I wish you had kept reading. In the end, you nitpicked about precisely the thing which the author ended up explaining.

The point about using infinite memory to store an irrational number is just a rhetorical device / presentation style to keep the novice reader engaged, and following along. And then you say, "Next we will explain how to solve this impossibility...," and such.

It's a good technique for writing, and presenting, but doesn't work if your audience already knows where you're going and gets impatient with you!

Thank you.

I did keep reading the article, as I say in my edit. I differ from you in my interpretation of this facet of the essay. To me, it reads as internal contradiction — the author says that it is impossible to represent 1/120 in memory and then goes on to describe a strategy for doing just that.

The author does not present this conflict as "it may seem impossible…" but rather as "is is impossible." That's an inaccuracy, not a writing technique.

Mate, I mean this with the best of intentions: please reflect on your requirement for exactness and consider whether it serves you well.

I did so with mine and I concluded it did not. Perhaps you will conclude differently. Perhaps the same. In either case, I think you will find it worthwhile.

The claim is only missing a qualification that the impossibility exists within the framework of ieee 754. It can be easily inferred from context and it is important to note because thats what the major CPU architectures use to store native floating point numbers. Therefore, the point is, we need a non-native (wrt the cpu) way to store and calculate with irrationals and some rationals if mathematical precision is required.

Thanks for the comment. For the record, I am very aware of iee 754, And I am aware that some numbers that have infinite repetition in decimal can have an exact representation. I actually thought about pointing that out. However I decided that it didn't add much to the post. It was written for more of a general audience and diving into those details, while would make the post more technically correct, would not actually add any value for the average reader.

I don't want to come off as combative, but I would argue that a correction would add quite a bit to the post. What you have is straight-up incorrect and wouldn't even take that much effort to fix. "For a computer to store the decimal representation a number that repeats forever would require an infinite amount of memory, so the number is approximated" would be correct and it's just a couple words more.

Agreed.

I agree. This is a small change that makes the statement true.

There are uncountably infinite real numbers and countably infinite rational numbers, so we cannot represent all of them using a {0, 1}^n where n is a finite number. No matter which numbers you can exactly represent, there will always be uncountably many real numbers which are not exactly representable, and, conversely, the set of real numbers exactly representable using a finite number of bits will have measure zero.

With all due respect, you sound like you don't understand the difference between rational and real numbers.

Also, the 'decimal' part of your comment is not needed, number base is irrelevant here.

The issues of "repeating decimal" and exact representation as a floating point number are orthogonal. Because the mantissa in a floating point number (which is fixed width) is in base 2, only fractions with denominators that are powers of two can be exactly represented. This means that simple base-10 decimals like 0.1 cannot be exactly represented by a float. For this reason it's at least a bit misleading to talk about how some numbers cannot be represented with finite memory, and then go straight into a demonstration of how they can in fact be represented in finite memory (as a fraction).

As for the comment about a general audience, if (some of) your target audience is HN readers, I think it's reasonable to expect many readers to be familiar with computer science.

If this were my article, I would replace the paragraph in question with a discussion of the error introduced in floating point calculation — consider perhaps that many programming languages will tell you 0.1 + 0.2 = 0.30000000000000004 [0].

[0]: https://0.30000000000000004.com/

Thanks again for the refresher. But again I am keenly aware for how floating point works. As for audience, this is not a hacker new exclusive. It just happens to be here rite now. It will probably make rounds in other forums as well. If you want to write an post an article on how floating point works. I’ll be happy to read it!

I know this sort of criticism can be annoying, but despite commenting on HN I would surely qualify as a 'lay reader' in this context, and I have a strong preference for authors taking care to either tell the whole truth or flag that a potentially misleading simplification has been made -- even if the falseness seems like a technicality to the author. If I notice it then I lose faith in the author's credibility, and if I don't then now I've 'learned' something false.

I enjoyed the article. Please ignore the nitpicking that goes on around here.

You can represent most useful irrational numbers on computers via programs that calculate them to the required degree of precision:

You can represent pi:

  4*atan(1)

e:

  log(1)

Now I feel like you could make arguments about non-computable numbers, although I feel like you could still “represent” them.

Upvoted for the sentiment. Although using log(1) to represent e gives me nothing (pun intended).

log(1) = 0

I think you meant exp(1).

Darn, as much as I hate editing posts that old I do wish I could correct that.

Cripes, you're clearly smart enough to understand that the author meant "to [literally] store [all the digits/bits of] a number that repeats forever [in the given base] would require an infinite amount of memory."

Talk about splitting hairs.

> The number result is 0.008333333333 with the 3 repeating forever. For a computer to store a number that repeats forever would require an infinite amount of memory

This is clearly a literary technique to create interest. Like saying something like "I told you a billion times not to exaggerate". It should be clear to most readers that this is not an attempted mathematically precise comment.

A floating point number with an infinitely sized mantissa could represent it exactly, which would require infinite memory.