# Question related to IntegerPart

I am having trouble understanding the following result:

Through[{IntegerPart,FractionalPart}[{100 4.02}]]

{{401}, {1.}}


Shouldn't it read {{402}, {0.}}?

I understand that in the Documentation one reads

"IntegerPart[x]+FractionalPart[x] is always exactly x. "

which is actually the case here, but one also reads:

"IntegerPart[x] in effect takes all digits to the left of the decimal point and drops the others. "

(emphasis added) which isn't the case here.

I was using IntegerPart[100#[[1]]]& as a second argument in GatherBy over a long list of entries where the first entry was a real number like 4.02 above.

It is actually the case that modifying the example produces the following

Through[{IntegerPart,FractionalPart}[{100 4.021}]]

{{402}, {0.1}}


Is this bug material or should this behavior be expected?

• Have look at FullForm[100 4.02] and FullForm[Through[{IntegerPart, FractionalPart}[{100 4.02}]]]. The number 0.02 cannot be represented exactly in machine precision. – Henrik Schumacher Apr 14 '18 at 9:23
• thanks for the hint! got it! – user42582 Apr 14 '18 at 9:27
• You're welcome! – Henrik Schumacher Apr 14 '18 at 9:29
• However, Through[{IntegerPart, FractionalPart}[{Rationalize@(100 4.02)}]] returns what you are expecting. I assume that Rationalize converts 0.02 to the rational number that when multiplied by 100 is an integer. – José Antonio Díaz Navas Apr 14 '18 at 13:48

As was kindly suggested by @Henrik Schumacher, 100 4.02 is not the same thing as 402, which is what I was assuming IntegerPart was seeing as input.
Taking FullForm[100 4.02] reveals that the input to IntegerPart evaluates to 401.99999999999994', which in turn-as should be expected-evaluates to {{401}, {1.}} by Through[{IntegerPart,FractionalPart}[{100 4.02}]].
So, the answer is, "no, the result is not a bug; instead IntegerPart operates exactly as it is expected to in the documentation. What appears as a discrepancy is due to the failure to realize how not all numbers can be represented exactly."