol. 00:01:09.940 --> 00:01:17.570 And then if I have 10 things, so 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 00:01:17.570 --> 00:01:19.290 What symbol do I use? 00:01:19.290 --> 00:01:21.400 I've already used up my 10 digits. 00:01:21.400 --> 00:01:24.090 We only have 10 digits in a base 10 system. 00:01:24.090 --> 00:01:25.860 So we start reusing them. 00:01:25.860 --> 00:01:30.340 So what we do is we introduce this idea of number places. 00:01:30.340 --> 00:01:34.250 You say that I have 1 ten and 0 ones. 00:01:34.250 --> 00:01:40.730 So you say you have 1 ten, and then 0 ones. 00:01:40.730 --> 00:01:44.600 We call this 1, we say it's in the tens place. 00:01:44.600 --> 00:01:50.990 This is literally saying 1, this is saying 1 tens. 00:01:50.990 --> 00:01:54.597 This is 1 tens plus 0 ones. 00:01:54.597 --> 00:01:57.180 So that's what this is saying, but we didn't have to reuse it. 00:01:57.180 --> 00:01:59.580 We could have had maybe more symbols. 00:01:59.580 --> 00:02:01.880 Maybe this was a symbol, or maybe we 00:02:01.880 --> 00:02:04.130 would have created a new symbol, instead of, you know, 00:02:04.130 --> 00:02:05.740 all of these had their own symbol. 00:02:05.740 --> 00:02:07.531 So instead of having to reuse the old ones, 00:02:07.531 --> 00:02:11.816 maybe we could have made the symbol star for 10. 00:02:11.816 --> 00:02:13.190 And then when you go to 11, maybe 00:02:13.190 --> 00:02:14.814 would have had another symbol for that. 00:02:14.814 --> 00:02:17.180 Let me go to 11 just to hit the point home. 00:02:17.180 --> 00:02:22.950 So 2, 3, 4, 5, 6, 7, 8, 9, 10, 11. 00:02:22.950 --> 00:02:28.570 So 11 in our number system, we say that this is 1 ten-- 00:02:28.570 --> 00:02:34.210 let me write it this way-- and then this is also, it's 1 ten 00:02:34.210 --> 00:02:35.035 and then 1 one. 00:02:37.770 --> 00:02:41.120 So it's 1 ten plus 1 one. 00:02:41.120 --> 00:02:43.490 I know this is kind of strange to see it this way, 00:02:43.490 --> 00:02:46.200 but it represents this number of objects. 00:02:46.200 --> 00:02:49.634 If we had a base 11, or I guess we could say base 12 number 00:02:49.634 --> 00:02:51.300 system, maybe we would have had a symbol 00:02:51.300 --> 00:02:53.330 for this instead of reusing our old digits. 00:02:53.330 --> 00:02:56.009 Maybe a symbol could have been something wacky. 00:02:56.009 --> 00:02:57.550 Maybe it would've been a smiley face. 00:02:57.550 --> 00:02:58.970 Who knows what it would have been? 00:02:58.970 --> 00:03:01.390 And I'll introduce higher number base 00:03:01.390 --> 00:03:04.300 systems in kind of future videos, where we see what, 00:03:04.300 --> 00:03:06.962 kind of, the symbols that are actually used. 00:03:06.962 --> 00:03:08.420 But what I want to do in this video 00:03:08.420 --> 00:03:10.460 is think about, how would we count, 00:03:10.460 --> 00:03:14.190 or what symbols would we use, if we had fewer digits? 00:03:14.190 --> 00:03:17.190 And in particular, how would we count things 00:03:17.190 --> 00:03:19.020 if we only had two digits? 00:03:19.020 --> 00:03:21.654 If we only had 0 and 1. 00:03:21.654 --> 00:03:23.570 And that's essentially what we're going to do, 00:03:23.570 --> 00:03:27.620 is think about how we would represent numbers in base 2. 00:03:27.620 --> 00:03:31.060 Our traditional number system is a base 10 number system. 