Practical Numbers - Numberphile
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- Опубліковано 17 тра 2023
- Featuring Dr James Grime. More links & stuff in full description below ↓↓↓
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Practical Numbers (AKA panarithmic numbers) on the OEIS: oeis.org/A005153
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“Anti-primes” are an objectively cooler name than “highly composite numbers”
And to be fair, the Wikipedia redirect was created on 2006, way before they talked about antiprimes in this channel!
no
the numbers hated by mathematicians: the "anti-grimes"
@@DDbenkoDD Fine, "unprimes" it is.
its why the name sticks.
How about being able to both add and subtract the factors to make the numbers? You can imagine it as putting the subtracting weights on the opposite side of the adding weights. In this case, the imbalance of the scale is the result you're looking for, meaning that if you put a weight equaling the result of the computation on the side where the subtracting weights are, you get equilibrium.
As an example, you can make all numbers up to 14 with its factors {1, 2, 7, 14}:
1 = 1
2 = 2
3 = 2 + 1
4 = 7 - 2 - 1
5 = 7 - 2
6 = 7 - 1
7 = 7
8 = 7 + 1
9 = 7 + 2
10 = 7 + 2 + 1
11 = 14 - 2 - 1
12 = 14 - 2
13 = 14 - 1
14 = 14
I'd call these (positive integer) numbers semi-practical numbers if they don't have a name already.
Edits and results:
• The lowest non-prime, non-practical number that is also non-semi-practical is 22.
• The lowest non-prime, non-practical number that is semi-practical is 10.
• The lowest non-semi-practical number is 5.
• All practical numbers are semi-practical numbers.
• All powers of 3 are semi-practical and play a role similar to the powers of 2 in practical numbers. They also give the minimal size set possible that allow the greatest total to be reached. Their expressions are also unique. (credits to @wiskundeboi). This system is also called "balanced ternary".
• Not all semi-practical numbers are even or multiples of 3. Example: 5005 = 5*7*11*13 is a semi-practical number.
• If all numbers up to half of the number checked can be obtained, then the number is semi-practical.
• If n is semi-practical, then 2n and 3n are also semi-practical. (credits to @Zeke)
Find the beginning of the sequence of semi-practical numbers and submit it to the OEIS.
I was just about to come here to say this. Merchant scales have two plates, not one.
(Any scale with just one plate doesn't need a set of weights at all.)
@@ragnkja First we should look for a number that is neither practical, nor semi-practical, nor a prime.
@@lonestarr1490 I started doing some quick tests and found out that 22 is the lowest of such numbers. Also, there's a grand total of two prime numbers in the sequence, being 2 and 3.
In such semi-practical numbers, the powers of 3 play a similar role as the powers of 2 do for practical numbers: they are both the minimal sized sets that allow the greatest total to be reached. Furthermore, the rule for practical numbers is if you can reach d-1 (for divisor d) using only smaller divisors; the analogous rule for semi-practical numbers would be if you can reach ⌊d/2⌋.
I love formulas like these that discover constants other than pi and e. If we sent out signals of mathematical constants for aliens to recognize, pi says "we were engineers." e says "we understood change." Phi says "we appreciated balence." But a number like this says "We were a society." Because to discover the practical number frequency constant is to show that we were looking not merely for beauty or truth but for convenience in useful numbers.
I love idea of giving each constant an official motto. And a thematically appropriate coat of arms.
BRUH
Interesting 🤔
Nah man, if I was an alien and I saw this constant being broadcasted, I would think they were trying to say “We were bored”
Parker and Grime are the only two people I need in my life.
And Brady to film them😄
Although they do have great channels on their own too
Your rwin Grimes conhecture?
(this is what I get for writing bad jokes in the shower)
Brady too
A related topic are Egyptian fractions, which are a way to represent fractions as the sum of various unit fractions (like 1/2 + 1/10 + 1/20 = 13/20). There was a conjecture that every positive rational number has an Egyptian fraction representation in which every denominator is a practical number and was recently proven in 2021. These fractions were developed in the Middle Kingdom, so 4000 years later new discoveries are being made from them.
Ooh.
The ancient Egyptians were apparently very practical.
