another stupid question (math)

[shhhnake_eyes]

yea im pretty stupid but
how come a negative number times a negative is a positive?

looking at this from a stupid standpoint, as i do most things: 7 * 9 = 63 but 8*8= 64.. 7 is one less than 8 and 9 is one more than 8.. i know it has to do with 1*1=1 but dont completely understand why this works

[MidGe]

3 * 3 = 3 + 3 + 3
3 * -3 = (-3) + (-3) + (-3)
- 3 * - 3 = -( 3 * -3) = -( (-3) + (-3) + (-3) )

[hmkpoker]

You drink five beers an hour for five hours. You have drunk 25 beers.

You decide you need to purge those beers, and have amazing vomit control, vomitting exactly one beer with each purge. You throw up five beers each hour for five hours. Now you have zero beers.

Now you're sober, but drat, there's no more beer. And the party's going to end soon, whatever shall we do! Luckily the bucket into which you were vomitting is still mostly beer. It also happens that you have a magical, foolproof way to make time run backward! Yes! So after you blow some special K, time is going in the reverse direction, and now you can simply proceed with drinking your vomit back at the rate of five beers an hour for five hours, and you suddenly have twenty-five beers again!

Get it?

[hmkpoker]

[ QUOTE ]
3 * 3 = 3 + 3 + 3
3 * -3 = (-3) + (-3) + (-3)
- 3 * - 3 = -( 3 * -3) = -( (-3) + (-3) + (-3) )

[/ QUOTE ]

Or you could use some stupid equations. Whatever.

[Windchasers]

[ QUOTE ]
...and you suddenly have twenty-five beers again!


[/ QUOTE ]

That's sick. /images/graemlins/grin.gif

[madnak]

I'm not sure I understand your example.

But.

7*9 is one less than 8*8 for a good reason. 7*9 is just 7*8 plus another 7. 8*8 is just 7*8 plus another 8. So if 7*8 is "x", then 7*9 = x+7. And 8*8 = x+8. x+8-x+7=1. So 8*8 is going to be 1 more than 7*9.

A proof might look like:

(n-1)(n+1) = n^2 - 1
n^2 + n - n - 1 = n^2 - 1
n - n - 1 = -1
-1 = -1
1 = 1

[madnak]

Think of "negative" as meaning "the opposite of."

Now, say the opposite of cold is hot. And the opposite of hot is cold.

So -hot = cold. And -cold = hot. Now, what is the opposite of the opposite of cold? We know that -cold=hot, so -(-cold)=-hot. And we know that -hot=cold. So -(-cold)=cold.

In other words, "the opposite of the opposite of cold is cold."

[infidel323]

[ QUOTE ]
You drink five beers an hour for five hours. You have drunk 25 beers.

You decide you need to purge those beers, and have amazing vomit control, vomitting exactly one beer with each purge. You throw up five beers each hour for five hours. Now you have zero beers.

Now you're sober, but drat, there's no more beer. And the party's going to end soon, whatever shall we do! Luckily the bucket into which you were vomitting is still mostly beer. It also happens that you have a magical, foolproof way to make time run backward! Yes! So after you blow some special K, time is going in the reverse direction, and now you can simply proceed with drinking your vomit back at the rate of five beers an hour for five hours, and you suddenly have twenty-five beers again!

Get it?

[/ QUOTE ]

How do you figure the probability that your friends will help you drink the 25 beers the second time around?

[BruceZ]

[ QUOTE ]
yea im pretty stupid but
how come a negative number times a negative is a positive?

[/ QUOTE ]

Proof that (-a)*(-b) = ab for a,b positive or negative real numbers:

(-a)*(-b) = (-1)*a*(-b) =

-[a*(-b)] =

-[a*(-1)*b] =

-[-ab] =

ab

I have used only the definition of -x = -1*x, the associative property of multiplication, and the commutative property of multiplication. Actually, if you believe that pos*neg = neg, then it's pretty much over at the second step, since -[neg] = pos.

[holmansf]

-x = -1 * x is not a definition.

Here's a proof:

1 + (-1) = 0 (Definition of -1 as additive inverse of 1)

x * (1 + (-1)) = 0 (for any x since x*0 = 0)

x + (-1) * x = 0 (Distributive Law, definition of 1)

(-1) * x = -x (adding -x to both sides)

[Sharkey]

[ QUOTE ]
...7 * 9 = 63 but 8*8= 64.. 7 is one less than 8 and 9 is one more than 8....

[/ QUOTE ]

8 is greater than 7 by more (proportionally speaking) than 9 is greater than 8.

[BruceZ]

[ QUOTE ]
-x = -1 * x is not a definition.

Here's a proof:

1 + (-1) = 0 (Definition of -1 as additive inverse of 1)

x * (1 + (-1)) = 0 (for any x since x*0 = 0)

x + (-1) * x = 0 (Distributive Law, definition of 1)

(-1) * x = -x (adding -x to both sides)

[/ QUOTE ]

Amazing, I already had exactly that proof on my paper where I went back later and "derived" this before I saw yours. In fact, mine looks exactly like this:

a*(-1 + 1) = 0

-1*a + a = 0

-1*a = -a


I chose to take this as a given before, which is what I meant by declaring it true by my own non-standard "definition" (of multiplication by -1).

Here is a combined proof of (-a)*(-b) = ab which incorporates this idea:

-1*(-1 + 1) = 0

-1 * -1 - 1 = 0

-1 * -1 = 1

-1 * a * -1 * b = ab

(-a)*(-b) = ab


Here is the shortest proof that I came up with, but you have to believe that (-a)*b = -ab:

-a*(-b + b) = 0

(-a)*(-b) - ab = 0

(-a)*(-b) = ab

[Meromorphic]

I have nothing to add from a mathematics standpoint. For a cognitive perspective, see the book Where Mathematics Comes From by Lakoff and Nunez. I'm too lazy to look up the chapter, let alone page numbers, but you can find it pretty easily if you're interested.

[Mik1w]

if you take 5 apples from a shelf every hour, then after 4 hours you'll have taken 20 apples.

if you put 5 apples on a shelf every hour, then after 4 hours you'll have take -20 apples

if you put 5 apples on a shelf every 20 hours, then 4 hours ago you'll have 20 more apples than you do now... you'll have taken -(-20) apples = 20 apples.

[Nottom]

[ QUOTE ]

looking at this from a stupid standpoint, as i do most things: 7 * 9 = 63 but 8*8= 64.. 7 is one less than 8 and 9 is one more than 8.. i know it has to do with 1*1=1 but dont completely understand why this works

[/ QUOTE ]

This is just based on basic polynomial algebra:
(x+y)(x-y)=x^2-y^2

so put in x=8 and y=1 and you get 7*9 = 8^2 - 1^1 = 63
x=10 and y=3 gives 13*7 = 10^2 - 3^2 = 91

This can be a useful trick for doing quick multiplication in your head if you happen to know the squares. So if someone asks you what is 34*26, you can quickly just subtract 16 from 900 and say 884.

[Borodog]

It's cooler to take square roots in your head, like the square root of 884 is 29 & 11/15.

pardon me I am drunk