Why is approximately equal not a proper kind of sameness in mathematics?
Nudge a number by a tiny amount, then again, then again, and each step looks almost equal to the last while the ends drift far apart. That failure of transitivity is exactly why mathematicians refuse to treat approximately equal as an equivalence relation, the formal version of being the same in some respect.
An equivalence relation is a relation between pairs of elements that satisfies three rules. It is reflexive, since everything relates to itself; symmetric, since if a relates to b then b relates to a; and transitive, since if a relates to b and b to c, then a relates to c. Plain equality of numbers is the simplest case. Others include triangles being similar or congruent, angles sharing a cosine, real numbers with the same absolute value, directed line segments of equal length and direction, and people who share a birthday.
Every such relation carves its set into non-overlapping pieces called equivalence classes. Two elements are equivalent exactly when they land in the same piece, so the birthday relation sorts humanity into one bin per calendar date. A set together with an equivalence relation on it is sometimes called a setoid.
Counterexamples show why each rule matters. Greater than or equal to is reflexive and transitive but not symmetric: 7 is at least 5, yet 5 is not at least 7. Sharing a common factor bigger than 1 is reflexive and symmetric but fails transitivity, since 2 and 6 share one, as do 6 and 3, while 2 and 3 do not. The empty relation passes symmetry and transitivity vacuously but is not reflexive unless the set is empty. Approximate equality can be rescued, though: defining two functions as close near a point when their difference tends to zero there gives a genuine equivalence.
Related ideas sit nearby. A partial order swaps symmetry for antisymmetry, and equality is the only relation that is reflexive, symmetric and antisymmetric at once. Equal things can replace each other anywhere in an algebraic expression, but merely equivalent things cannot; only whole classes stand in for each other. Congruence relations, equivalences that respect an algebraic structure, let mathematicians build quotient structures, and on groups they correspond to normal subgroups.
Source: Equivalence relation
