Distributive Laws/Arithmetic

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:

$m \paren {n + p} = m n + m p$

$\paren {m + n} p = m p + n p$

This is demonstrated in these pages:

$\blacksquare$

$\ds 2 \times \paren {3 + 6}$

$=$

$\ds 2 \times 9$

$\ds $

$=$

$\ds 18$

$\ds $

$=$

$\ds 6 + 12$

$\ds $

$=$

$\ds \paren {2 \times 3} + \paren {2 \times 6}$

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.

Beware:

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
1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): multiplication
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
2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): multiplication
2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): distributive