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One of the pieces is actually 0.33333....4
Technically no
0.3333.... repeats infinitely. The 0.333...4 is not an infinitely repeating number. And since 0.333... is, there's no room to add that 4 anywhere
Which is why adding them up you get 0.999..... which is exactly and completely equal to 1
If you cut perfectly, which is impossible because you won't count or split atoms (and there is a smallest possible indivisible size). Each slice is a repeating decimal 0.333... or in other words infinitely many 3s. (i don't know math well that's just what i remember from somewhere)
If the number of atoms is a multiple of 3, then you can split it perfectly.
For example say there’s 6 atoms in a cake, and there’s 3 people that want cake. Each person gets 2 atoms which is one third of the cake.
But if the cake has 7 atoms, better get cover on a nuclear bunker just to be safe.
The main problem is simply that math is "perfect" and reality isn't. Since math is an abstract description of causality while reality doesn't/can't really "do" infinity.
But if you really wanted to, you could bake a cake in a lab with a predetermined number of atoms and then split that cake into 3 perfect slices. However, once you start counting multiples(like atoms in a cake) you would no longer get 1/3 or 0.3 because you are now dividing a number bigger than 1(the number of atoms) so you would't get a fraction(0.3) You would get a whole number.