Proof that the last non-zero digit to the right of the decimal point in a non whole number float is always 5?

Question

Is it true that the last non-zero digit to the right of the decimal point in a non whole number float is always 5? Can/has someone prove[ed] this formally? Also, how many 0s in a row to the right of the decimal point are needed to conclude that there are no more non-zero digits farther to the right--I have seen #s such as 1.23450678500000 (an example, probably not a real float), so I know it is at least two.

Answer 1

Is it true that the last non-zero digit to the right of the decimal point in a non whole number float is always 5?

In a binary floating-point format, if a number is not an integer, the last non-zero digit of its decimal numeral is 5.

Generally, a floating-point format represents numbers as some integer M multiplied by power e of a fixed-base b: M•b^e, with various limits on M and e. If b is two, we call this a binary format. Binary floating-point formats are currently the most common. Some platforms provide decimal formats, where b is 10. Other bases are mathematically possible but rarely used, although 16 was somewhat used in the past.

To see the last digit is 5, consider the form M•b^e. The only way this can be a non-integer is for e to be negative. If M is even, we can reduce the form to (M/2)•b^e−1, and, if M/2 is also even, we can reduce further and continue reducing until we have an odd integer multiplied by a negative power of 2. This is equivalent to dividing by a positive power of 2. Then we can see that the last digit is 5:

• If we are dividing by 2¹, the number is .5, 1.5, 2.5, 3.5, and so on.
• If we are dividing by 2², the number is half of one of those, so it is .25, .75, 1.25, 1.75, and so on.
• If we are dividing by any greater power of 2, we are always dividing some number that ends in 5 by 2, and that means we carry the ½ fraction from the position where the 5 is to the next position, producing half of 10 there, and half of 10 is 5.

Alternately, we can see that multiplying a numeral by 2, even repeatedly, can never make a digit position 0 unless it is 0 or 5. Multiplying 1, 2, 3, 4, 6, 7, 8, or 9, will produce 2, 4, 6, 8, 2, 4, 6, and 8, respectively, and multiplying those by 2 produces a digit from the same set. So, if a non-integer multiplied by a power of 2 produces an integer, its last non-zero digit must be 5.

Also, how many 0s in a row to the right of the decimal point are needed to conclude that there are no more non-zero digits farther to the right…

This depends on the specific floating-point format. A high-precision format can represent numbers that are very close to some number .xxxx500 but not equal to it. For any given floating-point format, there must be some limit to the number of zeros there can be before a non-zero digit, but it may require some work to figure it out. If you have concerns about finding the number of digits required to represent a floating-point number accurately, there are other ways to approach it.

Answer 2

Regarding the 2nd question (narrated): >>> v 1.5000005 # OK

>>> print("%f") % (v)
1.500001
**# Maybe not...**

>>> print("%.15f") % (v)
1.500000500000000
**# Ends in 5, so OK now, right?  No! It should've rounded to even.  Huh?**

>>> print("%.16f") % (v)
1.5000005000000001
**# "16 decimal digits is accuracy", I learned in school** 

>>> print("%.31f") % (v)
1.5000005000000000698889834893635
**#NOT SO FAST; but now I found the real value since it ends in 5--see, "I'm smart, and I want respect" (Fraedo)**

>>> print("%.2000f") % (v)
1.50000050000000006988898348936345428228378295898437500000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000
**# I should have listened to mom, and been a Mail Man! (are there yet more lurking non-zero digits???)**