Integer Multiplication Distributes over Addition

Theorem

The operation of multiplication on the set of integers $\Z$ is distributive over addition:

$\forall x, y, z \in \Z: x \times \paren {y + z} = \paren {x \times y} + \paren {x \times z}$

$\forall x, y, z \in \Z: \paren {y + z} \times x = \paren {y \times x} + \paren {z \times x}$

Corollary

The operation of multiplication on the set of integers $\Z$ is distributive over subtraction:

$\forall x, y, z \in \Z: x \times \paren {y - z} = \paren {x \times y} - \paren {x \times z}$

$\forall x, y, z \in \Z: \paren {y - z} \times x = \paren {y \times x} - \paren {z \times x}$

Proof

Let us define $\Z$ as in the formal definition of integers.

That is, $\Z$ is an inverse completion of $\N$.

From Natural Numbers form Commutative Semiring, we have that:

All elements of $\N$ are cancellable for addition

Addition and multiplication are commutative and associative on the natural numbers $\N$

Natural number multiplication is distributive over natural number addition.

The result follows from the Extension Theorem for Distributive Operations.

$\blacksquare$

Sources

• 1951: Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ... (previous) ... (next): Introduction $\S 5$: The system of integers
• 1964: W.E. Deskins: Abstract Algebra ... (previous) ... (next): $\S 2.5$: Theorem $2.23: \ \text{(iii)}$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $20$. The Integers
• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $2$: Integers and natural numbers: $\S 2.1$: The integers: $\mathbf Z. \, 5$
• 1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization
• 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras ... (previous) ... (next): $\S 1$