Rhind Mathematical Papyrus
en.wikipedia.org
en.wikipedia.org
For example, you don't have rational numbers, so you are forced to represent everything as awkward sums. Even computing, say, three fifths of a number is a challenge under that system!
You don't have \pi to represent "ratio of circumference to diameter", so you have to use ad-hoc approximations.
You don't have notation for polynomials, so even basic equations are ad-hoc problems.
Another example: (almost) the entire work of Diophantus, the man diophantine equations are named after, is basically obsoleted by \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
Mind you, it's not that these people were dumb. That they were solving those problems with limited tools is proof enough that they were very smart.
It's like trying to write a program in 80s Basic, at a certain point the incidental complexity is eating you alive.
I had a similar deep insight when studying both theoretical physics and computer language / compiler design: for some areas of human endeavour, the only limit is the human brain. The machines ceased being the limiting factor a long time ago, but the human brain is essentially unchanged over the timescale of millennia.
In areas like this, incidental complexity is the key limit holding us back.
Once you've reached 100% of the brain's capacity, the only way to "get more out of it" is to waste less of its capacity on non-useful things.
You mentioned pi as a significant advancement. Yes, it is. But tau would be better. Every time I (or anyone) mentions this, there's dozens of people coming out of the woodwork saying that it doesn't matter, that it's just notation, that there's an established standard, blah, blah, blah.
Meanwhile, basically nothing in trigonometry really clicked for me until I saw Vihart's rant about why we should use tau. None of it. Not really, because the pi notation is nonsense, an accident of history, nothing more. It filled my brain with incidental complexity: that factor of two everywhere that made no sense to me.
The case for me of this is current flowing from - to +. Franklin didn’t know better and had a 50/50 chance of getting it right, but happened to get it “backwards”.
A small thing but one that always makes me skip a beat while following a circuit.
> Once you've reached 100% of the brain's capacity, the only way to "get more out of it" is to waste less of its capacity on non-useful things.
Queue modern day social media... It has its uses, but I feel like the brainrot and lessening of attention span it induces (and not to mention everything else like polarization, cult-like behaviours etc) is basically not worth the small amount of benefit it actually provides.
Really, the only problem with 2pi is that if you multiply it by a constant, it disappears. so people might write 2*2pi as 4pi. Sometimes you may not want to do that.
Anyway you don't need to fully transition to tau in order to benefit from it. Just like the planck constant has a reduced planck constant (which is probably more useful), and both are used, tau can be already used when it makes sense, without ever fully replacing pi.
Einstein tensors are a great example of notation as an enabling technology - it’s a mistake to think we have thought of all the notation tricks we might need.
I wonder if, for example, in attempting to reason about neural networks in terms of ‘gradients’ and ‘matrix operations’ we might look, to future mathematicians, like poor ancients struggling to handle arithmetic without a clean notation for rational fractions…
A few examples:
https://arxiv.org/abs/2303.08829 (ZX shows equivalence between approaches that used to be thought of as different)
https://arxiv.org/abs/1704.08670 (ZX shows how to find new ways to implement fault tolerant operations)
https://arxiv.org/abs/2206.12780 (using ZX to build a 4-body parity measurement out of 5 2-body parity measurements instead of 6, and to show the previous constructions are easily related)
People had been doing derivatives (at least sort of) before Newton and Leibniz, Cardano (sort of) used complex numbers before Euler, algebra before Galois, categories before Mac Lane.
But these people "connected the dots", as it were. It's not merely a trick -- it's a fundamental contribution.
The new mathematics at least could be initially tiny, and then the notation provides the leg work for huge forthcoming discoveries.
The fully expanded coordinate version is based on an arbitrary choice of spacetime coordinates.
Ideally you'd use a computer language/environment with a rich enough basic formalism to let you write the 1 formula in a human-legible way and have your compiler expand out all of the coordinate version as appropriate for the particular problem for the sake of doing floating point arithmetic.
The quadratic formula wasn't interesting to Diophantus because he didn't believe in negative or irrational or imaginary numbers; he considered those "impossible".
This is nowhere close to an accurate statement, even if we only consider works that have survived to the present (most of his work has been lost). Have you tried to read the work of Diophantus or any modern summary?
