Why does multiplying negatives result in a positive?

[bozo the clown]

can anyone explain why for example
-7 * -7 = +49

after all isn't -7 * -7 = to saying - 7 + -7 seven times

now if we take -7 + -7 sven times it would be -49

 

[hello3719]

here is a proof that -1*-1 = 1

-1*0 = 0
since 0 = 1-1
-1*(1-1) = 0
by distributivity we get
-1*1 + -1*-1 = 0
(-1) + (-1*-1) = 0
since -1 + 1 =0
then -1 *-1 = 1

 

[Gokul43201]

after all isn't -7 * -7 = to saying - 7 + -7 seven times

now if we take -7 + -7 seven times it would be -49

No... look carefully again at what you claim. -7*-7 should be the same as adding -7 to itself "minus" seven times.

What you are doing is adding it seven times. So that is the same as-7*7, which is why you get -49

 

[Chen]

(-7) * (-7) = -[7 * (-7)] = -(-49) = 49.
You can show this with complex numbers.

 

[killerinstinct]

can anyone explain why for example
-7 * -7 = +49

after all isn't -7 * -7 = to saying - 7 + -7 seven times

now if we take -7 + -7 sven times it would be -49

It isn't -7 + -7 times. THis doesn't make sense, but its acutally -7 + -7 negative seven times...
(really doesn't make sense!)

 

[phoenixthoth]

this is how i would explain it; it's not intended to be a proof...

let's work out why (-3)(-3)=9.

(3)(3)=9
(2)(3)=6
(1)(3)=3
(0)(3)=0

the pattern is the answers go down by 3 when the first number is decreased by 1.

(-1)(3)=-3
(-2)(3)=-6
(-3)(3)=-9

you can do something similar to get the following results:

(-1)(2)=-2
(-2)(2)=-4
(-3)(2)=-6

(-1)(1)=-1
(-2)(1)=-2
(-3)(1)=-3

(-1)(0)=0
(-2)(0)=0
(-3)(0)=0.

look what happens at (-3)(x): when x goes down by 1, the answer goes UP by 3. therefore:

(-3)(-1)=3 (0+3)
(-3)(-2)=6 (3+3)
(-3)(-3)=9 (6+3)

 

[quartodeciman]

Finally, there is a choo-choo train illustration.

------

A train track runs from station A past station B onto station C, where A is to the west of B and C is to the east of B. Also, the track is perfectly straight and the distance AB equals distance BC. It's numerically 49 distance-units for this example.

Positive velocity of a train is in the eastward direction and negative velocity is in the westward direction (assuming the train can run bidirectionally).

Positive time durations are later and negative time durations are earlier.

Positive distances are to the east and negative distances are to the west.

OK.

Suppose a train at B runs eastward at velocity +7 (distance-units/time-unit) and will run for +7 time-units. Then the distance will be (+7)(+7)=+49, or 49 distance-units to the east of the station B. That is where station C is.

Suppose a train at B runs eastward at velocity +7 (distance-units/time-unit) and already ran for -7 time-units. Then the distance was (+7)(-7)=-49, or 49 distance-units to the west of the station B. That is where station A is.

Suppose a train at B runs westward at velocity -7 (distance-units/time-unit) and will run for +7 time-units. Then the distance will be (-7)(+7)=-49, or 49 distance-units to the west of the station B. That is where station A is.

Suppose a train at B runs westward at velocity -7 (distance-units/time-unit) and already ran for -7 time-units. Then the distance was (-7)(-7)=+49, or 49 distance-units to the east of the station B. That is where station C is.

 

[hello3719]

here is a proof that -1*-1 = 1

-1*0 = 0
since 0 = 1-1
-1*(1-1) = 0
by distributivity we get
-1*1 + -1*-1 = 0
(-1) + (-1*-1) = 0
since -1 + 1 =0
then -1 *-1 = 1

-7*-7 = 7(-1*-1)

This is a PROOF

 

[Warr]

(-1) + (-1*-1) = 0
since -1 + 1 =0

somehow you set (-1*-1) = +1, even though that is what you were trying to prove!

here is one way to get your head around it.

-7 * -7 = (-1)(7)(-7) or -(7 * -7)

seven -7s is -49, so

-(7*-7) = -(-49)

the negative of a negative number is always positive

therefore -(-49) = 49

 

[Icarus]

Warr - You have accused hello3719 of something he did not do, and then you turn around and do the very thing yourself:

the negative of a negative number is always positive

This is what you are trying to show!

To see that hello3719 did not commit this error, let me expand on the proof:

          -1*0 = 0       Anything times 0 = 0
      -1*(1-1) = 0       Substitution of 1-1 for 0
  -1*1 + -1*-1 = 0       Distributive Law
(-1) + (-1*-1) = 0       Simplify the first term.
         -1*-1 = 1       Add 1 to both sides of the equation.

 

[hello3719]

somehow you set (-1*-1) = +1, even though that is what you were trying to prove!

no you didn't follow well
(-1)(0) = 0
(-1)(-1 + 1) = 0
by distr
(-1)(-1) + 1(-1) = 0
1(-1) = -1
so -1 + (-1)(-1) =0
since -1 + 1 = 0 (eq 1)
and that we have -1 + (-1)*(-1) = 0 eq 2)
then if we substract eq 2 form eq 1 then we get (-1)*(-1)=1

the negative of a negative number is always positive

LOL .that IS what we have to prove, negative of a negative means
(-1)(-1)

 

[Warr]

sorry, I reread. My mistake

 

[geometer]

Here is Hello's proof in more general terms:

The proof will be in two steps. First, we will prove that x*(-y) = - k where x, y, and k are all integers

Let a and b be any arbitrary integers

Since the integers represent a number field, they are closed under multiplication
Thus a*b = c where c is another integer.

Since the integers are also closed under addition, we can find an integer d such that d-1 = b. (You could also cite Peano's Postulates for this.) We note that d must be greater than b.

Substituting these into our original equation: a*(d-1) = c

By the distributive property of the integers then, we can write:

a*d + a*(-1) = c

But notice: Since d > b, then a*d > c. Obviously then a*(-1) must be less than 0.

Now, multiply a*d + a*(-1) = c through by -1. Based on our results from above we get:

-a*d + (-a)*(-1) = -c

Again since d > b, then -a*d must be less than c. Therefore, (-a)*(-1) must be greater than 0.