Natural Number Multiplication Distributes over Addition/Proof 1
Jump to navigation
Jump to search
Theorem
The operation of multiplication is distributive over addition on the set of natural numbers $\N$:
- $\forall x, y, z \in \N:$
- $\paren {x + y} \times z = \paren {x \times z} + \paren {y \times z}$
- $z \times \paren {x + y} = \paren {z \times x} + \paren {z \times y}$
Proof
\(\ds \paren {x + y} \times z\) | \(=\) | \(\ds +^z \paren {x + y}\) | Definition of Natural Number Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {+^z x} + \paren {+^z y}\) | Power of Product of Commuting Elements in Semigroup equals Product of Powers | |||||||||||
\(\ds \) | \(=\) | \(\ds x \times z + y \times z\) |
$\Box$
\(\ds z \times \paren {x + y}\) | \(=\) | \(\ds +^{x + y} z\) | Definition of Natural Number Multiplication | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {+^x z} + \paren {+^y z}\) | Index Laws for Semigroup: Sum of Indices | |||||||||||
\(\ds \) | \(=\) | \(\ds z \times x + z \times y\) |
$\blacksquare$
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers: Theorem $16.9$