@@vigilantcosmicpenguin8721And evidently, pretty rational
That moment when you discover that James Bissonette is also a Numberphile supporter, in addition to being a supporter of History Matters. The man is everywhere.
Dr. Grime's enthusiasm is contagious. I often wind up with the urge to spin up a Scheme REPL and play around with the concepts he presents.
Kind of unbelievable how Numberphile's been teaching me about numerical fun facts for over 12 years now. Can't get enough of them
True. For me it's been ... 13 years? I feel like it was around the time I started uni. Can't check because UA-cam watch history doesn't go back that far anymore.
@@bsharpmajorscalethe first video of numberphile was on nov 8 2011 iirc
A 4d (four old pence) coin was called a “groat”. I don’t know how common it was in use, but common enough to get a nickname.
It was the GROAT
There's a bit in the history section of the groat page on wikipedia which says that they were in circulation in Scotland until the 20th century.
If you want lots of weird practical numbers, take a wander through German numismatics from the 1500s-1700s. Osnabruck alone had coins of 1, 1½, 2, 3, 4, 5, 6, 7, 8, 9, 10, and 12 pfennigs around 1600.
James Grime reacts to numbers the way I react to meeting puppies. Just sheer enthusiasm.
It appears that every positive integer has not only an unique prime factorization but it also an unique flavoor
There was a medieval four pence (4d) coin called a groat. In modern times there is an annual religious ceremony in which the monarch distributes a small number of sets of special Maundy money, which includes specially minted 1,2,3 and 4 pence coins.
“Medieval”? It was minted until 1856 in Britain, and even longer in British Guiana, where it remained in circulation until that territory’s currently went decimal in 1955.
6:11 If you are using a balance (and the weights imply this) then 4 is simply 7 on one side with 2 and 1 on the other side with the item being weighed. Similar reasoning will take you up to 7 and 14, so it is possible to get utility from these impractical numbers as long as you approach it the right way 😄
Not all impractical numbers can be used this way. Try with 22, for example.
Does that make 14 a half-practical number?
@@hansnorleaf
Ismy has suggested the term “semi-practical”.
@@ragnkja 22 was not the example. but I take your point!
@@ragnkja somewhat-practical numbers
As this is about adding weights and coins, I'd like to mention a fact I noticed one day and have never had any use for. Adding the denominations of Australian coins gives one $3.85, or 385c. This is also the sum of the square numbers from 1^2 to 10^2.
Powers of 3 seems the most practical IMO. By putting it on either side of the weighing balance (or, equivalently, either giving that coin to the cashier or receiving it as change) you can make the most numbers with the fewest items. It boils down to being able to write any number in trinary but were each digit is {-1, 0, 1} rather than the standard {0, 1, 2}. Of course, powers of 3 aren't so great for mental arithmetic.
I love how you also call it trinary
For those curious, this system is called "balanced ternary."
Yes, yes. As a graduate of grade -1, I was going to say this too.
@@SpencerTwiddy I didn't even think about it to be honest. tri- as a prefix for three seems much more natural than ter- to me. Both the Latin and Greek word for three start with tri-. Who decided it should be ternary anyway?!?
Agree - a set of weights 1oz, 3oz, 9oz and 27oz will provide the ability to measure any integral number of ounces from 1 to 40.
Its nice to have a fun and easy to understand numberphile video
There WAS a four-penny coin, I believe... it was called a groat, if memory serves? I'm 58 and I was raised on Imperial (except in science lessons) but I now prefer metric for most things, and it's quite handy to be able to "speak both systems fluently" :)
Always a good day when theres a Numberphile upload!!
If you have traditional weighing scales then more numbers are practical because you can, for example, make 4 by doing 7 on one side and "2+1" on the other.
I always look forward to one of Dr. Grimes' videos.
I just re-watched the 11 11 11 video since it's now 11 years old and I'm now convinced you're a mad scientist who's realized the mathematical formula to prevent aging
He'll occasionally age in negative numbers
After I watched the video I thought "are there abundant numbers (numbers whose divisors add up to more than the number) that are not practical numbers?", and there are! 70 is the first abundant-non-practical number. I think these ones deserve to be called "impractical numbers".
