Product of Negative Numbers

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


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

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

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

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

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

To prove why the quotient of two negative real numbers is a positive real number, just treat either AA or BB 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, 1 month ago

No vote yet
1 vote

  Easy Math Editor

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.
  • Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

MarkdownAppears as
*italics* or _italics_ italics
**bold** or __bold__ bold

- bulleted
- list

  • bulleted
  • list

1. numbered
2. 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 1

paragraph 2

paragraph 1

paragraph 2

[example link]( 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 2×3 2 \times 3
2^{34} 234 2^{34}
a_{i-1} ai1 a_{i-1}
\frac{2}{3} 23 \frac{2}{3}
\sqrt{2} 2 \sqrt{2}
\sum_{i=1}^3 i=13 \sum_{i=1}^3
\sin \theta sinθ \sin \theta
\boxed{123} 123 \boxed{123}


Sort by:

Top Newest

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

Calvin Lin Staff - 6 years, 1 month ago

Log in to reply

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.

Steven Zheng - 6 years, 1 month ago

Log in to reply


Problem Loading...

Note Loading...

Set Loading...