Transcription
Chuck, can you count to 10? You know, sometimes I wonder. I try it: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. What's the difference between nine and 10?
10 has a zero. Yes, it has two. It has two numerals in it. Yeah, and I didn't start from zero. That's fine. That's fine. I'm just saying that one through nine all have one numeral, right? Then you ran out of numerals.
Yes, and had to start them over again with...
Okay, okay. So go: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Now you start doubling up on the numerals that you've already uttered. You have a one and a zero, a one and a one, a one and a two, a one, and you already used those numerals. Sounds pretty efficient.
Therein is the foundation of base 10. Base 10, it's not the only way to count. You can count other bases, and it works the same way.
All right, okay. So base five would have how many numerals?
1, 2, 3, 4, 5, and then zero.
So six. If base 10 has 10 numerals, base five has five.
That's what I thought I heard you say that.
Okay, so what are the five?
You don't... it’s got to be 0, 1, 2, 3, 4.
Yes, that's what it would have to... five would be 0, 1, 2, 3, 4, and then when you got to five, there is no symbol for it.
Symbol for five?
And so five in base five would be what?
Uh... uh... 40?
Wait, okay: 0, 1, 2, 3, 4. Start over: 1, 2, 3, 4.
0, 1, 2, 3, 4. What's next?
Uh... 0, 1, 2, 3, 4.
No, I'm really not getting this.
You're overthinking it.
I'm overthinking this big time. I'm telling you, ready? I want to count in base five: 0, 1, 2, 3, 4.
It's one Z.
1, 1, 1, 2, 13, 14.
Oh my God, now I'm at 4! I don't have any one... anything left.
Anything left?
So what's my next number?
2!
2, 0?
21, 22, 4!
3, 0, 3, 0, 31, 32, 33, 4Z.
No! 3, 4!
3, 4, 4, and I just keep doing that!
You just keep doing that. You with your five numbers.
Wow, you're 0, 1, 2, 3, 4. That's base five.
Let's go bigger than base 10.
Okay, how about base 16?
So base 16, you use up your 10 numerals and now you need how many more?
Six more.
You need six more? Where are you going to get them from?
They just make letters.
So I will count to 16 in base 16. You ready? 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F.
16! That's base 16.
Okay, what comes after F?
Um, give me a second.
0, 1.
You go back. You go back to...
Wait, oh, you go back to zero.
Go back all the way: 1, 2, 3, 4, 5, 6.
Okay, base 16 now. Wow!
Jesus, I'm trying to see this: A, B, C, D, E, F, right? And then we're at the end.
End of all? You used up all your numerals?
Okay, so that's... so F0, you go back to the beginning.
So F starts the beginning.
Zero, you got to double up.
I'm doing the same thing again!
Yes, you are! I'm doing the exact same thing again.
Wait a minute, what?
You got to just double up like you've been doing with base 10 and base five!
I'm still missing.
You have 16 digits to work with, right?
Okay, you used them all up.
Okay, now you got to start doubling up on them.
That's what I'm going... wouldn't it be... oh, wait a minute...
So then it be... oh man, you're getting...
What did you do when you got to nine?
I went to... I went back and it's one zero.
What did you do when you got to four?
I went back to five.
Uh, well when I got to four, I go back to... as five, uh, I... well, when I got to four I go back to zero.
But 4Z, it was one Z, I mean one zero for five in base five, right?
Okay, now we're in base 16. We got up to F. What do you do now?
We used up all the digits. I got to go back and start over again by doubling them up.
That's what I said, so it wouldn't it be F1, 2, 3?
But no, no! It wouldn't... F is not the beginning of the one; it's the beginning now.
Double up!
Oh! 1, 1?
No, 1, 0!
1, 0!
Oh, it's the same thing! It's the same thing!
It's the same thing! I'm just...
It's the same... you're overthinking it.
I'm overthinking it! It's the same thing! So it's not 1, 0, 2, 3, 4, 5, A, B, C, D, E, F.
No! No, it is 0 through F.
Now it's... okay, 1, 0!
Right!
And now 1, 1, 1, 2, 1, 3, 1, 4, 16, 17, 18, 1n.
Uh oh!
1A.
Oh, that's 1A! There we go! Give me some fist love there!
Next one, B!
1C, 1F, 1F, next 2, 0.
Thank you!
2, 1, 2, 2, 3, 3, all the way up to 2A, 2B, 2... yes! F!
Okay! Then 3, 0, 3, 1, 32, 3... 3, 3, 4, all the way up to 3A through F.
40, 41, 42, 4!
You see how many numbers we are packing?
Why is this so hard for me?
I don't know.
Okay, do you see how many numbers we're packing into this?
Because... oh, I can get so many more numbers out of my number line!
My number line now... oh my gosh, just increase... oh my gosh!
Number line. I thought I could get a ton of the numbers out of 1 through 9 or 0, 1...
Oh! n 0, 1 through F!
Oh my gosh!
Okay, forget it, forget it!
So now, watch: every computer in the world...
Mhm.
Has a MAC address, a media access control address.
Just call it the MAC address!
We use the decimal system; that's heximal!
Heximal!
