Proposition: Multiplication of Real Numbers Is Cancellative

The multiplication of real numbers is cancellative, i.e. for all real numbers \(x,y,z\in\mathbb R\), with \(z\neq 0\) the following laws (both) are fulfilled1:

Conversely, the equation \(x=y\) implies

for all \(x,y,z\in\mathbb R\) with \(z\neq 0\).

Proofs: 1

  1. Proposition: Contraposition of Cancellative Law of for Multiplying Real Numbers

Proofs: 1 2
Propositions: 3 4
Sections: 5

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  1. Note that this proposition would be obviously wrong if we allow \(z\) to equal \(0\), e.g. for \(z=0, x=\sqrt 2, y=\sqrt 3\) we would get \(0\cdot \sqrt 2=0\cdot \sqrt 3\), but \(\sqrt 2\neq \sqrt 3\).