Hm, well that's two interesting things I know about 70 now. Thanks.
@@hughcaldwell1034 what's the other interesting thing you know about 70?
@@JonWilsonPhysics The 24th square pyramidal number is 4900, aka 70^2. This is the only non-trivial solution to the canonball problem.
3:45: Regarding trying to find a set of weights for perfect numbers: I recognize that this is a math exercise, but from an engineering standpoint, some duplication would actually be more efficient. Take 20 for example. Instead of weights/coins/whatever of [20,10,5,4,2,1], you could instead do [20,10,5,2,2,1] and still determine each number 1 to 20. Either way uses six items, but the latter uses fewer materials and requires one fewer standard to align to, one fewer production line to produce, and so on. This is perhaps why there are two-cent coins, two-euro coins, two-dollar bills, etc., but not four cents/dollars/euros. It's just more practical to double up the twos.
i love James Grime! his happiness and enthusiasm is so infectious
Here's an attempt at a generalisation: For a natural number n, define the practicality degree pdeg(n) as the least amount of copies of the set of divisors of n you need to express every number m
I'm surprised the number of practical numbers less than X is proportional to the number of primes less than X. My first instinct is that practical numbers would become MORE common the higher you go, not less. I guess I need to think about that for a while to correct my gut feeling.
I wondered this too.
Less than 100: 25 prime numbers, 29 practical numbers (16% more)
Less than 1000: 168 prime numbers, 197 practical numbers (17% more)
Less than 10,000: 1229 prime numbers, 1455 practical numbers (18% more)
I notice big random-ish numbers tend to have big clunky factors (hence not practical), e.g.
90210 = 2x3x5x31x97
2023 = 7x17x17
5212023 = 3x19x61x1499
Any natural number where its prime factorization includes only powers of the first k primes is also practical, so 150 (2 * 3 * 5^2) for example. This was hinted at when James mentioned powers of 2, primorials, factorials, and highly composite numbers, where this holds true.
2:37 "42" mention! Shoutout to the answer to life, the universe, and everything
Knew we could count on you
Great to see that it only took 12 years of numberphile videos to get to some practical numbers
I will be going for a maths and computing program in college after a few months.
Thanks to numberphile 🎉
Why waiting? I don't even have my school graduation and am already sitting in introduction to algorithms I in Germany.
@@hello_world4859 actually I am sitting for more and more engineering entrance exams for getting into college 😅.
Then there's a long formal procedure to allot branches.
So it will take a few months.
Binary seems kind of optimal for weights if you put the product on one side and weights on the other. But what if you also can put weights on the product side? E.g. in binary to weigh 15 you need the set 1,2,4,8,16, 32, ... , if you can also add weights to the product side then you can also use left: 16, right: product+1. I think you can possibly find a set with fewer weights in this way. I appears that you can get by with 1,3,9,27,... unless I'm making a mistake.
Yep, it's the powers of three. You're essentially writing down a number in trinary, but using the digits {-1, 0, 1} rather than {0, 1, 2}.
This is called "balanced ternary" and was used in real life by merchants and some early computers. It's pretty neat.
Yes, when trying to weigh with minimal weights, if at first you don't succeed, tri tri tri again.
While Practical Numbers clearly are sufficiently interesting to justify a lot of number theory studies, I immediately turned to the question of efficiency: with, for instance, 20, there are multiple ways to achieve some outcomes (7=4+2+1=5+2), which suggests that the set of numbers contains some redundancy; for pure efficiency, I don’t see anything outstripping the powers of 2.
An instant classic numberphile vid. I shall watch this again a practical number of times.
The “antiprimes” have always been my favorite numbers. Glad i finally know a “formal” name for them 😄
A little while ago I made a spreadsheet to play around with similar ideas for currencies.
E.g. Between US and UK money, which on average requires fewer coins for any given value up to 100?
UK, slightly
Of course, this measure would also say the optimal solution is to have a unique coin for each value. So we could also take into account efficiency in that sense per number of denominations of currency.
In which case the USA system is slightly ahead.
I also worked out that, if you can only have two denominations, the most efficient way to make any number up to 100 is 1 and 10.
