So the Bing API has limitations of modern computer science as you have described above in addition to the rounding and converting most likely in the API.

Bingo. And not just the Bing API, but everything ever run on a computer has the same limitations. That's why computers typically deal differently with two types of number. The set of integers, in maths, referred to as

**ℤ** are all the whole numbers, anything which can be written without a fractional part (e.g. -19, 0, 120). The set of real numbers,

**ℝ** includes all the whole numbers (i.e. members of

**ℤ**), plus all fractions, plus all the irrational numbers such as π. In maths,

**ℝ** is a superset of

**ℤ**. In a computer they are treated separately.

Integers within a certain range can be stored exactly on computers. Operations which combine integers to form other integers, such as addition and multiplication are always exact, but the results may exceed the storage capacity and overflow. Integer division is allowed, but the fractional part of the result is ignored, so 8/3 = 2. That's just the way it is.

Real numbers can sometimes be stored exactly, e.g. 0.5, because as noted before, that's 1/2 and requires only 1 bit in a binary computer, but oftentimes, real numbers are only approximations in a computer because the exact representation exceeds the 53 bits that you have available. Again, just the way it is.

Computers typically have a dedicated processor (either on a separate chip or integrated into the CPU chip) just to deal with floating point numbers, usually called a floating point unit (FPU) or a maths co-processor. The integer stuff is dealt with by the separate arithmetic logic unit (ALU).

In a perfect mathematical world, we wouldn't need to go to all this effort, we'd just need a single processing unit to deal with all numbers and an infinite amount of memory to store these numbers in, but computers are part of the real world, so have to be engineered to work around these issues as well as possible.

But let's get back to Bing and put this in perspective. The real numbers we deal with here, due to the inherent limitations of our computers, probably have capacity of around 16 significant figures (decimal notation), so for a maximum distance possible in our measurements of 20,000km (1/2 way around the world), that means 11 decimal places are available e.g. 20,000.00000000001 (16 in all). The "1" here is 10 nanometres, so we're saying Bing's values are mathematically accurate to 10 nanometres in 20,000 km. To be fair, since we're combining several steps to get the final result and each step can have an inaccuracy, the result may be out by a bit more than 10 nanometres, but that's for an extreme distance. For much shorter distances, the calculated values will be much closer to the theoretical mathematical result.

Microsoft, as we've demonstrated, round their results to the nearest 1cm, meaning that on this scale, the inaccuracies introduced by the computer itself are utterly irrelevant.