Speaking from experience, if you exclusively deal with integers internally and then apply the appropriate formatting when outputting to the user (which you need to deal with regardless of internal representation), it makes dealing with monetary data so much easier. Integers are generally better supported, faster, and more convenient for most languages, databases, and serialization formats.
Floating point (up to a certain amount of absurdity) you can describe with integers (value, scale).
Real numbers you can describe with integers (numerator, denominator, scale).
Irrationals are more difficult. That's why we try not to use them :). Hopefully your company's commission tiers are not defined in terms of circles or euclidean distances.
Accounting rules often care more about consistency than precision per se. As long as the code follows accepted accounting rules (say bankers rounding) choosing a known precision works. Often GAAP and other standards don’t define all details for, say amortization, but if you use a scheme the code should always use that method.
