The first part of this video is a refresher about how to translate FROM other bases. Then most of the video is about a technique for translating INTO other bases (which I haven't covered before). Timestamps of different parts: 0:00 - the two directions we could translate 1:34 - the patterns of "basic bases" and the easier direction to translate 5:33 - the technique to solve the harder direction to translate Consider supporting on Patreon: www.patreon.com/comboclass See the video description for more links
Blackpenredpen has a video all about this! He says, "Let's say you're converting from base a to base b. We should have the base-b 0 to b-1 times table written out, so we don't go crazy." (In the example video, b=7) If a < b, he says, "take the number you're starting with and convert it into a polynomial in a, putting it in Horner form. Evaluate from the inside out, and convert to base b of course, using the times table, if the result ever exceeds b-1." If a > b, we handle it a little differently. Sure we turn the number into a polynomial, but now of course a > b, so we handle it differently. Change the coefficients (and value of a) to all be base-b friendly. Evaluate as before, and... you're done!
For converting to binary, there's a method I like that requires at most one subtraction per bit, as opposed to requiring a whole division per bit. 1. Let the running remainder be the number you're trying to convert to binary 2. Take the largest power of two less than or equal to the running remainder (e.g. the largest power of two less than seventy is sixty-four) 3. in the solution, place a 1 in the place of that power of 2. 4. Subtract the power of two from the running remainder to get the new running remainder 5. If the running remainder is nonzero, GOTO 2. Otherwise, place a 0 in every place we haven't placed a 1. Example, converty 70 from decimal to binary The largest power of 2 less than or equal to 70 is 64, so there's a 1 in the 64's place, and a 0 in the infinitely many places to the left of it. 70 - 64 = 6. The largest power of 2 less than or equal to 6 is 4, so there's a 1 in the 4's place. 6 - 4 = 2. The largest power of 2 less than or equal to 2 is 2, so there's a 1 in the 2's place. 2 - 2 = 0. So there's a 1 in the 64's, 4's, and 2's place, and a 0 in all other places. 1000110
This most-significant-digit first method is also applicable to other bases, but it requires more memorization or lookups than it does in binary, for diminishing returns - In general, you look at the largest *multiple of a power of the base* less than the running remainder, which gets more dense as the base gets bigger. Converting seventy to seximal for example: The largest multiple of a power of six less than seventy is thirty-six, so there's a 1 the 36's place 70-36=34 The largest multiple of a power of six less than 34 is 30, so there's a 5 in the 6's place. 34-30=4, so there's a 4 in the 1's place, so 70 in seximal is 154.
I was first taught this trick in my computing class when we were learning binary and hexadecimal and how to convert base 10 into them. For binary, its slightly easier to just find the largest power of 2, subtract and then repeat on the leftover, since the digits are only 0 or 1, so we don’t need to work out any coeffecient in front of it. But for hex, using this division-remainder method is easier conpared to finding the largest power of 16 and then finding the coefficient infront of it Great video nontheless
@@815TypeSirius You certainly could use only letters, but it's already an established convention to start with digits for the first ten, so that's what people normally stick with, in order to avoid confusion. When and why that became the convention, I don't know. Probably people figured it would be easier to learn to read e.g. hexadecimal if you only had to memorize the numeric values of 6 new symbols instead of all 16.
Stopping by base ten in the middle lets you do the division using base ten arithmetic, which is way easier for a modern human than trying to do the division directly in another base's arithmetic. Also it lets you know what quantity it represents.
To use this method directly from one base to another, you would need to know arithmetic (division) in another base. If you want to use base ten arithmetic (which will typically be easier) you essentially stop by base ten in between (which is like the step of figuring out what quantity it represents).
Ah yes, Purrrrrfect powers. Neat trick for recalculating the bases. As an extra, if you have decimals. You simply need to do the same. And place the decimal back in with the same ammount of numbers it had on the right. Little thing to watch out is that if there are remainders of zero, right at the most right. You need to keep an eye out for those as well.
Unexpectedly, this turned out to not be new to me. I basically worked out the same method in the past while writing a simple timer that properly showed things like minutes, hours, and days.
@@ComboClass Thanks! I have not watched the full video yet but I use to work on this problem that I started in highschool and your videos on bases just reminds me of it. The problem was, how could you easily write out a base 10 fraction converted to base b using its digits. I managed to find a relationship but it's not easy to explain on a youtube comment.
