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> decimal numbers are just a subset of fractions

If you're talking about general decimal numbers, this is provably false. Pi, for example, cannot be written as a fraction.

If you're talking about floating-point numbers on a piece of finite hardware, they can all be written as fractions, and, depending on whether you use multiple-precision math, you might be able to represent all the fractions your computer can work with as decimal numbers.

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> Pi, for example, cannot be written as a fraction.

Pi cannot be written as a decimal either.

Decimals _are_ a subset of the rationals. Any rational whose denominator has a prime factor other than 2 or 5 cannot be written using decimals.

I don't think we're agreeing on the basic definitions, and until we do it's pointless to continue.

Under what definition of "decimal" is Pi a decimal number?

When you use it as a very informal synonym for 'real', or as a short form for 'infinite decimal'. (This is even more understandable when you realize every decimal is an infinite decimal, and our convention of truncating an infinite string of zeroes has little mathematical reality.)

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