For 4 denominations I think it was 1, 5, 10, and 20.
I see Grimes, I click like.
What about videos like this one, where there’s only one Dr Grime?
Hearing Dr James Grime speak makes me happy
I mean, the thing with those 1,2,7,14 weights is that you could also subtract them by putting them on the other side of the scales. And with this approah you can actually get all the numbers from 1 to 14. getting 4 by sutracting 1 and 2 from 7, 8 by only subtracting 2 from seven, and so on.
Watching James talk about numbers makes me happy! 🙂
It’s always a good day when Numberphile posts a new video
Congratulations on the "antiprime" entry on Wikipedia :)
Coming back to the binary thing. I know it’s going off at a tangent but I liked the ternary thing where you can count a weight negatively or positively (as you can put it in the same pan or opposite pan to thing you are weighing, on a pair of scales).
So with 1,3,9,27 lb you can weigh any integer number of lb up to 40. Highly practical!
I don't know if it's just because it's Dr Grimes in the video or if it was the "recreational math" aspect of the subject, or both of those things, but this video reminded me of 2018/2019 Numberphile, when I first got hooked on the channel.
I work in calibration and a standard weight set would normally contain for example 100g 200g 200g and 500g i assume this is because its cheaper to have 2x 200g rather than 200g and 400g which i would say makes this set more practical despite it not being quite as sexg due to not needing to reuse 200
This is related to Golomb ruler, wich states the minimal number of weights (or lenght) needed to get all measures btw 1 and a certain value by using différences (and not sums) between them.
If 8:50 isn't a sly reference to the famous Abbott & Costello bit, I'll eat my hat.
Also, Numberphile has been going for well over a decade, and yet there's still new, easy to describe sequences like this. Brady can't be allowed to retire until he's done a video about all of them 😁
Which Abbott & Costello bit?
@@benwisey Rather than trying to post a link... just search Abbot And Costello 7x13=28.
i'm glad someone else caught that.
In old analogue electrical control systems, a common range of measure was 4 to 20 milliamperes - a range of 16. (Zero is not used as the “bottom” to be able to determine a circuit was off or failed.)
This allowed for being able to easily divide into ranges such as halfs, quarters, and eights of the full range.
This is quite interesting because I always wondered the reasoning behind the imperial system since metric system seemed so easy to my decimal brain.
But practical numbers make the inches, feet, pounds, etc make some sense.
The reasoning is mostly people cobbling stuff together over thousands of years (Imperial measures were largely based on Roman measures).
But, yes, hundreds of years ago, having things that broke into convenient fractions was useful for practical purposes. Which is why you see things like 12, 16, 24 pop up a lot. Similarly, they tended to be based on approximations of common objects or body parts. A foot was about the length of a typical adult male foot.
But it is also not at all consistent. For example, a pound can be 12 or 16 ounces today depending on what you are measuring. And historically there were many more pounds than the Troy and avoirdupois commonly used todaym
In response to Dr. James Grimes,
I work as a repair specialist for physical measurement testing machines. I have to use calibrated weight sets to calibrate electronic load cells. My weight sets follow this numbering setup. From 0.1 to 100 Newtons of force:
0.1N
0.2N
0.4N
0.5N
1N
2N
4N
5N
10N
20N
50N
100N
Our weight sets do include a few redundant weight values to make some values easier to create and so we have less weights on a hanger. Also our weights are custom manufactured to meet ASTM and ISO calibration standards.
In the 14 example, With balance scales, you could put seven on one side and the two and the one on the other side to make a difference of four.
If using an actual scale, you can put the weights on both sides of the scale and thus can also create negative values using the actual weights.
There are a nice set of numbers (i think it were the squares, but i don't remember) where you can use at most 3 numbers of the set and all numbers can be made with at most 3 numbers of the set (using only addition and negative addition)
With the series of 14 for weights, James says you can’t get to 4. But you can if you imagine the other side of the scale. You place the 7 on the side you want to be “4”, then place the 2 and 1 on the other side of the scale. You can get to 5 and 6 by this method as well.
