# Proposal: 1) Number (integer or decimal) to Array 2) Array to Number (integer or decimal)

Jeremy Martin jmar777 at gmail.com
Mon Mar 11 16:33:01 UTC 2019

```With respect, it's still not clear how you want to interact with the array
of values once you've destructured a Float into your array format.

If all you have is an array of single-digit numbers that represent the
values in the tenths/hundredths/etc. positions of the source number, how
does this actually circumvent the challenges of representing Decimal values
that aren't exactly representable as a Float?

To illustrate this challenge, let's use the classic example we've all seen
hundreds of times:

> .1 + .2
0.30000000000000004

For a long time, all the reading I would do about *why* this produced a
weird result would *sort* of make sense and *sort* of confuse me. That is,
I could understand *why* 3/10ths isn't representable as a Float, but then I
would get confused by the fact that I could type `.3` into a REPL, and it
would *actually work *(??!):

> .3
0.3

I mean, who's lying? How come `.3` works fine when I just type it straight
in, and `.1 + .3` works just fine, but there's just these specific cases
like `.1 + .2` where all of a sudden `.3` decides not to be representable
again?

I admit this is conjecture, but maybe that's part of the confusion
motivating this proposal? And maybe the idea is that if we can break `.1`
and `.2` into some sort of an array structure (e.g., [0, 1] and [0, 2]), then
we can add the individual parts as integers (giving us something like [0,
3]) which we can then just convert back into a single numeric value at the
end as 0.3, and voila, no 0.30000000000000004 shenanigans?

The problem is that this all builds on a fundamental misunderstanding of
what's going. Let's revisit the basic example of entering a value into the
REPL:

> .3
0.3

This, as I stated earlier, contributed greatly to my own hangups in
understanding what was going on here. What I personally didn't understand
was that the `0.3` value you see above isn't actually the *Decimal* value
0.3. It's just a *very* close approximation of 0.3. (so close, in fact,
that 0.3 is the *closest* Decimal value that it can be rounded to, so
that's what gets emitted).

So, going back to our earlier example, why do we get a *different* output
when we're dealing with the result of a mathematical operation, as opposed
to getting the *same* very close approximation of 0.3 that we get when we
simply type it into the REPL?

> .1 + .2
0.30000000000000004

The answer lies in the fact that 0.1 and 0.2 *also* can't be represented
exactly as Floats. Just like we saw with 0.3, we can type them into the
REPL and see a value that looks the exact same being emitted back at us:

> .1
0.1
> .2
0.2

...but also like we saw with 0.3, they only *look* like accurate
representations. Once again, 0.1 and 0.2 are just the closest Decimal
values that the underlying Float values can be rounded to for display.

This rounding behavior, then, is what causes us to get 0.30000000000000004
when we add them together, because the *slight* rounding error with 0.1 and
the *slight* rounding error with 0.2 compound to result in a Float that no
longer rounds closer to 0.3, and instead closer to the "wrong" Decimal
value that we see emitted before.

It's worth noting that this same behavior applies to, e.g., 0.1 + 0.3, even
though that *looks* like it produces the correct result of 0.4. In reality,
however, this is just a case where the rounding errors have the effect of
(almost) canceling each other out, such that the resulting Float rounds
closer to 0.4 than any other value for display purposes (despite being only
negligibly more accurate than our 0.30000000000000004 result was).

Ok, so why am I trying to explain all this? Because I'm trying to
illustrate why it sounds like this proposal doesn't actually solve the
problem that you want it to. Is it possible to standardize a system for
transformations and math operations like the following?

const arg1 = numberToArray(0.1) // [0, 1]
const arg2 = numberToArray(0.2) // [0, 2]

const arraySum = addArrayNumbers(arg1, arg2) // [0, 3]

const result = arrayToNumber(arraySum) // 0.3

Sure, and at the very end, you actually get a value that *looks* right (0.3,
yay!). But *it's still not actually 0.3.* So what become the motivation for
this? You have a solution that, in terms of memory and CPU cycles, is
orders of magnitude more costly to calculate than `0.1 + 0.2` as a plain
JavaScript expression, and in return you get a result that is, *at best*,
infinitesimally more accurate than the alternative when carried all the way
out to the quadrillionths place or greater.

