A Geometric Representation of the Distributive Property

The distributive property of multiplication over addition states that for all real numbers $latex a$, $latex b$ and $latex c$, then

$latex a(b + c) = ab + ac$.

In this short post, we are going to see the visual representation or ‘visual proof’ of this property where it is represented as area. However, one limitation of this representation is it does not represent negative values for $latex a$, $latex b$ or $latex c$. This means that this is only good for positive numbers.

A Geometric Representation of  the Distributive Property

In the diagram below, two rectangles (red and blue) are placed adjacently. The have the same height $latex a$. The red rectangle has width $latex b$ while the blue rectangle  has width $latex c$.  Continue reading

The Product of Negative Number and a Positive Number is Negative

This is the third part of the Basic Algebra Theorems Proof Series. In this post, we use the Axioms of Real Numbers to show that the product of two a negative number and positive number is negative. That is, we show that the product of –a and b is –ab. Please refer to the the preceding link to verify the axioms used below.

Theorem

For any a, b = (-a)b = –ab.

Proof

We know that –ab is a unique solution to the equation xab  = 0, therefore it is sufficient to show that

ab + (-a)b = 0

But

ab + (-a)b = (a + (-a))b

by the  Distributive Property of Real Numbers (Axiom 5A) and 

a + (-a) = 0

by Axiom 5A (the existence of Additive Identity).

Therefore,

ab + (-a)b = (a + (-a))b = 0b = 0

and we are done.

The theorem above give to 2 corollaries.

Corollary 1

For any number b, (-1)b = –b.

If we take a = -1, then (-1)b = – (1b) = –b by the existence of multiplicative identity (Axiom 5M).

Corollary 2

(-1)(-1) = 1

Proof: Left as an exercise.