Heximal! Okay, you got it! That's very cool!
All right, if you only have 10 digits from 0 to 9, right?
How many possible numbers can you make that are two digits long?
If you have 10 digits from 0 to 9, 99!
99?
It's 99!
Yeah, actually, it's 100! If you add the zero.
If you add the zero... okay!
All right, so three digits: how many can you make?
199, 299, 9... 99!
So a thousand!
So what you're doing is you're multiplying 10 by 10 by 10!
All right, two digits you get 100, three digits you get 1,000, four digits you get 10,000!
Right! If it's hexadecimal, you're starting with 16 digits.
You got 16 times 16! That's 256 possible numbers in hexadecimal with two digits!
All right, now the fun part: we did base 16, base 10, base five.
Okay, how about base two?
All right, let's start it off: 0, 1. Boom! Those—that's all you got to work with!
That's your two.
Now you got to start over again! Start over again!
1Z!
1—no! No! No! It's 0!
1!
One!
No, you start with one set of digits; the next round is two sets: 0, 0, 1.
No, two digits: 0, 1 is first—you have zero, then one!
Right! What number comes next?
We got to start all over again!
Yes! Which is 0!
01!
Okay, am I doing it again?
You're all for four on this.
Zero!
Right! One!
Next number—there's only two numbers I know! So what's the next number?
Z!
One!
Start over again!
Start over again?
No! We exhausted the single digits, correct?
Right now, you got to go to double digits.
So 0, 1... what's the lowest double-digit number?
00?
No! That's just zero! We already have a zero!
Okay, okay!
0, 1!
Right!
We just used up all our numbers! Okay, now we go back like every other base; you start over again with two digits: 1, 1, Z!
Right!
Okay, next! We ran out of numbers again!
No!
01!
1Z?
Yes! What's after that?
01?
No! A1!
0?
1?
0?
No, so it can't be 10, and it can't be 11!
That's correct, because there are other numbers less than that that you haven't gotten to yet!
You got to use up the numbers in sequence!
So, this one—zero?
What else?
1; zero!
1!
1, 1!
1!
No! Just one, one!
We're counting 0, 1, 1...
Zero!
One!
One!
Okay, what's next?
Is there one...?
Two?
No, because two is not in base two!
So what's after one, one?
It's got to be one...
Okay, one, one!
And then one, one!
There's a number less than that: one, zero!
Yes!
And then you go 101!
Thank you!
Next one: 110!
Give me a number bigger than that than 110!
Yes! It's staring at you in the face! There’s 101, 1110, and 111, 111!
Yeah, now you just maxed out!
Now that's one-one is like 99 or 999!
Now I have to add another zero!
So now I got to go... one, zero?
1... no! Just 110 next!
Then 101!
No! One too many zeros!
Okay, 1001!
Okay, next! 1011, no!
No! 1, 0, 10!
1, 0!
1, 0!
Okay! One... yeah?
10!
10!
Okay! Next!
One... no!
1Z!
1, 1!
1, 0!
Perfect!
Next: 1, 1!
0!
Next: 11!
1, zero!
Next: 111!
I think he’s got this!
So now one, Z!
No! No!
1, 000000!
Thank you!
Next!
Now one, 0000!
No! Too many Zs! Too many zeros!
But you know what I mean, so you know what I’m saying!
I play that!
Okay, here's my point: I can't—I can't do this without—I got... there’s a bigger story here!
I got to write it down! That's so weird! There’s a bigger story here!
Go ahead!
We have 10 fingers, right? We have base 10!
All our numbers are on this system!
If aliens came and they had eight fingers, if they counted based on how many fingers they have, all their numbers would not match our numbers, right?
They'd be counting in base eight!
Yes! And you have to be able to convert!
If they gave us the digits of pi in base eight, how are we going to know?
So we go to... we have to be sensitive to the biases built into what it is to be human!
Gotcha!
So, um, base 2 is the simplest of all counting systems, because it only has two!
And uh, computing and chips and things, it’s binary!
When I say it’s binary, that’s what’s going on! Only two: is it on or off?
Now, in quantum computing, it doesn’t use bits—it uses qubits!
Cubic?
Cubic, where it’s not just one or zero or on or off!
It could be any combination of the two!
You can be partly one and mostly zero or partly zero and mostly one or half and half of each!
Quantum bits—okay! Don’t even ask me to count!
Don’t you dare ask me to count quantum bits!
So now, here’s something interesting: our timekeeping is base 12 and base 60!
Okay!
What happens when you get to 12?
You start over!
You start over again!
And the same thing when you get to 60!
Okay!
You start over again!
Right?
Yeah!
Teaching Chuck how to count in base two?
Yes!
Without paper, it’s not going to happen!
All right! Just to redeem yourself, count fast in base five, go!
Uh... 0, 1, 2, 3, 4—five?
No! I’m sorry! 0, 1, 2, 3, 4!
1, 2, 3, 4—okay!
0, 1, 2, 3, 4!
0, no! 0, 1, 2, 3, 4—you're done!
Right! Next pair of numbers!
Now 0, 1!
Wait!
Oh God!
All right, Neil deGrasse Tyson here, keep looking up!