Do you actually have a use case for mathematical operations that are fault
tolerant enough to represent Decimal values as Floats, but that fault
tolerance is sensitive to very, very specific rounding behavior at the
quadrillionths level? I can't even imagine what that use case would be.

On Mon, Mar 11, 2019 at 10:06 AM guest271314 <guest271314 at gmail.com> wrote:

> JS numbers are specified to be in terms of IEEE-754 doubles, so tenths,
>> hundredths, and so on cannot be precisely represented. [1] So there is no
>> way to increase precision here beyond the above that Tab showed, assuming
>> each of those operations are accurate to the bit.
>
>
> Not sure what the message is trying to convey?  The code at the first post
> already overcomes the issue of
>
>     i % 1 // 0.5670000000000073
>
> described by Tab. All of the input numbers are converted to array then
> back to number without losing any precision or adding more numbers to the
> input.
>
> The proposal suggests that each discrete digit of any number a user can
> get and set and be clearly defined with a consistent name, or reference.
> Converting the number to an array is a simple means of processing each
> digit independently.
>
> On Mon, Mar 11, 2019 at 10:41 AM Isiah Meadows <isiahmeadows at gmail.com>
> wrote:
>
>> JS numbers are specified to be in terms of IEEE-754 doubles, so tenths,
>> hundredths, and so on cannot be precisely represented. [1] So there is no
>> way to increase precision here beyond the above that Tab showed, assuming
>> each of those operations are accurate to the bit.
>>
>> [1]:
>> https://www.exploringbinary.com/why-0-point-1-does-not-exist-in-floating-point/
>> On Sun, Mar 10, 2019 at 13:26 guest271314 <guest271314 at gmail.com> wrote:
>>
>>> So this would help with precision?
>>>
>>>
>>> To an appreciable degree, yes, without the scope of JavaScript
>>> floating-point number implementation.
>>>
>>> The gist of the proposal is to formalize, standardize, or whatever term
>>> specification writers want to use, the *naming* of each method or operation
>>> which can get and set each discrete digit of a number - without using
>>> String methods.
>>>
>>> For input
>>>
>>>     1234.567
>>>
>>> Each digit has a formal name which developers can get and set, whether
>>> in an array, object or number format.
>>>
>>> On Sun, Mar 10, 2019 at 5:17 PM Michael Theriot <
>>> michael.lee.theriot at gmail.com> wrote:
>>>
>>>> So this would help with precision?
>>>>
>>>> On Sunday, March 10, 2019, guest271314 <guest271314 at gmail.com> wrote:
>>>>
>>>>> (If you really wanted this as an integer, it's not well-founded; .567
>>>>>> isn't exactly representable as a double, so JS doesn't know that you
>>>>>> "meant" it to have only three digits after the decimal point, and thus
>>>>>> want 567 as the answer. You'll instead get some very very large
>>>>>> integer that *starts with* 567, followed by a bunch of zeros, followed
>>>>>> by some randomish digits at the end.)
>>>>>
>>>>>
>>>>> The code at the first post solves that problem.
>>>>>
>>>>> But the question is still "what would someone use this information
>>>>>> for?"
>>>>>
>>>>>
>>>>> That question has been answered several times in the posts above. This
>>>>> users' motivation was and is the ability to manipulate JavaScript
>>>>> floating-point numbers  (which could be considered "broken", as you
>>>>> described above) in order to solve mathematical problems (in this case,
>>>>> directly calculating the *n*th lexicographic permutation) with the
>>>>> number or decimal being represented as an array, without having to be
>>>>> concerned with not getting the same value when the array is converted back
>>>>> to a number.
>>>>>
>>>>> Felipe Nascimento de Moura mentioned several other applications.
>>>>>
>>>>> The work has already been done. This proposal is essentially to
>>>>> standardize the naming conventions. Whether a Number method is used
>>>>>
>>>>>     i.getTensMethod
>>>>>
>>>>> or an array is used
>>>>>
>>>>>    arr["integer"] // 1234
>>>>>
>>>>> or an object where values are arrays is used
>>>>>
>>>>>     o["fraction"] // .567
>>>>>
>>>>> Having mentioned Intl.NumberFormat earlier in the thread, if the issue
>>>>> devoting resources to a *new *proposal, Intl.NumberFormate can be
>>>>> extended; e.g. a rough draft in code
>>>>>
>>>>>     function formatNumberParts(args) {
>>>>>       return Object.assign({sign:0, fraction:[0], integer:[0]},
>>>>> ...args.filter(({type}) => type === "integer" || type === "fraction" ||
>>>>> type === "minusSign").map(({type, value}) => ({[type === "minusSign" ?
