Why does division by zero in IEEE754 standard results in Infinite value?

It's a nonsense from the mathematical perspective.

Yes. No. Sort of.

The thing is: Floating-point numbers are approximations. You want to use a wide range of exponents and a limited number of digits and get results which are not completely wrong. :)

The idea behind IEEE-754 is that every operation could trigger "traps" which indicate possible problems. They are

• Illegal (senseless operation like sqrt of negative number)
• Overflow (too big)
• Underflow (too small)
• Division by zero (The thing you do not like)
• Inexact (This operation may give you wrong results because you are losing precision)

Now many people like scientists and engineers do not want to be bothered with writing trap routines. So Kahan, the inventor of IEEE-754, decided that every operation should also return a sensible default value if no trap routines exist.

They are

• NaN for illegal values
• signed infinities for Overflow
• signed zeroes for Underflow
• NaN for indeterminate results (0/0) and infinities for (x/0 x != 0)
• normal operation result for Inexact

The thing is that in 99% of all cases zeroes are caused by underflow and therefore in 99% of all times Infinity is "correct" even if wrong from a mathematical perspective.

I'm not sure why you would believe this to be nonsense.

The simplistic definition of a / b, at least for non-zero b, is the unique number of bs that has to be subtracted from a before you get to zero.

Expanding that to the case where b can be zero, the number that has to be subtracted from any non-zero number to get to zero is indeed infinite, because you'll never get to zero.

Another way to look at it is to talk in terms of limits. As a positive number n approaches zero, the expression 1 / n approaches "infinity". You'll notice I've quoted that word because I'm a firm believer in not propagating the delusion that infinity is actually a concrete number :-)

NaN is reserved for situations where the number cannot be represented (even approximately) by any other value (including the infinities), it is considered distinct from all those other values.

For example, 0 / 0 (using our simplistic definition above) can have any amount of bs subtracted from a to reach 0. Hence the result is indeterminate - it could be 1, 7, 42, 3.14159 or any other value.

Similarly things like the square root of a negative number, which has no value in the real plane used by IEEE754 (you have to go to the complex plane for that), cannot be represented.

In mathematics, division by zero is undefined because zero has no sign, therefore two results are equally possible, and exclusive: negative infinity or positive infinity (but not both).

In (most) computing, 0.0 has a sign. Therefore we know what direction we are approaching from, and what sign infinity would have. This is especially true when 0.0 represents a non-zero value too small to be expressed by the system, as it frequently the case.

The only time NaN would be appropriate is if the system knows with certainty that the denominator is truly, exactly zero. And it can't unless there is a special way to designate that, which would add overhead.