Im trying to use base 10,077,696 and this video useless because if i use a number smaller than 10077696 it leaves a remainder which is just the vase number with zero as the number for the next step, usefull for bumbers larger than it though
That would still work in the pattern. If your base is larger than your number, your result is 0 with a remainder of that number, and that a single-digit character for that remainder would the only digit of the answer
You can do division in another base's arithmetic if you want to "cut out the middle man" but if you want to use the base ten arithmetic you're familiar with, it's easier to stop by base ten in the middle (which also lets you know what quantity it represents).
Wait now, please! Why wouldn't 0 (zero) work as a base? Then we have two thingies: The impossible, and the definition of the impossible. And thus having two things, we can start the count down, and can go hunt primes together.
Hmm.... Some digit.... ANY digit? Like, it DOESN'T matter the Digit, if we know the Base, we can Parse the Representation, if we know what quantity the Digit is? My new digit is 'Iophiel'.
I hate looong division! and I don't even know what it is supposed to be. If I were tought it in school, I've forgotten it. I too count from the front, much like I have six. And 1200+something is a good start for the lazy.
Does this guy just get to study math between videos? I think I really need to break into this market lol.
Just don't burn your fingers trying.
When I first started you were speaking python code to me but now I actually understand what you're saying! Hawking awesome!
The first part of this video is a refresher about how to translate FROM other bases. Then most of the video is about a technique for translating INTO other bases (which I haven't covered before). Timestamps of different parts:
0:00 - the two directions we could translate
1:34 - the patterns of "basic bases" and the easier direction to translate
5:33 - the technique to solve the harder direction to translate
Consider supporting on Patreon: www.patreon.com/comboclass
See the video description for more links
You should pin this comment!
Blackpenredpen has a video all about this! He says, "Let's say you're converting from base a to base b. We should have the base-b 0 to b-1 times table written out, so we don't go crazy." (In the example video, b=7)
If a < b, he says, "take the number you're starting with and convert it into a polynomial in a, putting it in Horner form. Evaluate from the inside out, and convert to base b of course, using the times table, if the result ever exceeds b-1."
If a > b, we handle it a little differently. Sure we turn the number into a polynomial, but now of course a > b, so we handle it differently. Change the coefficients (and value of a) to all be base-b friendly. Evaluate as before, and... you're done!
Be a rebel, use base pi.
eeeeeee 😆
Domotro doesn’t get enough credit for soundtracking ❤
ohhh so close to 50k .. might even hit it tonight hahaha
burning paper with text on it as a video editing technique looks really cool
For converting to binary, there's a method I like that requires at most one subtraction per bit, as opposed to requiring a whole division per bit.
1. Let the running remainder be the number you're trying to convert to binary
2. Take the largest power of two less than or equal to the running remainder (e.g. the largest power of two less than seventy is sixty-four)
3. in the solution, place a 1 in the place of that power of 2.
4. Subtract the power of two from the running remainder to get the new running remainder
5. If the running remainder is nonzero, GOTO 2. Otherwise, place a 0 in every place we haven't placed a 1.
Example, converty 70 from decimal to binary
The largest power of 2 less than or equal to 70 is 64, so there's a 1 in the 64's place, and a 0 in the infinitely many places to the left of it.
70 - 64 = 6.
The largest power of 2 less than or equal to 6 is 4, so there's a 1 in the 4's place.
6 - 4 = 2.
The largest power of 2 less than or equal to 2 is 2, so there's a 1 in the 2's place.
2 - 2 = 0.
So there's a 1 in the 64's, 4's, and 2's place, and a 0 in all other places. 1000110
This most-significant-digit first method is also applicable to other bases, but it requires more memorization or lookups than it does in binary, for diminishing returns - In general, you look at the largest *multiple of a power of the base* less than the running remainder, which gets more dense as the base gets bigger.
Converting seventy to seximal for example:
The largest multiple of a power of six less than seventy is thirty-six, so there's a 1 the 36's place 70-36=34
The largest multiple of a power of six less than 34 is 30, so there's a 5 in the 6's place. 34-30=4, so there's a 4 in the 1's place, so 70 in seximal is 154.
Thanks domotro
Well presented!🌞 Nice job, brother.
You should look up James Tanton's exploding dots. I feel you'd get a kick out of it.
Converting between bases 2 and 16 is super easy, just group digits.