We should discuss the most impractical numbers, This occurs in base 0. Every number can be represented by 0/0 in base 0, but not a single number is distinguishable from any other number... which makes base 0 the most impractical numbering system
you can use numbers like 1,2,4,8
to get any numbers between 1-15
you can also get more numbers if you follow the pattern 16,32,64 i think this is more practical
edit: i just saw he showed it in the vid
Yes, powers of 2 are all practical.
An interesting idea. Maybe we could use this to create some kind of information storage, then implement that to build a device to watch moving pictures on.
@@gargravarr2 i knew this before i knew practical numbers i needed it to code something and thought using these numbers was less laggy than using large if chains
You can make quite a lot more weights for the impracticle numbers by putting weights on the other side of the scale to cancel some out - ie make 5 by using the 7 on one scale and the 2 on the other. That is often how merchant scales were used.
Pleaseeeeeeeeeeeee I need my favourite mathematicians explaining the "animation vs math" short!!! PLEASEEEEEEEEEEEEEE YOU GUYS WOULD EXPLAIN THAT VIDEO SO WELL!!! Thank you for your content!
The E-series of resistors is a generalization of 1,2.5,10 ... its practical for adding up any number. 60 is practical for dividing, as it contains many factors: 1,2,3,4,5,6,10,12,15,20. So does 12. That's why a circle has 360° and our time is based on 12h with 60 minutes. Historical it's Babylonic but practical, too.
Back in time where the value of coins were determined by gold/silver the dividing was more important than the ability to add up. Thus some traditional currency systems were based 12 or 60.
Another great video featuring Dr Grime (my favorite!)
Assuming you are using an old style balance scale with two pans, one usually for the sample and the other for the standard weights, you could fill in missing weights/numbers by putting a small weight on the side with the sample. This would essentially subtract from the standard weights on the other side. This would work a bit like Roman numerals, I, II, III, IV, V, VI... or 1, 2 , 2+1, 5-1, 5, 5+1...
Numberphile has a large enough general reach that if you decide to name something, that's what the name will end up being. Antiprimes, Parker squares, etc.
Believe it or not we use them in construction for the sides of nails, but very few people know that history anymore.
Ah. I now see lots of poeple have said this!
With balances you can put a 5 weight in Bowl A and a 1 weight in Bowl B and then add your ingredient to Bowl B until the bowls balance. Thus measuring out a 4 weight of the ingredient. With that proviso, then at 5:34 in the video you can make 4 weight from 7-2-1, which I presume is what that set of weights is designed to do.
Dang. Practical numbers are really living up to the name we gave them with how much we're finding they have in common with other types of numbers like primes and perfects. Maybe _they're_ the real key to a lot of mathematical mysteries.
5:00
28... Something in between... 100.
Said like a true physicist.
I’m always ready for videos like this! Also, is this Brady’s house they always film in? 😁😁
when we are talking practical practical (as in the smallest number of weights for a travelling merchant), don't forget that you can do subtraction on the scales. you can get 4, 5 and 6 by subtracting from 7.
On a balance scale you can do subtraction 7-(1+2) to get 4 so you can have a different sort of practical number set.
I like that the thumbnail is just the factors of 20 💀💀💀💀
A fourpence coin actually did exist at one point, sometimes called a "groat." It wasn't much used though. British currency has existed with all the following denominations, and probably others too. Of course, they didn't all exist at the same time.
