{"items": [{"author": "Jim", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787743024542", "anchor": "fb-787743024542", "service": "fb", "text": "Make it base 9 (-4 to +4) and it'll be unbiased rounding, too! (In base 10, we need hacks like \"banker's rounding\" because otherwise if you round a bunch of numbers in the usual way and then add them up, they're biased slightly away from zero.)", "timestamp": "1463930973"}, {"author": "Michael", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787743024542&reply_comment_id=787753623302", "anchor": "fb-787743024542_787753623302", "service": "fb", "text": "→ The trouble with base 9 is if you're using fractional numbers, 1/2 comes up a lot. Changing base won't make 1/2 easier to round, but it has an infinite decimal representation in base 9 (0.44444... or 0.-4-4-4-4-4...).", "timestamp": "1463934104"}, {"author": "Andrew", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787745389802", "anchor": "fb-787745389802", "service": "fb", "text": "Yes, there are mistakes in the tables. Getting past that, assuming that the point of truncation or rounding is simplification, I wonder about how much more difficult this rounding process is, compared to truncation. I might be inclined to truncate to two places instead of rounding.", "timestamp": "1463932274"}, {"author": "Jonah", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787745489602", "anchor": "fb-787745489602", "service": "fb", "text": "Laura Sheppard-Brick and I think that it's clunky to use the negative sign within a multi-digit number and that you should instead just write the numbers backwards.", "timestamp": "1463932339"}, {"author": "Jan-Willem", "source_link": "https://plus.google.com/100580955183019057735", "anchor": "gp-1463935667707", "service": "gp", "text": "The crucial question: Which will come first, killer robots ruling the planet or biased decimal arithmetic?", "timestamp": 1463935667}, {"author": "Jonathan", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787757585362", "anchor": "fb-787757585362", "service": "fb", "text": "It seems unfortunate that this system encourages rounding 5 to 10.", "timestamp": "1463936439"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787757585362&reply_comment_id=787762904702", "anchor": "fb-787757585362_787762904702", "service": "fb", "text": "→ What should you round 5 to?", "timestamp": "1463939001"}, {"author": "Jonathan", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787757585362&reply_comment_id=787762919672", "anchor": "fb-787757585362_787762919672", "service": "fb", "text": "→ 5 should be rounded to itself.", "timestamp": "1463939050"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787757585362&reply_comment_id=787763209092", "anchor": "fb-787757585362_787763209092", "service": "fb", "text": "→ And 4? 6?", "timestamp": "1463939293"}, {"author": "Jonathan", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787757585362&reply_comment_id=787763393722", "anchor": "fb-787757585362_787763393722", "service": "fb", "text": "→ 4 should round to 4, 6 should round to 6.

Also, 4.5 (or 10 - 5 -.5) should round to 5 and not 10.", "timestamp": "1463939408"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787757585362&reply_comment_id=787764451602", "anchor": "fb-787757585362_787764451602", "service": "fb", "text": "→ Sometimes you want to round to something other than the 1s place. For example, if we have two significant digits we'd express 0.334 as 0.33 and 334,000 as 330,000.", "timestamp": "1463940155"}, {"author": "Michael", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787757585362&reply_comment_id=787764960582", "anchor": "fb-787757585362_787764960582", "service": "fb", "text": "→ 5.000... is a pretty special case. 5.00001 is better rounded to 10 than 0, so if you're just looking at the next digit, better to round 5 up", "timestamp": "1463940505"}, {"author": "Jonathan", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787757585362&reply_comment_id=787771931612", "anchor": "fb-787757585362_787771931612", "service": "fb", "text": "→ I agree that 15 should round to 20 and not 10, but asking whether 5 should round to 0 or 10 is a wrong question, the closest 1 significant digit number to 5 is 5.", "timestamp": "1463942390"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787757585362&reply_comment_id=787772460552", "anchor": "fb-787757585362_787772460552", "service": "fb", "text": "→ Sorry, yes, I can't think of any reason to round 5 on its own.", "timestamp": "1463942590"}, {"author": "David Chudzicki", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787757585362&reply_comment_id=787828089072", "anchor": "fb-787757585362_787828089072", "service": "fb", "text": "→ Related: in a context where single digit numbers are very common and negative numbers aren't, this system gets you a lot of awkwardness that doesn't help with anything (by turning 6 be through 9 into two digit numbers).", "timestamp": "1463968597"}, {"author": "David Chudzicki", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787825334592", "anchor": "fb-787825334592", "service": "fb", "text": "There must be some psychology studies backing this up...? (The claim that numbers which will round up are perceived as unreasonably low.)", "timestamp": "1463967353"}, {"author": "David Chudzicki", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787827096062", "anchor": "fb-787827096062", "service": "fb", "text": "Next step is suggesting some pronunciations.", "timestamp": "1463968285"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787827096062&reply_comment_id=787840833532", "anchor": "fb-787827096062_787840833532", "service": "fb", "text": "→ nive, nour, nee, nou, nunit, zero, one, two, three, four.