>>>>> "sign" : type]: type !== "minusSign" ? [...value].map(Number) : -1})));
>>>>>     }
>>>>>
>>>>>     let number = -123;
>>>>>
>>>>>     let formatter = new Intl.NumberFormat('en-US');
>>>>>
>>>>>     let res = formatter.formatToParts(number);
>>>>>
>>>>>     formatNumberParts(res);
>>>>>
>>>>> If the concern is that the proposal would not be useful, consider what
>>>>> you would *name* various uses of Math.trunc and remainder operator
>>>>>
>>>>>
>>>>> On Sun, Mar 10, 2019 at 3:58 PM Tab Atkins Jr. <jackalmage at gmail.com>
>>>>> wrote:
>>>>>
>>>>>> On Sat, Mar 9, 2019 at 11:10 AM Felipe Nascimento de Moura
>>>>>> <felipenmoura at gmail.com> wrote:
>>>>>> >
>>>>>> > Personally, I don't think it would be THAT useful...
>>>>>> > but...I think there is something behind this proposal that makes
>>>>>> sense.
>>>>>> >
>>>>>> > I do believe it could be useful for developers to have an easier
>>>>>> > Perhaps something like:
>>>>>> >
>>>>>> > const i = 1234.567;
>>>>>>
>>>>>> Can you provide a scenario in which these would do something useful,
>>>>>> such that it would be worth adding them over just using the math
>>>>>>
>>>>>> > console.log( i.float ); // 567
>>>>>>
>>>>>> i % 1
>>>>>>
>>>>>> (If you really wanted this as an integer, it's not well-founded; .567
>>>>>> isn't exactly representable as a double, so JS doesn't know that you
>>>>>> "meant" it to have only three digits after the decimal point, and thus
>>>>>> want 567 as the answer. You'll instead get some very very large
>>>>>> integer that *starts with* 567, followed by a bunch of zeros, followed
>>>>>> by some randomish digits at the end.)
>>>>>>
>>>>>> > console.log( i.abs ); // 1234
>>>>>>
>>>>>> Math.trunc(i)
>>>>>>
>>>>>> > console.log( i.thousands ); // 1
>>>>>>
>>>>>> Math.trunc(i / 1000)
>>>>>>
>>>>>> > console.log( i.million ); // 0
>>>>>>
>>>>>> Math.trunc(i / 1e6)
>>>>>>
>>>>>> > console.log( i.hundred ); // 2
>>>>>>
>>>>>> Math.trunc(i / 100) % 10
>>>>>>
>>>>>> > console.log( i.hundreds ); // 12
>>>>>>
>>>>>> Math.trunc(i / 100)
>>>>>>
>>>>>> > console.log( i.ten ); // 2
>>>>>>
>>>>>> Math.trunc(i / 10) % 10
>>>>>>
>>>>>> > console.log( i.tens ); // 123
>>>>>>
>>>>>> Math.trunc(i / 10)
>>>>>>
>>>>>> > console.log( i.tenth ); // 5
>>>>>>
>>>>>> Math.trunc(i % 1 * 10) % 10
>>>>>>
>>>>>> > console.log( i.tenths ); // 5
>>>>>>
>>>>>> Math.trunc(i % 1 * 10)
>>>>>>
>>>>>> > console.log( i.hundredth ); // 6
>>>>>>
>>>>>> Math.trunc(i % 1 * 100) % 10
>>>>>>
>>>>>> > console.log( i.hundredths ); // 56
>>>>>>
>>>>>> Math.trunc(i % 1 * 100)
>>>>>>
>>>>>>
>>>>>> Some of these are easy to remember and use; others take some thinking
>>>>>> to deploy. But the question is still "what would someone use this
>>>>>> information for?", such that the benefit to developers is worth the
>>>>>> cost to all parties involved (spec writers, implementors, testers, and
>>>>>> then developers having to navigate a larger stdlib).
>>>>>>
>>>>>> ~TJ
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>>>>>>
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--
Jeremy Martin
661.312.3853