# Product of Negative Numbers

Prove why the product of two negative real numbers is a positive real number.

Solution

Let $A$ and $B$ be two positive real numbers. Subsequently, $-A$ and $-B$ are the respective additive inverses.

$-A \cdot -B+ -A\cdot B = -A \cdot (-B+B)$ $-A \cdot (-B+B) = 0$

We then add $A \cdot B$ to both sides of the equation:

$-A \cdot -B+ -A \cdot B+A \cdot B = 0+A \cdot B$ $-A \cdot -B+(-A + A) \cdot B = 0+A \cdot B$

which simplifies to $-A \cdot -B = A \cdot B.$

To prove why the quotient of two negative real numbers is a positive real number, just treat either $A$ or $B$ as the multiplicative inverse (division is the multiplication of reciprocals).

Check out my other notes at Proof, Disproof, and Derivation

Note by Steven Zheng
6 years, 11 months ago

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

• Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
• Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
• Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold
- bulleted- list
• bulleted
• list
1. numbered2. list
1. numbered
2. list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1paragraph 2

paragraph 1

paragraph 2

[example link](https://brilliant.org)example link
> This is a quote
This is a quote
    # I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
# I indented these lines
# 4 spaces, and now they show
# up as a code block.

print "hello world"
MathAppears as
Remember to wrap math in $$ ... $$ or $ ... $ to ensure proper formatting.
2 \times 3 $2 \times 3$
2^{34} $2^{34}$
a_{i-1} $a_{i-1}$
\frac{2}{3} $\frac{2}{3}$
\sqrt{2} $\sqrt{2}$
\sum_{i=1}^3 $\sum_{i=1}^3$
\sin \theta $\sin \theta$
\boxed{123} $\boxed{123}$

Sort by:

You should include the definition of a positive number, and explain why the product of 2 positive numbers is also positive.

Staff - 6 years, 11 months ago

This problem assumes that you only know that the product of two positive reals is a positive real, and that there exists an additive inverse element for every real number. Maybe I should include that, but I assume the reader knows algebra. They just have to prove why this is so.

- 6 years, 11 months ago