Why can't decimal numbers be represented exactly in binary?

Question

There have been several questions posted to SO about floating-point representation. For example, the decimal number 0.1 doesn\'t have an exact binary representation, so it\'

Answer 1

It's the same reason you cannot represent 1/3 exactly in base 10, you need to say 0.33333(3). In binary it is the same type of problem but just occurs for different set of numbers.

Answer 2

I'm surprised no one has stated this yet: use continued fractions. Any rational number can be represented finitely in binary this way.

Some examples:

1/3 (0.3333...)

0; 3

5/9 (0.5555...)

0; 1, 1, 4

10/43 (0.232558139534883720930...)

0; 4, 3, 3

9093/18478 (0.49209871198181621387596060179673...)

0; 2, 31, 7, 8, 5

From here, there are a variety of known ways to store a sequence of integers in memory.

In addition to storing your number with perfect accuracy, continued fractions also have some other benefits, such as best rational approximation. If you decide to terminate the sequence of numbers in a continued fraction early, the remaining digits (when recombined to a fraction) will give you the best possible fraction. This is how approximations to pi are found:

Pi's continued fraction:

3; 7, 15, 1, 292 ...

Terminating the sequence at 1, this gives the fraction:

355/113

which is an excellent rational approximation.

Answer 3

If you make a big enough number with floating point (as it can do exponents), then you'll end up with inexactness in front of the decimal point, too. So I don't think your question is entirely valid because the premise is wrong; it's not the case that shifting by 10 will always create more precision, because at some point the floating point number will have to use exponents to represent the largeness of the number and will lose some precision that way as well.

Answer 4

The problem is that you do not really know whether the number actually is exactly 61.0 . Consider this:

float a = 60;
float b = 0.1;
float c = a + b * 10;

What is the value of c? It is not exactly 61, because b is not really .1 because .1 does not have an exact binary representation.

Answer 5

This is a good question.

All your question is based on "how do we represent a number?"

ALL the numbers can be represented with decimal representation or with binary (2's complement) representation. All of them !!

BUT some (most of them) require infinite number of elements ("0" or "1" for the binary position, or "0", "1" to "9" for the decimal representation).

Like 1/3 in decimal representation (1/3 = 0.3333333... <- with an infinite number of "3")

Like 0.1 in binary ( 0.1 = 0.00011001100110011.... <- with an infinite number of "0011")

Everything is in that concept. Since your computer can only consider finite set of digits (decimal or binary), only some numbers can be exactly represented in your computer...

And as said Jon, 3 is a prime number which isn't a factor of 10, so 1/3 cannot be represented with a finite number of elements in base 10.

Even with arithmetic with arbitrary precision, the numbering position system in base 2 is not able to fully describe 6.1, although it can represent 61.

For 6.1, we must use another representation (like decimal representation, or IEEE 854 that allows base 2 or base 10 for the representation of floating-point values)

Answer 6

The root (mathematical) reason is that when you are dealing with integers, they are countably infinite.

Which means, even though there are an infinite amount of them, we could "count out" all of the items in the sequence, without skipping any. That means if we want to get the item in the 610000000000000th position in the list, we can figure it out via a formula.

However, real numbers are uncountably infinite. You can't say "give me the real number at position 610000000000000" and get back an answer. The reason is because, even between 0 and 1, there are an infinite number of values, when you are considering floating-point values. The same holds true for any two floating point numbers.

More info:

http://en.wikipedia.org/wiki/Countable_set

http://en.wikipedia.org/wiki/Uncountable_set

Update: My apologies, I appear to have misinterpreted the question. My response is about why we cannot represent every real value, I hadn't realized that floating point was automatically classified as rational.