I have been enjoying James Tanton’s website. In this video, Tanton explains a foolproof method for creating Egyptian fractions:

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This is a neat video — I’d heard that this method always leads to a finite Egyptian fraction, but never done the actual calculations and hadn’t realized how straightforward they are. Thanks for sharing it!

I have a couple tangential questions you might know the answer to, since you’re talking about Egyptian fractions. 🙂 First, I didn’t think that the Egyptians always used the largest possible fractions at each step. It makes sense that they would, but I was under the impression that they weren’t completely consistent, despite what this video implies. Do you know more about this?

Second, I’ve been looking for examples of Egyptian numbers from ancient Egypt, and aside from the Rhind papyrus do you know any photos of historical objects with numbers on them? I haven’t looked hard; mostly I’ve been glancing at sites on Egyptian numbers and they’re almost always drawn and not examples from, say, temples somewhere. But I figure there have to be examples somewhere, right? Or else we wouldn’t know what the Egyptians did in the first place?

Thanks! [And I’m really just wondering if you know the answers offhand; I’m not trying to take your time to dig up what I haven’t yet!]

No, this is not the method the Egyptians used, at least not all the time. You can often get “nicer” fractions if you don’t start greedy. For example, 3/10 = 1/5 + 1/10, which is easier to actually cut or measure than the greedy algorithm’s 20ths.

Also, the Egyptians had a special notation for 2/3 and seemed to prefer using it whenever possible, so they might use 5/7 = 2/3 + 1/21, rather than the greedy algorithm’s 1/2 + 1/5 + 1/70.

What’s cool about the greedy algorithm is that we can prove it will always give a finite answer. It works, but I think the Egyptians relied as much on insight as on any algorithm.

I have a couple of papyrus images in Egyptian Math Puzzles, and Google has a few more. And the Egyptian Mathematics Papyri site is very interesting.

Thanks for both answers 🙂 I also just found a color photograph of a tablet in a book about the museum in Cairo (so lots of pictures, though not a math book) that had some numbers in it. [I like being able to see photographs of where numbers were actually used, and had just been thinking about how few examples I knew of with Egyptians numbers when I read this post.]