Rational Identities and Exact Arithmetic

For a rational identity, first exclude the zeros of every original denominator. Factor denominators and combine fractions over a common denominator; cancel factors only after multiplication has been made explicit. For t 0,-1, the elementary decomposition follows by transforming one side: 1t- 1t+1= (t+1)-tt(t+1) = 1t(t+1). The numerator is one because the complete second numerator is subtracted. Brackets prevent the minus sign from being applied to only one term. Substituting t=2 checks 12- 13= 16 exactly. This checks one case; the algebra above proves the identity for every permitted value. An expression can agree with a simpler expression wherever the original is defined and still have a different domain. Always carry the original exclusions into the final statement.