Numbers limit on digits?

HI Ian,

It has to do with floating point binary, and how many bits are available.

Until recently, most 'personal' computer systems used 64 bit binary numbers for their calculations, which limited them to about 19 places precision for the 'same' number expressed in base 10. Apple set the maximum display precision at 15 significant digits, with the last of those rounded up or down depending on the (undisplayed) value of the 16th.

Apple (and others) recently upped their math engine to 128 bit calculations, which is, I think, mentioned in the promotional bumph for the most recent version of Numbers, but the effects of that may not be seen in machines out the door prior to that change.

That's the gist of it. A search for articles on 'floating point binary' might provide the details.

Regards,

Barry

" I thought that integers converted exactly into binary…"

You thought correctly. Integer values in base ten can be converted into exactly the same integer values expressed in binary. The only limitation is the number of 'fingers' the computer has to count on.

With a single 'finger', a computer (or you) can can count from zero to 1 ( 2^1 - 1)

With two fingers, you can count to 3: 00, 01, 10, 11 ( 2^2 - 1)

With four fingers, you can count to 15: 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101. 1111 (2^3 - 1)

Etc.

And all of those translations from base 10 integers to binary integers are exact, Until the computer runs out of fingers. Beyond that limit, the computer cannot express any larger integers. (Well, it can, but it requires a shift in the stating point,)

"…and the inaccuracy was only with fractional numbers."

The accuracy is fine. It's the precision that has limitations.

But that's true of base 10 as well. Here's a simple exercise for you.

Using pencil and paper, calculate the exact base 10 expression of the common fraction 3/7.

Have fun!

Regards,

Barry

PS: "There is a limit of 15 digits in Numbers"

That's the display limit. Internally, the calculations are carried out to a few more places than that, then rounded to fifteen places for display.

B.

When I entered the numbers you entered I noticed something:

just typing in 15 digits produces a cell entry Numbers thinks is a number. however, it seems that entering 16 digits produces a cell entry numbers thinks is text.

I also think you are hitting the limit of resolution for the value stored in the cell:

And to add to confusion (or precision?) in the discussion.

"Degree of Accuracy is how precise a measurement is, often shown as the number of decimal places or significant digits."

https://www.mathsisfun.com/accuracy-precision.html

SG

My favourite examples are a couple from the time when Canada was in the process of converting from Imperial measure to metric.

Alberta Motor Association published newsletter that included the distance from Edmonton to Calgary in miles and in kilometres. The miles figure was the one that had been available for ages in distance tables on their roadmaps: 187miles.

The distance in Kilometres, they said, is 300.947 km

I sent them a letter.

The other was a sign at the entrance to the underground parking beneath the central library in Edmonton:

Low clearance

    3.556m

Both are 'accurate' translations of the Imperial distances, and both imperial unit measurements were likely accurate within the precision intended, but neither translation accounts for the precision of the first being to the nearest mile (not to less than 1/3 the length of a 'standard' automobile) and of the second being to the nearest inch—(preferably to the inch less than the measured height (not to the thickness of a dime).

The clearance sign lasted for several years (and may still be unchanged—I've not been near that location recently. The AMA corrected theirs in the next publication.

Regards,

Barry

Similar, but slightly different, results in Excel. Here I entered 1234567890123456 in each.

Excel defaults to Scientific.

SG

Ian, strange behaviour indeed.

Data, formatted as number, seem to be limited to 15 digits, the 16th digit being used to round the 15th digit.

When I enter 16 digits (cell format = Automatic) the entered "number (?)" moves to the left: the entered digits appear to be considered as text!?

Paul.

The accuracy is fine. It's the precision that has limitations.

And just when I thought I had a more or less competent grasp of the concepts of accuracy and precision!

I had convinced myself that precision is a necessary, but not sufficient, condition for accuracy. Is that not true?

My teachers used to torture me by being still sticklers for significant figures (we called them significant digits).

SG