Distributive Laws/Arithmetic
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Theorem
On all the number systems:
- natural numbers $\N$
- integers $\Z$
- rational numbers $\Q$
- real numbers $\R$
- complex numbers $\C$
the operation of multiplication is distributive over addition:
- $m \paren {n + p} = m n + m p$
- $\paren {m + n} p = m p + n p$
Proof
This is demonstrated in these pages:
- Natural Number Multiplication Distributes over Addition
- Integer Multiplication Distributes over Addition
- Rational Multiplication Distributes over Addition
- Real Multiplication Distributes over Addition
- Complex Multiplication Distributes over Addition
$\blacksquare$
Also known as
This result is known as the Distributive Property.
As such, it typically refers to the various results contributing towards this.
At elementary-school level, this law is often referred to as (the principle of) multiplying out brackets.
Also see
- Modulo Multiplication Distributes over Modulo Addition
- Matrix Multiplication Distributes over Matrix Addition
- Euclid's proofs:
Beware:
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $1$. Scalar and Vector Quantities
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.1$. Number Systems: $\text{VII}.$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 16$: The Natural Numbers
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): distributive law
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): distributive
- 2008: David Joyner: Adventures in Group Theory (2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if...
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): distributive
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): distributive