In case this wasn't clear, the binary values I gave are perfectly exact.
If you let the digits repeat forever, then it is exact. If you truncate
the number at any point, then it is not exact.
I calculated the binary values of 1 / 10 and 1 / 100 using long
division, on paper. I checked the values by doing this in Python:
# 0b0.000110011 should be appoximately 0.1.
# We need to shift left until there is no decimal place.
# Shifting left by one digit is equal to multiplying by 2.
# 0b0.000110011 --> 0b110011 (we shifted 9 digits)
# 0.1 --> 0.1 * 2 ^ 9 = 51.2
# 0b110011 and 51.2 should be almost equivalent.
# Let's check:
>>> 0b110011
51
# Yup, that's pretty close. It is off by 0.2 because
# I truncated the binary after 9 decimal places. If I used
# more decimal places then it would be closer to exact.
As you add more and more digits, you get closer and closer to exact
equality. This is a convergent series. This is also an example of a
mathematical limit. With infinite digits, the equality is exact.
Zak Fallows
On 4/14/13 8:44 PM, Zak wrote:
The Wikipedia page on repeating decimals
<http://en.wikipedia.org/wiki/Repeating_decimal> is relevant to this
discussion. I will use "bracket notation" as defined on that page.
First, let's explain bracket notation:
One third = 1 / 3 = 0.3333... = 0.(3)
One eleventh = 1 / 11 = 0.090909... = 0.(09)
As you can see, the sequence of digits inside parentheses is repeated
forever.
To denote numbers in different bases, I will use Python notation. Thus
"1234" is in base 10, "0x12ab" is in base 16, "0o1234" is in base 8,
and "0b101101" is in base 2. You can use these prefixes (0x, 0o, and
0b) in Python code, did you know that?
Now on to the results:
1 / 10 = 0.1 = 0b0.0(0011) = 0b0.000110011001100110011...
1 / 100 = 0.01 = 0b0.00(00001010001111010111) =
= 0b0.00000010100011110101110000101...
As you can see, one hundredth has a period of 20 digits in base 2.
There is a block of 20 digits that is repeated forever.
Here is a great fraction:
1 / 81 = 0.(0123456789) = 0.01234567890123456789...
Why is 1 / 81 so fantastic? Well, we are in base 10. Base 9 is one
off. 9 squared is 81. That is why 1 / 81 is so fantastic. Base 10 and
base 9 have a very special relationship, look up "casting out nines"
for more mind-blowing but simple math.
Zak Fallows
On 4/14/13 7:30 PM, Algis Kabaila wrote:
Just to add a little fuel to this fire:
First, by hand try to convert the "exact" 0.1 then 0.01 decimal to
their binary equivalent. Send to this list the binary value. Is it exact?
Second, who would care about 1 cent in $10 000 ?
Al.
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