Question

why is the product of a negative integer and a positive integer a negative integer? I know basic math skills and I know that (-)x(+)=(-) but how do I explain to someone why (-)x(+)=(-)?

Answer 1

depends on what you get to assume

Answer 2

like if you were dealing with someone who knew nothing about math whatsoever, how would you teach them and get them to understand that a (-)x(+)=(-) and (-)x(-)=(+)?

Answer 3

the connection between plus and minus (which live in the world of addition) and multiplication always comes from the distributive property

Answer 4

okay so I was able to find a way to explain that, but how would you explain that the product of two negative numbers is positive?

Answer 5

in particular, how would you convince a skeptic that -5x-6=30?

Answer 6

you could look at
\[5\times 6+(-5)\times 6+(-5)\times (-6)\] and prove that \(5\times 6)=(-5)\times (-6)\] it might be more trouble than it is worth, but it is not hard

Answer 7

\[5\times 6+(-5)\times 6+(-5)\times -6)\] can be rewritten as
\[5\times 6+(-5)\times (6+(-6)]\] by the distributive law, and since \(6+(-6)=0\) this is the same as
\[5\times 6\]
at the same time it is
\[5\times (6+(-6)+(-5)\times (-6)=0+(-5)\times (-6)=(-5)\times (-6)\]

Answer 8

i seem to have missed some parentheses there
in any case you get that it is both \(5\times 6\) and \(-5)\times (-6)\) so they are equal`

Answer 9

Think it as walking forward or backward and facing positive or negative direction.

When you just walk forward at positive direction, you just go forward, which is just \((+)(+)=+\)

When you walk backward and facing positive direction, you basically go backward \((-)(+)=-\)

When you walk forward facing negative direction, you also go backward. \((+)(-)=-\)

BUT when you walk backward facing negative direction, you end up moving in positive direction. That's \((-)(-)=+\)

Answer 10

Just explanation in layman's terms.

Answer 11

thank you all so much! so in terms of (-)x(-)=(+), i have this problem that is confusing me...

Look for a pattern: Multiply 3x-6, 2x-6, 1x-6, 0x-6. Based on the pattern you see, what should -1x-6 be? What about -2x-6? -3x-6? -5x-6?

Answer 12

Start with easier pattern.

\(~~~3\times-1=-3\)
\(~~~2\times-1=-2\)
\(~~~1\times-1=-1\)
\(~~~0\times-1=0\)
\(-1\times-1=1\)
\(-2\times-1=2\)
\(-3\times-1=3\)

Answer 13

oh duh! I completely overlooked that and was just overwhelmed with all the numbers!

Answer 14

Another explanation is when I say "Eat!" I am encouraging you to eat (positive)

But when I say "Do not eat!" I am saying the opposite (negative).

Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).

Answer 15

thank you so much! :) great explanation!!

Answer 16

Welcome!

Answer 17

We can understand that negative times negative is hard concept to grasp.

Answer 18

*