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 \(t\ne0,-1\), the elementary decomposition follows by transforming one side:

\(\begin{aligned}\dfrac1t-\dfrac1{t+1}&=\dfrac{(t+1)-t}{t(t+1)}\\&=\dfrac1{t(t+1)}.\end{aligned}\)

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 \(\dfrac12-\dfrac13=\dfrac16\) 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.