00:03:31.060 --> 00:03:34.040 We have 10 digits, 0 through 9. 00:03:34.040 --> 00:03:36.560 How would we count in base 2? 00:03:36.560 --> 00:03:38.970 So if you have zero things, you'd still probably say, 00:03:38.970 --> 00:03:41.910 hey, I have zero, I can use the digit 0. 00:03:41.910 --> 00:03:43.780 If I have one thing, I could still say, 00:03:43.780 --> 00:03:47.061 hey, I have one thing, because we have the digits 0 and 1. 00:03:47.061 --> 00:03:48.060 So let me make it clear. 00:03:48.060 --> 00:03:55.850 The digits here, the digits in base 2 can be 0 or 1. 00:03:55.850 --> 00:03:59.300 So if I have one thing, I can still use the number 1. 00:03:59.300 --> 00:04:02.030 But now all of a sudden, I have these two objects here, 00:04:02.030 --> 00:04:06.890 and I'm saying that I'm limited to only these two digits 00:04:06.890 --> 00:04:07.520 over here. 00:04:07.520 --> 00:04:09.220 So how could I represent it? 00:04:09.220 --> 00:04:11.270 Well, instead of having a tens place, 00:04:11.270 --> 00:04:13.440 I could create a twos place. 00:04:13.440 --> 00:04:15.980 And I know it might sound a little bit counterintuitive, 00:04:15.980 --> 00:04:17.938 but I think you'll get used to it a little bit. 00:04:17.938 --> 00:04:23.180 So over here in base 10, we said we had 1 ten and 0 ones. 00:04:23.180 --> 00:04:30.450 So in base 2, why can't we say that we have 1 two and 0 ones? 00:04:30.450 --> 00:04:32.200 Let me make that clear. 00:04:32.200 --> 00:04:40.434 So this right here is saying 1 two and 0 ones. 00:04:40.434 --> 00:04:42.600 I want to make sure you understand the analogy here. 00:04:42.600 --> 00:04:45.430 In base 10-- let me write a larger number in base 10. 00:04:45.430 --> 00:04:49.960 So if I write the number 256 in base 10. 00:04:49.960 --> 00:04:53.350 So this is base 10 over here. 00:04:53.350 --> 00:04:54.290 What is this saying? 00:04:54.290 --> 00:04:59.850 This is saying 2 hundreds, so 2 times 100-- or maybe 00:04:59.850 --> 00:05:01.510 I should write down the word, because I 00:05:01.510 --> 00:05:06.850 don't want to confuse the symbols-- 2 hundreds plus 5 00:05:06.850 --> 00:05:09.440 times, or maybe I should say, 2 hundreds 00:05:09.440 --> 00:05:19.902 plus 5 tens plus 6 ones. 00:05:19.902 --> 00:05:21.110 That's what I represent here. 00:05:21.110 --> 00:05:22.610 And the way we know that, is that we 00:05:22.610 --> 00:05:24.780 know that if we go two places to the left, 00:05:24.780 --> 00:05:28.580 this is the hundreds place. 00:05:28.580 --> 00:05:30.490 This is the tens place. 00:05:30.490 --> 00:05:31.930 And this is the ones place. 00:05:31.930 --> 00:05:33.520 And if you know from your exponents, 00:05:33.520 --> 00:05:36.330 this is equal to 10 times 10. 00:05:36.330 --> 00:05:40.260 This right here is equal to 10 times itself only once. 00:05:40.260 --> 00:05:42.370 And this is equal to 10 times itself, 00:05:42.370 --> 00:05:44.400 I guess you could call it, zero times. 00:05:44.400 --> 00:05:48.090 Or if you know your exponents, this is 10 to the second power. 00:05:48.090 --> 00:05:50.330 This is 10 to the first power place. 00:05:50.330 --> 00:05:52.080 And this is 10 to the 0-th power place. 00:05:52.080 --> 00:05:53.323 And if you added another digit here, 00:05:53.323 --> 00:05:54.739 that would be the thousands place, 00:05:54.739 --> 00:05:56.780 which would be 10 times 10 times 10. 