1⁄16d (quarter farthing, Ceylon only)
1⁄12d (third farthing, Malta only)
⅛d (half farthing, Ceylon only)
¼d (farthing)
½d (halfpenny)
¾d (three-farthing)
1d (penny)
1⅕d (new halfpenny)
1½d (three halfpence)
2d (twopence)
2⅖d (new penny)
3d (threepence/threepenny bit)
4d (fourpence, groat)
4⅘d (2p)
6d (sixpence, half shilling)
12d (shilling, bob, 5p)
18d (quarter-florin)
20d (gold penny, quarter noble)
24d (florin [a different florin], 2s, 10p)
30d (half crown, 2/s)
36d (3s)
40d (original half noble, original half-angel)
45d (later half-angel)
48d (double florin, 4s, 20p)
50d (later half-noble)
60d (crown, 5s)
63d (quarter guinea)
66d (still later half-angel)
72d (florin, 6s)
80d (noble, angel)
84d (third guinea, 7s)
90d (later angel)
96d (still later angel, 8s)
120d (half pound, half sovereign, double crown, 50p)
126d (half guinea)
180d (15s)
240d (pound, 20s, £1, quid)
252d (guinea)
360d (fine sovereign)
480d (double sovereign, £2)
504d (double guinea)
600d (50s)
720d (treble sovereign)
1200d (£5)
2400d (£10)
4800d (£20)
12000d (£50)
sets of weigths of pwers of 3 were used a LOT in france for balance stuff (you CAN do every numbers by having it on either side of the balance as +side NOT have it or have it on - side ! powers of 3 will give you ALL the numbers AND are a set of weight that DO exist and WERE used in shops for check out balance)
there was a problem about these in a math competition recently. we were supposed to prove there are infinitely many such numbers of the form a^2+a+42. it's a fun challenge, i recommend solving it for yourself. and if i have time, i'll try to take it to the next level by finding out whether there's any polynomial that nontrivially has only a finite amount of practical values (the trivial cases are the ones where the values of the polynomial are never divisible by numbers from a certain set, such that all large enough practical numbers are divisible by at least one number in that set, so e.g. polynomials that only have odd values).
14 works if you allow negative weights, which could be done by putting them on the side with the thing you are weighing. Ex: 4=7-2-1, 18=14+7-2-1
You can derive 4 from 14,7,2,1
4 = 7 - (2+1)
Where subtraction is putting the weights on the other side of the scales.
A groat was a fourpence coin, back when a shilling was 12 pence (12d), and other coins included 6d, 4d, 3d, 2d, 1d and 1/2d. There's the historic example you didn't find.
If he only checked the denominations in use right before Decimal Day in 1971, he wouldn’t have found it, as it wasn’t minted in Great Britain after 1856, and the last territory to use it was British Guiana, which switched to decimal currency in 1955.
@@ragnkja You're right, of course - I never saw a groat. But in my UK childhood, one pound sterling was worth 4 crowns, 8 half-crowns, 10 florins, 20 shillings, 40 sixpences, 80 threepences 240 pennies, 480 ha'pennies or 960 farthings. That was handy, but it made some of our schoolroom arithmetic problems interesting ...
I think real systems tend to be hybrids. They have some elements of practical numbering to them, but because things like powers of 2 (which are ideal) are not very familiar to humans with their ten digits, they also throw in 10 and 5 to gum up the works (or ease usage, if you prefer) :)
14 is a practical number when using weights. To get 4, just add the 7 to one side of the scale and (1+2) to the other side.
To get 5 take the 1 away. To get 6, take the 2 away. This method repeats up to 17.
Thinking about 14 and scales and you can get it to work if you allow putting the weights on either side. (Subtraction of weight)
Things in quotes represent the weight of material you are measuring
4: 7='4'+2+1
5: 7='5'+2
6: 7='6'+1
11: 14='11'+2+1
12: 14='12'+2
13: 14='13'+1
That was my thought as well
While this works for 14, far from all impractical numbers can be used this way. Try the factors of 22, for example.
One thing I've learned since entering the world of code golf and competitive programming is there is no end to the number of sequences that can be formed by looking at prime factors, and hundreds of them have names.
If you're weighing things on a scale, and everything will have whole number values, and if you can take multiple measurements to determine if weight X is higher or lower than various weight combinations, with four weights (1, 3, 8, 23) you can determine X up to 28. For example, you could figure out a weight of X=7 because X(1+8). With five weights (10, 12, 13, 17, 51) you can go up to 56. (Ref: Integer sequence
A037255)
I would argue breaking days up into 12 (highly composite) hour chunks, 60 (highly composite) minutes hours and 60 sec minutes is an example of practical numbers. Not in this sense of the word practical but actually practical in terms of what is practical to use irl. As proof: we never stopped using it. And the french trying to stop it failed.
I remember the old pounds, shillings and pence. 12 pennies to the shilling and 20 shillings to the pound. Making the whole system very practical for when a pound was an actual weight of metal.
I'm convinced James Grime has got an ageing portrait of himself in his attic.
