I’ve got one for you Brady, I know you love a new type of number. Today we’re doing practical numbers. You like the sound of that? So apparently this is meant to be a good number to use when designing weights and measures. So things like weights or coins or you know lengths and things like that.
20 is a practical number and the idea is we’re going to design a set of weights, it’s based on the number 20 and the divisors of 20; divisors or factors if you prefer of 20. So imagine I’ve got a weight of 20 units - whatever we’re going to call them. The next weight is based on the divisor so it’s going to be 10 because it divides 20. All the other numbers that divide 20: 5 will be the next weight, and we’re going to have - what am I going to have next - 2? No 4. Then I’m gonna have 2 and then I have 1; because you always have 1 don’t you.
The idea is I’m going to try and make every value up to 20. So imagine I’m a merchant, I’ve got a set of scales. I’ve got this set of weights here, this is a complete set of weights, and I might want to measure whatever number up to and including 20. So let’s check I can do that, okay. The idea is we’re going to use each weight just once only, so we won’t need to have two sets of weights. The idea is I’m not- I don’t need to use my weight 4 twice, I should just be able to use one of each - so let’s do that.
Basically I’m just going to do the same thing again now but I’ve got a 10 weight. So it’s 10 plus 1, and then it’s 10 plus 2, and then it’s 10 plus 2 plus 1 and it’s 10 plus 4. 10 plus 5. 16: 10 plus 5 plus 1; 10 plus 5 plus 2; 10 plus 5 plus 2 plus 1. 19: 10 plus 5 plus 4 and then that gets me up to 20. And I can do every number up to the number itself and I can actually keep going until I get to like the sum of all the weights uh which would add up to 42 in this example; and then I stop because I can’t go any further because that’s the complete set of weights.
So that’s called a practical number because the idea is that you could design a set of weights that do that. The idea was that the old weights and measures are meant to be kind of superior, this is what they say, because they were based on these kinds of numbers. They were based on 16, like 16 ounces in a pound. In the old days it was 20 shillings to a pound, as in pound sterling - money. 12 is another practical number - inches in a foot. 28 is a practical number. So apparently the old weights and measures were based on these kind of numbers because you could design sets of coins or sets of weights and then you could use them like this in this combinations to make any number that you wanted until you got as big as you could get.
So that’s what they say, that’s why they’re called practical numbers. Oh because yeah the old weights and measures are far superior - much better than 10 and what we use now, our decimals and our metrics. Thing is, I tried to find a set of weights that actually do this; or a set of coins, don’t think it exists. I don’t think this was actually done. I just think this is mathematicians going “these numbers are far superior”. I don’t think this was actually ever used; with one exception which is 16 ounces in a in a pound, which is these weights. The ones we’ve talked about are powers of 2, perfect numbers, factorials, primorials, and highly composite numbers - also known as anti-primes, but that’s just Brady.These are all numbers where the divisors are fairly close together, so they are practical.You can also use the prime divisors test to determine if a number is practical or not.So, there are infinite practical numbers. In fact, a practical number multiplied by a practical number is a practical number. Additionally, a practical number multiplied by a divisor is also a practical number. As an example, if 6 is a practical number and 2 is a practical number, then 2 times 6 is 12 which is a practical number. Furthermore, 6 times 3 (where 3 is not a practical number, but is a divisor of 6) results in 18 which is a practical number. Thus, it can be seen that practical numbers can be created from other practical numbers, resulting in an infinite number of practical numbers. Additionally, these practical numbers are related to primes. For example, the twin prime conjecture (which states that there are infinitely many prime numbers that are two apart) has been proven for practical numbers, as there are infinitely many pairs of practical numbers that are two apart. Additionally, the Goldbach conjecture (which states that every even integer greater than 2 is the sum of two primes) has been proven for practical numbers, as every even number is the sum of two practical numbers. The prime number theorem (which states that the number of primes less than x is approximately x divided by the natural log of x) also has a result for practical numbers, with the number of practical numbers less than x being x over log x multiplied by a constant (calculated by Andreas Weingartner in 2020 to be 1.33607). Furthermore, it can be seen that these practical numbers become more sparse the higher one goes.
In order to minimise this sequence of practical numbers, a sequence of minimal practical numbers (also known as primitive practical numbers) can be created. This sequence is 1, 2, 6, 20, 28, 30, etc. and is sequence A267124 on the online encyclopedia of integer sequences. Additionally, the names of these numbers are Numberphile Patreon supporters, and if one would like to support Numberphile on Patreon, there is a link on the screen and in the video description.
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