People think floats are too magical. Calling it an approximation is sort of leaning into this. Floats have limited precision like every other fixed size representation of a number.
This is sort of saying that integers are an approximation because int(1.6) + int(2.6) = 5. What do you mean‽ Clearly 1.6 + 2.6 = 4.2 ~= 4!
Floating points can’t perfectly represent 0.1 or 0.2 much like integers can’t represent 1.6. So it is rounded to the nearest representable value. The addition is then performed perfectly accurately using these values and the result is the rounded to the nearest representable value. (Much like integers division is “rounded”). That result happens to not be equal to the nearest representable value to 0.3.
It is definitely a bit surprising. And yes, because of rounding it is technically approximate. But the way people talk about floating point makes it sound like it is some nebulous thing that gives a different result every time or just sort of does the correct thing.
I think specifically, they have amazing precision. But the boundaries just don’t fall perfectly on round numbers we humans would expect. That’s what gets people confused.
Rounding can resolve these problems, or don’t use float if you don’t need to.
The difference is that when you input a specific, precise floating point number, the number that’s stored isn’t what you entered.
When you enter integers and store them in ints, as long as the number is small enough, what’s stored is exactly what you entered.
If you tell your program that the radius of the circle is 0.2 units exactly, it says OK and stores 0.200000000000000011102230246251565404236316680908203125.
Of course everybody knows that there’s a limit to how many digits get stored. If you tried to store Pi, there’s obviously some point where it would have to be cut off. But, in life we’re used to cutting things off at a certain power of 10. When we say Pi is 3.14 the numbers after the 4 are all zero. If we choose 3.14159 instead, it’s the numbers after the 9 that are zero. If we represent these as fractions one is 314/100 the other is 314159/100000. The denominator is always a power of 10.
Since computers are base 2, their denominator is always a power of 2, so there’s a mismatch between the rounded-off numbers we use and the rounded-off numbers the computer uses.
People think floats are too magical. Calling it an approximation is sort of leaning into this. Floats have limited precision like every other fixed size representation of a number.
This is sort of saying that integers are an approximation because
int(1.6) + int(2.6) = 5. What do you mean‽ Clearly1.6 + 2.6 = 4.2 ~= 4!Floating points can’t perfectly represent 0.1 or 0.2 much like integers can’t represent 1.6. So it is rounded to the nearest representable value. The addition is then performed perfectly accurately using these values and the result is the rounded to the nearest representable value. (Much like integers division is “rounded”). That result happens to not be equal to the nearest representable value to 0.3.
It is definitely a bit surprising. And yes, because of rounding it is technically approximate. But the way people talk about floating point makes it sound like it is some nebulous thing that gives a different result every time or just sort of does the correct thing.
I think specifically, they have amazing precision. But the boundaries just don’t fall perfectly on round numbers we humans would expect. That’s what gets people confused.
Rounding can resolve these problems, or don’t use float if you don’t need to.
The difference is that when you input a specific, precise floating point number, the number that’s stored isn’t what you entered.
When you enter integers and store them in
ints, as long as the number is small enough, what’s stored is exactly what you entered.If you tell your program that the radius of the circle is 0.2 units exactly, it says OK and stores
0.200000000000000011102230246251565404236316680908203125.Of course everybody knows that there’s a limit to how many digits get stored. If you tried to store Pi, there’s obviously some point where it would have to be cut off. But, in life we’re used to cutting things off at a certain power of 10. When we say Pi is 3.14 the numbers after the 4 are all zero. If we choose 3.14159 instead, it’s the numbers after the 9 that are zero. If we represent these as fractions one is 314/100 the other is 314159/100000. The denominator is always a power of 10.
Since computers are base 2, their denominator is always a power of 2, so there’s a mismatch between the rounded-off numbers we use and the rounded-off numbers the computer uses.