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I encountered the process called "Anthyphairesis" which apparently was the basis of the Euclidean algorithm and was also described in Chinese writings under the name "reckoning".

A description of the "reckoning" algorithm:

The method for simplifying parts: Take the numerator from thedenominator; in turn take the denominator from the numerator. When thenumbers [on the sides of] the numerator and denominator are equal to oneanother, then you can go on to simplify.

I am trying to understand the idea behind it by using an analogy of movements along the number line.
E.g. if we want to apply the process for $8$ and $3$ then it could be something like:

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But I am confused on the process itself. I think the idea is that we move around in steps of the two numbers we are working on until we reach some limit, but I am not clear on how the process would work.

Could someone help on this?

Jim
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    I think it must be describing the original Euclidean algorithm: repeatedly subtract the larger number from the smaller until the two numbers are equal. This is the gcd. The algorithm described, if taken literally, wouldn't converge, would it? – saulspatz Jun 04 '21 at 18:22
  • @saulspatz: I was under the impression that by these subtractions/movements we would end up shifting along the number line till we either end in $1$ or $0$ and then it stops. E.g. if we end up in $1$ then $1 + 8 = 9 - 3 - 3 - 3 = 0$ but I am not clear on how to do this in the general way or if I am totally confused – Jim Jun 04 '21 at 18:57
  • I see that I said "subtract the larger from the smaller" where I meant, of course, "subtract the smaller from the larger". I don't see how the literal algorithm works. Say we have $(5,15)$. The algorithm seems to give $$(5,15)\to(5,10)\to(-5,10)\to(-5,15)\to(-20,15)\to\cdots$$ Perhaps one is supposed to take absolute values. Then it's Euclid's algorithm again. – saulspatz Jun 04 '21 at 19:06
  • @saulspatz: the quote I mentioned is from the TheSuàn shù shū Writings on reckoning commentary of Christopher Cullen p41 – Jim Jun 04 '21 at 20:07
  • In modern language it boils down to the subtractive (vs. mod / remainder) form of the Euclidean algorithm (and its manifestation in closely related objects, e.g. continued fractions, mediants, etc. But historical use an meaning is more complicated, e.g. David Fowler has written much on it, e.g. see Ratio in early greek mathematics or his book The mathematics of Plato's acacdemy: a new reconstruction). – Bill Dubuque Jun 04 '21 at 20:35
  • The 2nd book link above should be to this review – Bill Dubuque Jun 04 '21 at 20:41
  • @BillDubuque: The subtractive is using e.g. $133 = 1(85) + 48$? – Jim Jun 04 '21 at 21:21
  • The subtractive form repeatedly subtracts the smaller from the larger until they are equal (for positive integers), e.g. see here. – Bill Dubuque Jun 04 '21 at 21:51

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A number line may not offer a useful context for understanding anthyphairesis because it works with a single unit corresponding to the interval between 0 and 1. anthyphairesis and the Euclidean algorithm entail relations between multiple units: first a dividend unit and a divisor unit and also remainder unit(s) if the dividend is not a multiple of the divisor. So parallel line segments are better resources. But the units need to be explicitly labeled.

Let's imagine that line segments (or fraction bars) A and B are in the ratio of 10:7, but a student has not been told this.
The student measures A with B, finding it fits once, leaving a first remainder,R1. She then measures B with R1, obtaining a quotient of 2 and leaving a remainder, R2. Finally, she finds that R1 measures exactly 3 R2's.

If we assign the value of 1 to R2, we can infer that A = 10 and B=7.

Note also that the quotients obtained were (1;2,3). These also allow us to represent A/B as a simple continued fraction: A=1+ (1/(2+1/3)). It is useful to actually diagram these ideas to better understand how the quantities are interrelated.

DavidC
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