00:05:56.780 --> 00:05:59.070 We're going to do the exact same thing in base 2, 00:05:59.070 --> 00:06:01.750 but instead of using 10, we're going to use 2. 00:06:01.750 --> 00:06:03.770 So now this is the twos place. 00:06:03.770 --> 00:06:05.540 This is the twos place over here. 00:06:05.540 --> 00:06:07.070 This is the ones place. 00:06:07.070 --> 00:06:09.640 If we add more digits let me make it clear. 00:06:09.640 --> 00:06:13.840 So in base 2, let me write a number here in base 2. 00:06:13.840 --> 00:06:17.560 And remember, in base 2 I can only use zeroes and ones. 00:06:17.560 --> 00:06:23.235 So in base 2 maybe I have the number 1010. 00:06:23.235 --> 00:06:25.610 So when you think about it this way, if this was base 10, 00:06:25.610 --> 00:06:28.070 you would call this the tens place, the hundreds place, 00:06:28.070 --> 00:06:29.240 and the thousands place. 00:06:29.240 --> 00:06:31.020 But this is base 2 now. 00:06:31.020 --> 00:06:32.730 So let me be very clear. 00:06:32.730 --> 00:06:34.540 We are only using two digits. 00:06:34.540 --> 00:06:38.760 So in base 2, this right here, this is still the ones place. 00:06:38.760 --> 00:06:41.040 Now this is going to be the twos place. 00:06:41.040 --> 00:06:43.060 Remember in base 10 this was the tens place. 00:06:43.060 --> 00:06:45.170 Now this is the twos place. 00:06:45.170 --> 00:06:47.980 Now this would be, and you could take a guess here, 00:06:47.980 --> 00:06:49.930 hundreds was 10 times 10. 00:06:49.930 --> 00:06:53.470 When we go two spaces to the left in base 2, 00:06:53.470 --> 00:06:55.770 this should be the 2 times twos place, 00:06:55.770 --> 00:06:57.335 or this is the fours place. 00:07:00.110 --> 00:07:04.230 This over here is going to be the eights place. 00:07:04.230 --> 00:07:05.770 So if you wanted to kind of think 00:07:05.770 --> 00:07:14.660 about this in terms of base 2, this is 1 eight plus 0 fours 00:07:14.660 --> 00:07:19.810 plus 1 twos plus 0 ones. 00:07:22.940 --> 00:07:26.680 So if you wanted to represent this exact same number in base 00:07:26.680 --> 00:07:30.360 10, it's 1 eight plus 1 two. 00:07:30.360 --> 00:07:33.420 So in base 10 this would be-- let me write it 00:07:33.420 --> 00:07:35.960 over here-- in base 10, this would 00:07:35.960 --> 00:07:39.970 be an 8 plus a 2, which is just a 10. 00:07:39.970 --> 00:07:42.581 So this is it in base 10. 00:07:42.581 --> 00:07:44.080 This is how you would represent what 00:07:44.080 --> 00:07:47.580 we know as this many things, as 10 things, 00:07:47.580 --> 00:07:49.920 this is how you would represent it in base 2, 00:07:49.920 --> 00:07:53.700 this is how we know we would represent it in base 10. 00:07:53.700 --> 00:07:56.760 Now let's continue here, just to make sure we understand things. 00:07:56.760 --> 00:07:58.540 So, this many objects. 00:07:58.540 --> 00:08:02.130 Well, in base 2, we have 1 if you just 00:08:02.130 --> 00:08:05.330 have two objects, that's 1 two and 0 ones. 00:08:05.330 --> 00:08:08.870 Now, three objects would be one 2 plus 1 ones. 00:08:08.870 --> 00:08:09.990 So let me do it over here. 00:08:09.990 --> 00:08:12.990 So this would be 1 two plus 1 one. 00:08:12.990 --> 00:08:18.670 So this is three objects in base 2. 