> 1, 2, 6, 7, 8 and 9 loaves of bread (respectively, in each problem) are divided among 10 men. In each case, represent each man's share of bread as an Egyptian fraction.
I guess it shouldn’t be surprising, but wow, this form of question really goes back.
On the third hand, this is clearly a cultural concern as had I been born 100 years ago I wouldn't have cared in the slightest about removing a document from primitive people. And had I been born 100 years in the future I wouldn't have the slightest regard for the biological humans' concerns.
Perhaps it all works out.
But let's say that they are. What did the inhabitants of Egypt do about the Pharaonic era historical artifacts? They looted them time and again. Looted and sold them to the highest bidder.
Maybe you want to claim that people from Egypt don't "loot" the Ancient Egypt artifacts, they only "take possession" of them, since they are the rightful owners.
But if you claim that, then you are not in luck, my friend. This is what happened to the Rhind Papyrus. It was found (or looted) by the locals, who sold it to the highest bidder, who happened to be a guy from the Great Britain. The papyrus was not stolen, it was simply bought.
> was not stolen, it was simply bought.
We are talking about a thing that was made 3500 years ago. Who is its rightful owner?
It's a completely different situation than stealing from someone who is alive. How in the world do these things seem similar to you?
> One of the modern applications of Egyptian fractions is the request of a specific resistance value needed in the design of an electrical circuit, a problem called in the literature the 2- Ohm problem. College students know well from their physics class, that the equivalent resistance R of two parallel resistances and is given from a law very easy to deduce, based on equating the current passing through the fictitious equivalent resistance R with the two currents passing through both resistances while maintaining same potential. One direct application of this, suppose an engineer wishes to incorporate in one of his designs a resistor of so many ohms which the manufacturer does not produce; for it is impossible that the latter displays in the market all possible ohm-values for his resistors. First, the market cannot possibly sustain it, but more important, one cannot feasibly produce resistors with values as elements of a dense subset of the real line, being, as analysis taught us, an uncountable set. Rather, manufacturers display only in the market what they call an "E12 series", i.e. resistors in sets of 12 different values, namely
> 10, 12, 15, 18, 22, 27, 33, 39, 47, 56, 68, 82 ......
> 100, 120, 150, 180, 220, 270, 330, 390, 470, 560, 680, 820......
> 1000, 1200.......... Ohms, etc.....
> Now suppose an engineer needs in one of his designs a resistor of 7 ohms, then he would resort to a parallel combination from the fraction 1/7=1/10+1/56+1/100+1/120+1/150 in which all the resistors belong to the E12 series, i.e. he will replace his 7 ohms resistor with 5 parallel resistors; this he reaches using a special software (computer programmes exist for such designs, yet the exact solution is by no means trivial). What I did myself instead, is to resort to Ahmes 2/n table and wrote 2/7=1/4+1/28 or that 1/7=1/8+1/56. Decomposing further 1/8 into 1/12+1/24=1/12+1/48+1/48, but then I shall have to use instead of the 48 ohms resistor a resistor of 47 ohms from the E12 table. My final fraction is 1/7=1/12+1/ 47+1/47+1/56, i.e. my resistor of 7 ohms will be simulated by 4 parallel resistors instead of 5 (I am accepting equal fractions). My solution is both minimal and optimal based on Ahmes table. The relative error of my design will not exceed 0.6 per cent, being negligible; especially that any manufactured resistor will itself be subject to some allowed tolerance of the same order. https://web.archive.org/web/20130625181118/http://weekly.ahr... So, a thousands of years ago? Ancient Egypt and before? :) Also:
> The measured values of voltages and currents in the infinite resistor chain circuit (also called the resistor ladder or infinite series-parallel circuit) follow the Fibonacci sequence. The intermediate results of adding the alternating series and parallel resistances yields fractions composed of consecutive Fibonacci numbers. The equivalent resistance of the entire circuit equals the golden ratio.
Thisnis hyperbolic.
The number of abstract mathematically plausible values is countable, not uncountable, and the number of practical values under real world tolerances is finite. Real numbers have nothing to do with it, and Egyptian fractions are useful for compressing a large finite set into a small finite basis. They do nothing thay helps compress infinite sets.
https://web.archive.org/web/20130625181118/http://weekly.ahr...