A Negative Times a Negative Is Positive

Why is a negative times a negative positive? It might be best to say that multiplying by –1 rotates the number line by 180°.
Otherwise, starting from a set of axioms, it is possible to derive the usual properties of arithmetic, even including long multiplication and division. The proofs, though simple, are tricky.
This Demonstration shows two proofs by example that for any two positive integers and , , that is, a negative times a negative is a positive. The proofs can be adapted to a proper algebraic proof with letters replacing the variable numbers.
The statements on the right explain the steps taken. Those statements are either axioms or previously proved statements (i.e., theorems). For example, the fact that multiplication by zero is zero is not an axiom; it takes six steps to prove it.
The axioms needed for rational or real numbers are in this table (see the related links):
It is also necessary to talk about closure and to prove the uniqueness of the identities and of inverses, so the discussion here is not complete. The abbreviation is used to avoid some clutter. Division is not used in this Demonstration.



  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Related Curriculum Standards

US Common Core State Standards, Mathematics

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+