# Explanation about product of two negative numbers being positive. [duplicate]

I have been having a struggle finding an explanation why $-3 \cdot (-3) = 9$. Why does this question equal a positive number? Any explanations?

And btw, if $-3 \cdot (-3) = 9$. Why does $-3 + (-3) + (-3) = -9$ and not $9$?

• When I was young I remembered by thinking: If you think of a negative number as the opposite of a positive number, then multiplying two negatives gives you the opposite of the opposite of their product. Hence a positive number. – graydad Sep 10 '15 at 13:46
• Here is a set of equations which may help a bit $0=(-3)\times 0=(-3)\times (3-3)=-3\times (3+(-3))=(-3)\times 3+(-3)\times (-3)$ – Mark Bennet Sep 10 '15 at 14:25

The best intuitive explanation I came across is to think of $x=vt$ where $x$ is the displacement, $v$ is the velocity and $t$ is the time. Now suppose you are moving at a velocity of 3 m/sec backwards (hence $v=-3$) and you want to calculate where you were 3 seconds ago ($t=-3$).

$3\times-5=-15$

$2\times-5=-10$

$1\times-5=-5$

$0\times-5=0$

$-1\times-5=5$

$-2\times-5=10$

See how they form a pattern? It's because subtracting a negative number gives you a positive, and having a negative amount of negative numbers means your total is positive.

Sure, this isn't rigorous, but hopefully you can get some intuition.

$$(-3)+ (-3)+ (-3)=(-3)\times 3=-9$$ $$(-3)+ (-3)+ (-3)\neq(-3)\times(-3)$$

And your main doubt, as to why negative times negative is positive,

Let us assume $$(-3)\times (-3)=-9$$. We also know that $$(-3) \times 3=-9$$. Hence $$(-3) \times(-3)=(-3)\times 3$$, or $$-3=3$$, which is a problem.

There is no fundamental rule why negative times negative is positive. It is just that all our operators must be 'consistent'. Once we determine basic rules of addition of negative numbers, it becomes necessary to adopt specific rules for multiplication.

If we decided to adopt a system where negative times negative is negative, then the multiplication operator (and some other operators as well) will have a different meaning. It will no longer signify repeated addition. However, there is nothing wrong in doing so.

• I am pretty certain that you meant $(-3) + (-3) + (-3) = (-3)\times 3=-9$ – steven gregory Sep 10 '15 at 14:28
• @steven Yes, I did; careless of me.. – ghosts_in_the_code Sep 10 '15 at 14:31

The definition of $3\times 2$ is to sum the number 3 exactly 2 times, so

$$3\times 2 = 0 + 3 + 3 = 6$$ no sweat. Notice that I added a $0$ in the front, just to emphasize the sign of the first $3$. If you have $(-3)\times 2$, it's the same thing: sum the number $-3$ exactly two times. Now, the question is, what does it mean to sum a negative number? If you think about money, a negative number is an expense, while a positive is an income. To add an expense to your budget is the same as spending money, so it is the same as subtracting that amount (without the minus). Hence

$$(-3)\times 2 = 0 + (-3) + (-3) = -3 -3 = -6$$ So far so good. Now we get to the troubling part. What does it mean to compute $(-3)\times (-2)$? Well, if multiplying $-3$ by a positive number, say $2$, means to add the number $-3$ two times, it makes sense to think about multiplying $3$ by a negative number, sau $-2$, as subtracting the number $-3$ two times. Therefore,

$$(-3)\times (-2) = 0 - (-3) - (-3).$$

But we just shifted the question: what does it mean to subtract a negative number? Well, thinking again in terms of your money budget, if you remove an expense (i.e., subtract a negative number), you're effectively increasing your budget. As they say: money saved is money earned. Hence

$$(-3)\times (-2) = 0 - (-3) - (-3) = 0 + 3 + 3 = 6$$

Hope this helps. When thinking about negative numbers, examples with money always helped me.