Why not use Double or Float to represent currency?

While it's true that floating point type can represent only approximatively decimal data, it's also true that if one rounds numbers to the necessary precision before presenting them, one obtains the correct result. Usually.

Usually because the double type has a precision less than 16 figures. If you require better precision it's not a suitable type. Also approximations can accumulate.

It must be said that even if you use fixed point arithmetic you still have to round numbers, were it not for the fact that BigInteger and BigDecimal give errors if you obtain periodic decimal numbers. So there is an approximation also here.

For example COBOL, historically used for financial calculations, has a maximum precision of 18 figures. So there is often an implicit rounding.

Concluding, in my opinion the double is unsuitable mostly for its 16 digit precision, which can be insufficient, not because it is approximate.

Consider the following output of the subsequent program. It shows that after rounding double give the same result as BigDecimal up to precision 16.

Precision 14
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BigDecimalNoRound             : 56789.012345 / 1111111111 = Non-terminating decimal expansion; no exact representable decimal result.
DoubleNoRound                 : 56789.012345 / 1111111111 = 5.111011111561101E-5
BigDecimal                    : 56789.012345 / 1111111111 = 0.000051110111115611
Double                        : 56789.012345 / 1111111111 = 0.000051110111115611

Precision 15
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BigDecimalNoRound             : 56789.012345 / 1111111111 = Non-terminating decimal expansion; no exact representable decimal result.
DoubleNoRound                 : 56789.012345 / 1111111111 = 5.111011111561101E-5
BigDecimal                    : 56789.012345 / 1111111111 = 0.0000511101111156110
Double                        : 56789.012345 / 1111111111 = 0.000