Practical Numbers - Numberphile 

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)

“Anti-primes” are an objectively cooler name than “highly composite numbers”

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.

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.

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.

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.

Parker and Grime are the only two people I need in my life.

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.

Its nice to have a fun and easy to understand numberphile video

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.

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.

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 😄

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

Kind of unbelievable how Numberphile's been teaching me about numerical fun facts for over 12 years now. Can't get enough of them

Always a good day when theres a Numberphile upload!!

James Grime reacts to numbers the way I react to meeting puppies. Just sheer enthusiasm.

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.

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.

I always look forward to one of Dr. Grimes' videos.

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 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

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" :)

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.

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 will be going for a maths and computing program in college after a few months. Thanks to numberphile 🎉

Great to see that it only took 12 years of numberphile videos to get to some practical numbers

An instant classic numberphile vid. I shall watch this again a practical number of times.

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.

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.

Watching James talk about numbers makes me happy! 🙂

The “antiprimes” have always been my favorite numbers. Glad i finally know a “formal” name for them 😄

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".

Hearing Dr James Grime speak makes me happy

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.

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.

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.

It’s always a good day when Numberphile posts a new video

5:00 28... Something in between... 100. Said like a true physicist.

2:37 "42" mention! Shoutout to the answer to life, the universe, and everything Knew we could count on you

Congratulations on the "antiprime" entry on Wikipedia :)

I see Grimes, I click like.

Another great video featuring Dr Grime (my favorite!)

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!

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 😁

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!

Great video. The most practical numbers must be 2^n, where most practical is measured on the number of weights compared to their reach.

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.

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.

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

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

There is one practical number so useful that it's been given a special name, one dozen.

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.

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.

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.

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.

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.

Love to see more about factors from the certified dozenalist!

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

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.

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.

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.

Thanks!

Dr Grime means you know the video is immediately more interesting :)

Believe it or not we use them in construction for the sides of nails, but very few people know that history anymore.

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

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.

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.

Lovely video ❤

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.

I'm convinced James Grime has got an ageing portrait of himself in his attic.

Muchas bendiciones muchas gracias ❤❤❤❤❤

That unsolved Rubik's cube on your shelf drives me bonkers every time I see it. 😁

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)

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.

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.

I love your videos👍👍

(Wikipedia entry for Highly Composite Numbers) James is an even bigger numberphile fan than I am. Nice.

Awesome🎉

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

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.

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.

Oh this is kind of like learning about balanced tertiary!

If no one has started a band named "The Anti-Primes" yet, someone needs to get that done immediately.

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.

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

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.

Wonderful! :)

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.

As a twist, are there any impractical numbers if you could subtract also? These weights would traditionally be used on a hinged scales to compare something to the weights. In the "14" system, while you can't get a 4, you could still use a 7 and put the 1 and 2 on the other side with the item you're comparing.

2 is the nexus of the universe, being both practical AND prime. 😳🧐

Grime and Ben Sparks are by far my favorite presenters on this channel...

James is my fav

That's kinda funny, I'm participating in a correspondence seminar and three ago I had to hand in the first series and the third (of the four) question was: "The definition of a practical number (for some reason they called it a giraffe) and you had to prove that there are infinitely many natural solution for a^2 + a + 42 = n, where n is a giraffe"

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.

It's been a while since I was an active coin collector, but for a brief time in America in the mid to late 1800s there were in existence all at the same time a 1c (penny), 2c, 3c, 5c (nickel), 10c (dime), 20c, 25c (quarter), 50c (half dollar) and dollar coins in circulation. Perhaps it was due to their math practicality, however for ease of use in daily life the 2, 3, and 20 cent coins did not catch on (likely due to the informality of transactions back then, rounding a few cents didn't matter). I personally own the two and three cent examples, but the twenty cent coin is extremely rare and expensive!

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.

Hi, I have a question for any mathematician watching this video, in this video they talk about an approximation to the numbers of primes equal or inferior to a given numbers, it being an approximation does the gap between the exact number of primes and the approximated one get bigger when we take bigger numbers ? (That would not be very interesting right ? ) even though every little thing is already magic with primes numbers.

Immediately thinking about balanced ternary and a two-shoulder balance weights.

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)