Jackson, a 17 year old male from Imphal asks on April 26, 2011,

In my algebra, I learn that (-3 X -2) = +6 but (-3 X +2) = -6, why?

viewed 7694 times

There are lots of ways to answer this. Here is one:

If you start with a number like say 0, and then you add -3 and then add -3 again, then you have added -3 twice, or in other words 2 times -3. And where are you? You are at -6. So this is why 2 x -3 = -6.

Then, if you want to know about something like -2 x -3, you can think of -2 as -1 x 2, and so you have -1 x 2 x -3. The last part, 2 x -3, we already figured out is -6. So you have -1 x -6, which is +6.

And here is a somewhat fancier way to think about it:

You can think of addition and multiplication as "doing something" to the number line. For instance adding 4 slides the number line to the right by 4 units. If you started at 6, then after you add 4, you are at 10. If you started at -5, you will end up at -1.

(Likewise adding -4 slides to the left instead of the right.)

Multiplication, on the other hand, stretches the line out as if it's anchored at 0. Imagine the number line is elastic but pinned at 0. Multiplying by 2 stretches by a factor of 2: 1 goes to 2, 15 goes to 30, -12 goes to -24, 0 goes to 0, etc. The farther you are away from 0, the more you get moved by multiplication (unlike addition, where every number moves the same amount).

Now, multiplying by -1 is a little different: it flips the line around. (Still anchored at 0.) So 20 goes to -20, -20 goes to 20, etc.

Now, if you take a number and multiply by -3, that is like multiplying by 3 and also by -1, so you stretch by a factor of 3 and then flip the line. (Or flip first and then stretch -- the order doesn't matter.)

If you start at 2, you will end up at -6. If you start at -2, you will end up at +6. So -3 x 2 = -6, and -3 x -2 = 6.

Note: All submissions are moderated prior to posting.

If you found this answer useful, please consider making a small donation to science.ca.

- Canadian Nuclear Safety Commission Educational Resources
- National Inventors Hall of Fame
- JUMP Math
- Society for Canadian Women in Science and Technology
- SciQuest e-Solutions for Science
- Natural Sciences and Engineering Research Council of Canada
- Manning Awards for Innovation
- Royal Society of Canada
- Geological Survey of Canada
- Canadian Networks of Centres of Excellence
- Canadian Landscapes at Natural Resources Canada
- Canadian Association of Physicists
- A Century of Innovation
- Understanding Science
- AlphaGalileo
- National Film Board of Canada Youth Science
- PICS Climate Insights 101
- Canadian Association for Girls in Science
- Virtual Library for the History of Science
- The Chemical Institute of Canada
- Canadian Biotechnologist 2.0
- ISI Highly Cited Scientists
- Deep River Science Academy
- Journal of the History of Canadian Science
- Wikipedia
- Innovation Canada
- Mars Society
- Nobel Prize Archive
- science.gc.ca
- Online Science & Engineering Encyclopedia
- CurioCity
- Canadian Nuclear FAQ
- Association of Science Communicators
- Astrofiles
- Wilderness Astronomy