\" 193 -> two hundred nunity three
* 16 -> twenty nour
* 6999 -> seven thousand nunit
* 376 -> four hundred nouty nour", "timestamp": "1463974780"}, {"author": "David Chudzicki", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787827814622", "anchor": "fb-787827814622", "service": "fb", "text": "Note that with time speaking like this is very common.", "timestamp": "1463968354"}, {"author": "Michael", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787836267682", "anchor": "fb-787836267682", "service": "fb", "text": "I saw something very similar in the book \"Indian Mathematics \" by Ashok Jhunjhunwala on traditional methods for doing fast mental calculations.", "timestamp": "1463973339"}, {"author": "Mac", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787857255622", "anchor": "fb-787857255622", "service": "fb", "text": "Roman numerals actually have a subtractive function based on position rather than assigned negativity. A smaller number preceding a larger one is generally subtracted from the larger one. Hence IV = 4, CD = 400", "timestamp": "1463991141"}, {"author": "David Chudzicki", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787857255622&reply_comment_id=787865549002", "anchor": "fb-787857255622_787865549002", "service": "fb", "text": "→ Yeah. There's some commonality with Jeff's system, but Jeff's is a lot better.", "timestamp": "1464002217"}, {"author": "Charles", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=787905898142", "anchor": "fb-787905898142", "service": "fb", "text": "This is ridiculous when you have computers doing it. It doesn't improve anything just changes the method.", "timestamp": "1464021842"}, {"author": "Mars", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=788154200542", "anchor": "fb-788154200542", "service": "fb", "text": "This proposal doesn't round correctly.
Take the number 2-5-5, also known as 145. This should round to 100, but truncation rounds it to 200.", "timestamp": "1464043917"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=788154200542&reply_comment_id=788157733462", "anchor": "fb-788154200542_788157733462", "service": "fb", "text": "→ Yes, you're right, there's a problem. 145 needs to start \"15\" but \"5\" isn't allowed. Not sure how to fix this yet. Maybe by allowing both 5 and -5?", "timestamp": "1464045168"}, {"author": "David Chudzicki", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=788154200542&reply_comment_id=788165133632", "anchor": "fb-788154200542_788165133632", "service": "fb", "text": "→ Interesting how many people (including me) read this before someone noticed that.", "timestamp": "1464047808"}, {"author": "Nix", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=788154200542&reply_comment_id=788176800252", "anchor": "fb-788154200542_788176800252", "service": "fb", "text": "→ Using 5 as well would also allow the negation of any number to be written as the negation of all the digits of that number. For example, under the original system 15 was written \"2-5\" while -15 was \"-1-5\" (rounding positively instead of away from zero). Now it can be written \"-25\".", "timestamp": "1464052237"}, {"author": "Paul", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=788154200542&reply_comment_id=788195123532", "anchor": "fb-788154200542_788195123532", "service": "fb", "text": "→ I think you have to allow both 5 and -5. Interesting, thanks! Yes, best we just switch to balanced ternary I think.", "timestamp": "1464063171"}, {"author": "Mars", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=788154200542&reply_comment_id=788210881952", "anchor": "fb-788154200542_788210881952", "service": "fb", "text": "→ David, Jonathan noticed, but it got mixed up in his confusing narrative on rounding to 5.", "timestamp": "1464071704"}, {"author": "Mars", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=788154200542&reply_comment_id=788211181352", "anchor": "fb-788154200542_788211181352", "service": "fb", "text": "→ Allowing 5 as well as -5 would work, but it's not pretty. Watch me count from 145 to 155:
15-5
15-4
15-3
15-2
15-1
2-50
2-51
2-52
2-53
2-54
2-4-5

It's pretty confusing when to use the 5 and when not to.", "timestamp": "1464072096"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=788154200542&reply_comment_id=788283720982", "anchor": "fb-788154200542_788283720982", "service": "fb", "text": "→ @Vivian: how to handle 25 vs 3-5 depends on your rounding convention. If 0.5 always rounds up then 3-5 is correct and 25 isn't.", "timestamp": "1464089808"}, {"author": "Jeff Kaufman", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=788154200542&reply_comment_id=788535002412", "anchor": "fb-788154200542_788535002412", "service": "fb", "text": "→ added a correction", "timestamp": "1464198350"}, {"author": "opted out", "source_link": "#", "anchor": "unknown", "service": "unknown", "text": "this user has requested that their comments not be shown here", "timestamp": "1464103405"}, {"author": "David Chudzicki", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=788316809672&reply_comment_id=788349434292", "anchor": "fb-788316809672_788349434292", "service": "fb", "text": "→ hmm, the Roman system (which included subtraction) was at least natural enough to exist and be used for awhile...", "timestamp": "1464113532"}, {"author": "opted out", "source_link": "#", "anchor": "unknown", "service": "unknown", "text": "→ this user has requested that their comments not be shown here", "timestamp": "1464113811"}, {"author": "David Chudzicki", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=788316809672&reply_comment_id=788352707732", "anchor": "fb-788316809672_788352707732", "service": "fb", "text": "→ I'm kind of out on a limb, but... it's better on lots of ways. Maybe \"avoiding subtraction\" isn't really one of them.", "timestamp": "1464114780"}, {"author": "Ross", "source_link": "https://www.facebook.com/jefftk/posts/787741312972?comment_id=788973014632", "anchor": "fb-788973014632", "service": "fb", "text": "I know I'm late to the party, but it seems like another intuitive solution to make truncation be rounding is to round 184 to 150, 113 to 150, and 201 to 250. Thoughts?", "timestamp": "1464465452"}]}