Home » Blog » Multiplicative identity is its own multiplicative inverse

Multiplicative identity is its own multiplicative inverse

Prove that 1^{-1} = 1.


Proof. On the one hand since 1 is the multiplicative identity we have,

    \[   1 \cdot 1^{-1} = 1^{-1}.   \]

On the other hand, from Theorem I.10 (Exercise I.3.3, #1 part (d), we have (1^{-1})^{-1} = 1. Hence,

    \[   1^{-1} \cdot (1^{-1})^{-1} = 1 \qquad \implies \qquad 1^{-1} \cdot 1 = 1. \]

Therefore, since 1 \cdot 1^{-1} = 1^{-1} \cdot 1,

    \[   1 = 1^{-1}.   \qquad \blacksqaure\]

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):