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Comment by miffy900 | Hacker News Reader

I really wish people would stop saying N times smaller than something when making a comparison; that makes no sense - it's 1/9th (11.11%) the size. You don't get a smaller quantity by multiplying by a number greater than 1.0. You could instead reverse the subjects being compared - "the original model is 9x bigger than this new smaller, efficient model" or some such. That makes sense.

I keep seeing this being used when people talk about efficiency or performance gains and it's just very unintuitive language.

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I agree it makes little sense in a literal mathematical take but "it's 9x smaller" or is too much of linguistic advantage compared to "the original is 9x larger" or "it's 1/9th as large" to expect a change with. You don't have to invoke fractions, it keeps the thing in focus as the first subject, it matches the pattern of the inverse statement, and it's just plain short... so that's what people will adapt and interpret the meaning to be.

One other way to map both types of linguistic statement consistently to math is to interpret "9x" as "there is a 9 times difference between these two things" and then "smaller"/"larger" tells you which end of that separation the subject is (rather than specifying whether the multiplication builds up or down).

"it's 11% as large" avoids fractions but is much more clear (IMO) than "9x smaller" which my brain doesn't understand.

11% is also a really saying a fraction, 11 per-cent or 11/100, but there's nothing wrong with that feeling more natural to some and it is at least a nice shorthand way for the written form. "A ninth the size" is a similar alternative. All are really fine, there's always someone who has trouble with a given representation compared to another.

It's an idiom, much like "you are pulling my hair". Are you actually pulling my hair? Of course not; taken literally, many English phrases make no sense. But they are common elements of the language and everyone understands them, so they aren't problematic.

This is the same. Taken literally, "9x smaller" is nonsense, but everyone who hears that phrase knows exactly what mathematical operation you are referring to, thus it's a totally acceptable way to express that you mean to say 11.11% as large.

When I hear 9x smaller, I visualize 9 small items taken as much space as the big one, I find it quite clear.

I don't see the issue at all, consider whatever as a fraction, or just add a denominator of 1:

"9 times larger" => multiply the numerator of the fraction with 9

"9 times smaller" => multiply the denominator of the fraction with 9

Consider a bag of identical resistors with resistance R

Add 9 of them in series: the resistance is 9 times larger.

Add 9 of them in parallel: the resistance is 9 times smaller.

Its the series vs parallel dictionary wars all over again...

Check my other comment.

"The original is 9x larger"

This is another pet peeve that I have with the way we phrase these things: It should be 9x "as large" and 8x "larger."

The way I'd usually phrase this, to avoid ambiguity, is "this new model is 11% the size (of the original)." Or, the other way, "the old model is 9x the size".

If we were talking about speed instead of size, i think it would be perfectly reasonable to say 9x faster. I'm not sure I agree that 9x smaller is unintuitive. It makes sense to me.

"Nine times" literally means multiplied by nine, but here we're dividing by nine. It's not unintelligible (because the corrupted verbiage is so commonplace) but it is needlessly awkward. Like saying "resulted in a size reduction increase of 10 megabytes."

> "Nine times" literally means multiplied by nine

Rather, "nine times larger" means multiplied by nine, and "nine times smaller" means divided by nine. This is basic and not particularly awkward, certainly not more so than e.g. positive/negative correlation, or many much more awkward and more common linguistic constructions, IMO.

If you have to edit out words (i.e. context) to argue a phrase doesn't make sense... I am not sure what mental model you have for natural language, exactly, but it certainly isn't a very robust one.

Saying something is 9 times bigger is also a shorthand because “times” is a shorthand for repeated addition. You are adding a number “9 times” to make it “9 times larger”. But we don’t say the “adding” part.

Likewise if I repeatedly subtract a number 9 times to make it “9 times smaller” I can omit the subtracted bit.

That interpretation doesn’t make sense. Repeated subtraction does not correspond to division in the way repeated addition corresponds to multiplication.

Que? Division is the opposite of multiplication of course it does.

If you divide 81 by 9 you subtract 9 from 81 until you get to 0. The number of TIMES you do that is the result “81 divided by 9”.

This isn’t analogous to repeated addition giving multiplication. You’ve switched from applying an operation nine times to counting how many times an operation can be applied.

You chose 81 because it’s the one number that makes your argument appear to work. Try the same reasoning with 80 ÷ 9.

"faster" relates to speed. Speed is related to time and speed of a thing is usually defined by time. 9x faster speed translates to time/9. There is an extra step of related conversion there.

Conversely, 9x [filesize/natural number] is bigger. Every time. At least in the basic maths used by most people. There is no conversion into other units.

Therefore "9x smaller" when talking about a natural number like filesize is a nonsense statement in logic terms. If you strive for unambiguous phrasing - which is a significant part of the programming experience - this logical nonsense might well perturb you.

But english language is a flexible thing and if the phrase communicates your intent to your audience then that's fine by me.

I think more intuitively for speed - you are multiplying the things per period. If I’m making widgets 9x faster I have 9 x the previous number of units per period of time. To get the amount of time, you invert the ratio to period over units.