I was first taught this trick in my computing class when we were learning binary and hexadecimal and how to convert base 10 into them.
For binary, its slightly easier to just find the largest power of 2, subtract and then repeat on the leftover, since the digits are only 0 or 1, so we don’t need to work out any coeffecient in front of it.
But for hex, using this division-remainder method is easier conpared to finding the largest power of 16 and then finding the coefficient infront of it
Great video nontheless
For over 10 bases why not just use letters? Thats good to base 52?
I did use letters. I used B for 11 in one of my examples
@ComboClass like, only letters. So. 5 11 4 is ekd.
@@815TypeSirius You certainly could use only letters, but it's already an established convention to start with digits for the first ten, so that's what people normally stick with, in order to avoid confusion. When and why that became the convention, I don't know. Probably people figured it would be easier to learn to read e.g. hexadecimal if you only had to memorize the numeric values of 6 new symbols instead of all 16.
@@notnullnotvoid Mainly it's handy that you don't have to do any conversion for numbers that are single-digit in both bases.
@@815TypeSirius I don't think you gave this much thought: if e is 5, what would zero be?
Base pi sounds really good. Or some phi
tau! 😁
If you wanted to convert base a to base b, why would you convert to base ten (assuming neither a nor b are ten.)?
Stopping by base ten in the middle lets you do the division using base ten arithmetic, which is way easier for a modern human than trying to do the division directly in another base's arithmetic. Also it lets you know what quantity it represents.
How do i translate into and out of base 0 ?
I'll ask my NaN
What about direct conversions from one base to another
To use this method directly from one base to another, you would need to know arithmetic (division) in another base. If you want to use base ten arithmetic (which will typically be easier) you essentially stop by base ten in between (which is like the step of figuring out what quantity it represents).
Ah yes, Purrrrrfect powers.
Neat trick for recalculating the bases.
As an extra, if you have decimals. You simply need to do the same. And place the decimal back in with the same ammount of numbers it had on the right.
Little thing to watch out is that if there are remainders of zero, right at the most right. You need to keep an eye out for those as well.
Still the best when it comes to numbersystems! 🤗
6:31 Falling board jumpscare
For converting to balanced ternary, a quotient of n with a remainder 2, could also be considered a quotient of n+3 with a remainder of -1.
Unexpectedly, this turned out to not be new to me. I basically worked out the same method in the past while writing a simple timer that properly showed things like minutes, hours, and days.
It says the discord link is expired 😢
Whoops, I just replaced the link in this video description, so I think the link should work again
@@ComboClass Thanks! I have not watched the full video yet but I use to work on this problem that I started in highschool and your videos on bases just reminds me of it. The problem was, how could you easily write out a base 10 fraction converted to base b using its digits. I managed to find a relationship but it's not easy to explain on a youtube comment.
Im trying to use base 10,077,696 and this video useless because if i use a number smaller than 10077696 it leaves a remainder which is just the vase number with zero as the number for the next step, usefull for bumbers larger than it though
if you use a number smaller than the base, then you already have your answer, theres no next step
That would still work in the pattern. If your base is larger than your number, your result is 0 with a remainder of that number, and that a single-digit character for that remainder would the only digit of the answer
@@ComboClass but what if I get a remainder with a decimal
In excel I was doing basically the same thing. Divide and take whole part and repeat ( reduce() ).
Btw, shame you didn't use "dec" and "elf".
"Translate from base 6 to base 12 by using 10 as in intermediate"
Be a rebel. Cut out the middle man. Do math in base 13, if necessary.
You can do division in another base's arithmetic if you want to "cut out the middle man" but if you want to use the base ten arithmetic you're familiar with, it's easier to stop by base ten in the middle (which also lets you know what quantity it represents).
Wait now, please!
Why wouldn't 0 (zero) work as a base? Then we have two thingies:
The impossible, and the definition of the impossible. And thus having two things, we can start the count down, and can go hunt primes together.
Hmm.... Some digit.... ANY digit? Like, it DOESN'T matter the Digit, if we know the Base, we can Parse the Representation, if we know what quantity the Digit is? My new digit is 'Iophiel'.
Huh, the Denominator is like a Bas(is)...
I hate looong division! and I don't even know what it is supposed to be. If I were tought it in school, I've forgotten it. I too count from the front, much like I have six. And 1200+something is a good start for the lazy.
ua-cam.com/video/7PCkvCPvDXk/v-deo.html