Weight measurement can be a bad example in this regard. Because for 14; 4, 5 and 6 can be measured. If we put weights on different pans, this result can be obtained with 7-(2+1) for 4, 7-2 for 5, and 7-1 for 6. So you can weigh any number between 1-14 with its divisors.
4:16 I believe the old UK shillings were close.
They had a 1s, 2s, 4s, 5s, 10s, 20s (£1) however there was a 2/6s, and the 4s was very rare, being discontinued after 4 years of circulation.
You can do 4 with [14,7,2,1] by putting the [2 + 1] on the same side as the thing that you are measuring and weight against the [7] .
To get 4 with the 14 set place the 7 on one side of the scale then the 2 & 1 on the other so now you have the difference of 4. Very practical
Wasn't this in die hard part 3
The Australian currency system gets close, but doesn't use a 4 practical number. All levels of coins and notes use 1/2/5 and their orders of magnitude (although 1 and 2 cent coins are now out of circulation). The missing 4 means sometimes you need two of the same type, but that is it.
Great video. The most practical numbers must be 2^n, where most practical is measured on the number of weights compared to their reach.
There is one practical number so useful that it's been given a special name, one dozen.
You could have mentioned the problem of the minimal set of weights needed for a range of weights on a balance scale (subtraction also allowed when a weight is on the other side)
I'm thinking powers of three would be ideal for that.
How so?
@@skalderman Weights of 1, 3, 9, and 27 would get a measurement up to 40. There are three choices for a weight, left side of the scale, right side, or not used as all. 40 is 2222 base 3. Less than perfect number choices. 81 would be next and get up to 121.
I'm now thinking about how many "extra" weights you would need to "fix" an impractical number.
Dr Grime means you know the video is immediately more interesting :)
5:34 - "I can't make 4"
Actually, you could still weigh an amount of 4. You put the 7 on one side of the scale then put the 1 & 2 on the other side. Then you put the goods on the same side of the scale with the 1 & 2 weights, When the scale balances, the goods will weigh 4. Doing other combinations, you could also weigh 11, 12, & 13.
The point is he can't make 4 with just his set of weights, though. If he needed 4 ounces of weight at the end of a pendulum, for example, he'd be out of luck without duplicate weights.
Modern UK coinage + notes are based on 1,2,5,10,20,50,... which only omits the 4,40,... from the list James showed. If you look at the relative use of the 4-based values compared with the 1,2,5-based ones, as he composed the values from 1 to 20, you can see that the 4- values aren't needed as much. So I can imagine there is a simple practical argument for not bothering to mint 4-based coins or notes, and just assuming that there will be enough duplicates of the other values to fill the gap in everyday use.
Well, if you had a scale with 2 sides, you could use the 7, 2 and 1 weight to measure a weight of 4, by putting the 7 on the left, and both the 2 and 1 on the right to subtract from the 7 leaving a effective 4 weight on the left.
I love to see Antiprime again ❤, especially how prime numbers would be super impractical numbers.
Also, quite surprised to see 666 to be practical number.
You could put some of the weights in the other plate on your scale. For example, in your 14 example, 4 = 7 - 2 - 1.
British pre-dxecimal coinage was *almost* based on divisors of practical numbers.
So for 12d in a shilling, there was a penny, tuppence, thruppeny bit and sixpence, but no fourpence piece.
Similarly for 20 shillings in a pound, there was a shilling, 2 shilling piece, five and ten bob notes, but nothing for four shillings.
14 does work, assuming you’re using a balance as a scale, which is the whole point of using such weights. Put some of the weights on the opposite side of the scale thus “subtracting” that much weight. 4= 7-(2+1), 5= 7-2, 6=7-1, 7-7, 8 thru 10 are just 1 thru 3 + 7, 11 through 13 are 14 - 1, 2, or 3. It works all the way to 24.
In fact, for any set of n integers starting with 1, where the next set member is
Another way to prove 14 isn't practical is there simply aren't enough factors. Excluding 14 which doesn't make anything but itself, you have three different numbers which only make 8 distinct sums, 7 if you don't count 0. Of course, there's probably some correlation between the closeness and the number of factors, but I'm surprised one of them was never brought up.