Multiplying some scale by units per period makes sense and is both linguistically and mathematically sound.

It's simple

If 9 is "9 times greater" than 1 then 1 must be "9 times smaller" than 9

It'd help if you read "9 times" with the operator which is what's being flipped instead of with the number

Its complicated:

50 percent smaller means either half, or two-thirds the size, depending on Apple Marketing doing the math.

0.999 times smaller means size is nearly zero. 9 times smaller means you get negative memory from loading it.

This is widespread usage.

I don't see what makes it hard to understand.

But we're cutting drug prices 500, 800, 1700%! Numbers nobody thought were possible.

It was hilarious that this is the only time his, er, meta-imaginary-gains intensifier made the statement literally true.

Define the "parallel sum"

a∥b=((a*b)/(a+b)

A normal sum has the associative property:

a+(b+c)=(a+b)+c which justifies dropping the parentheses

a+(b+c)=a+b+c=(a+b)+c

It is just an exercise for the reader that the parallel sum ∥ operation is associative:

a∥(b∥c) = (a∥b)∥c

multiplication is typically defined as repeatedly adding:

a x b = b+b+...+b+b (a times b)

similarily one can define parallel multiplication xx :

a xx b = b∥b∥...∥b∥b (a parallel-times b)

The reader can verify that 9 parallel-times size

9 xx size = size∥size∥size∥size∥size∥size∥size∥size∥size = size / 9

So I just think the Bonsai designers meant it was 9 parallel-times smaller, which checks out...

x∥x=(xx)/(x+x)= x/2

x∥x∥x =(x/2x)/(2/x+x)=(x/2)/(3/2)=x/3

x∥x∥x∥x = x/4

Me too!! It's a huge pet peeve. And it's so hard to get people to see how it's linguistically AND mathematically WRONG.

I disagree completely and I'd be happy to be convinced otherwise

I wrote this elsewhere but I felt it was worth saying again. "9x smaller" is an idiom, much like "spill the beans" or "it costs an arm and a leg".

Idioms don't have to make literal sense or be linguistically/mathematically correct to be useful. All that matters is that other people know exactly what you mean when you say it.

And, pretty much universally, if I tell someone "the compressed file is 10x smaller than the original", they are going to know what I mean is that the byte size is 10% of the size of the original.

That makes it an idiom that is perfectly okay for everyday use.

They probably rephrased it from some more technical form like "we compressed the model by a factor of 9" or "we've improved the packing efficiency of the model by 9x". Where these are measurements of the transformation the model is undergoing, not measurements of the resulting model.

Eh, when I read smaller with an integer multiplier, I mentally switch to the reciprocal. Easier than convincing the world not to use "9x smaller". Do you feel the same way about "9x faster"? What you're actually measuring is time, and "faster" is the reciprocal of time, similarly to "smaller" being the reciprocal of size.

yes, actually I do think the phrase 'N times faster' is sensible and logical

if I say 'this Apple M5 chip is 3x faster' than this intel chip, it implies two things:

- the run time of most operations that runs on it is now reduced (so one quantity is smaller)

- but also: MORE WORK is being completed per unit of time compared to the intel chip (so this quantity is greater)

So yes, a greater quantity is being measured in the apple chip compared to the intel when you say apple is N times faster. i guess this a quirk with the word 'faster' - it actually measures two things, time and work performed per unit of time. the word smaller just measures size.

like, if I give a customer a cup of coffee one day, then give them a SMALLER cup of coffee the next day, for the same price, but declare "It's now 2X more space efficient!" i.e. it's now half the size, I'm certain the customer is gonna be pissed.

Ah, but in this case the customer is buying the coffee for the caffeine content. And you've doubled the caffeine content per volume! Twice as efficient a delivery mechanism, similar to this model!

I feel like this becomes less clear if you're talking about "one-time" events. "Our website loads 2x faster now" doesn't mean that they are sending more data per second, and the user won't be doing more website loads. The website load time going from 1 second to 0.5 seconds is "2x faster" and I think that that is a very common way of communicating such performance increases too.

It could be sending more data per second if it's the result of a network upgrade at their datacenter. :)

Also, X is 9 times larger than Y and X is 9 times as large as Y are two different statements.

to spell it out:

200% as large as 1 is 2.

200% larger than 1 is 3..

200% (or 2x) smaller than 3 is.. ?

10% smaller is 3*0.1=0.3 smaller than 3, which is 3-0.3=2.7

So 200% smaller is 3*2=6 smaller than 3, which is 3-6=-3

Which is why these kinds of statements usually don't make any sense at all. Usually something can't be more than 100% smaller, at which point it is completely gone.

This sounds like the same kind of error as writing "0.10 cents" because it's less than a dollar when the number is in dollars regardless of how big or small it is

Yes, 50 percent smaller means half the size. 1 times smaller means size zero, so nine times smaller is impossible.

I completely agree and this is a pet peeve of mine so it's nice to be validated :]

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