00:08:18.670 --> 00:08:20.120 Now when you go to this. 00:08:20.120 --> 00:08:25.870 So over here we have 1 four, 0 twos, and 0 ones. 00:08:25.870 --> 00:08:28.082 So now we're going to go to the four place, 00:08:28.082 --> 00:08:30.040 because we've essentially maxed out everything. 00:08:30.040 --> 00:08:32.375 If we increment more, we have to go to one more place. 00:08:32.375 --> 00:08:34.000 Just like we did in base 10, but now we 00:08:34.000 --> 00:08:36.730 could only use the digits 0 and 1. 00:08:36.730 --> 00:08:41.650 So now we'll have 1 four, 0 twos, 0 ones. 00:08:41.650 --> 00:08:43.141 Now when we add one more. 00:08:43.141 --> 00:08:44.390 We're going to add one more 1. 00:08:44.390 --> 00:08:50.150 So now we have 1 four, 0 twos, and 1 one. 00:08:50.150 --> 00:08:53.540 And just to be clear, this is this many things. 00:08:53.540 --> 00:08:56.010 This is this many things in base 2, 00:08:56.010 --> 00:08:59.240 this is the four place, 1 four and 1 one. 00:08:59.240 --> 00:09:01.370 If you wanted to convert this into base 10, 00:09:01.370 --> 00:09:08.690 you'd say, look, this is 1 four, 0 twos, and 1 one. 00:09:08.690 --> 00:09:10.910 So if you have a four and a one, we 00:09:10.910 --> 00:09:14.050 would represent that with the symbol 5 in base 10. 00:09:14.050 --> 00:09:16.930 We don't have that symbol available to us in base 2. 00:09:16.930 --> 00:09:18.100 Let's go to this. 00:09:18.100 --> 00:09:19.990 So now we're going to increment one more. 00:09:19.990 --> 00:09:21.656 So, how can we represent that in base 2? 00:09:21.656 --> 00:09:23.770 This is definitely, we're going to have 1 four. 00:09:23.770 --> 00:09:26.960 And then we're going to have 1 two. 00:09:26.960 --> 00:09:29.590 And then we're going to have 0 ones. 00:09:29.590 --> 00:09:31.230 And if you keep-- it's kind of fun 00:09:31.230 --> 00:09:32.680 to practice counting in base 2. 00:09:32.680 --> 00:09:34.350 You'll start to get the hang of it. 00:09:34.350 --> 00:09:38.699 So here we have to add 1 one to this, so we get 1, 1, 1. 00:09:38.699 --> 00:09:40.240 And now when we get to eight, there's 00:09:40.240 --> 00:09:42.020 no way to kind of increment any of these 00:09:42.020 --> 00:09:44.110 any higher, so we have to get a new place. 00:09:44.110 --> 00:09:45.650 We have to go to the eights place. 00:09:45.650 --> 00:09:50.710 So we have 1 eight, 0 fours, 0 twos, and 0 ones. 00:09:50.710 --> 00:09:53.000 This right here, it might look like 1,000 to you, 00:09:53.000 --> 00:09:55.930 but it would be 1,000 if we were in base 10. 00:09:55.930 --> 00:09:59.310 In base 2 this is this many objects. 00:09:59.310 --> 00:10:02.350 This is eight objects in base 2. 00:10:02.350 --> 00:10:05.100 When you increment it one, we'll have this many, 00:10:05.100 --> 00:10:07.000 we'll have 1 eight and then we'll have 1 one. 00:10:07.000 --> 00:10:10.660 So it would be 1001. 00:10:10.660 --> 00:10:15.210 And then I'll stop here at what we consider to be 10 objects. 00:10:15.210 --> 00:10:19.490 In base 10 you would say you have 1 eight, 00:10:19.490 --> 00:10:24.140 and you would need 1 two, so 0 fours, 1 two, and 0 ones. 00:10:24.140 --> 00:10:28.090 So this right here is 10 in base 2. 00:10:28.090 --> 00:10:30.480 This is 10 in base 10. 00:10:30.480 --> 00:10:33.330 Hopefully, that doesn't confuse you too much.