# Distributive Property of Multiplication

Distributive property of multiplication:

huge color(purple)(a times (b+c)=a times b+a times c)

Intuitively, we understand, that it is correct.

Indeed, multiplication is just a shorthand for addition.

If we want to multiply 4 by 5, this means that we want to use 4 in addtion five times.

We can separately add two 4's and three 4's, and then combine the result:

color(red)(4 times (2+3))=4 times 5=4+4+4+4+4=(4+4)+(4+4+4)=color(red)(4 times 2+4 times 3).

Distributive property of multiplication works for negative numbers (in fact, for real numbers) as well.

Example 1. 5 times (3+(-4))=5 times 3+5 times (-4)=-5.

Example 2. 1/2 times (-5+3)=1/2 times (-5)+1/2 times 3=-1.

We can even go in reverse direction.

Example 3. 5/8 times sqrt(2)+5/8 times 2=5/8 times (sqrt(2)+2).