Why a negative times a negative is positive

In summary, the conversation discusses two examples of real world objects, a tank and an hourglass, and how they can be used to demonstrate multiplication of signed numbers. The examples involve filling and draining at a constant rate, with the water level or sand flow being marked at regular intervals. The conversation also mentions using a video tape to record and play time in either a positive or negative direction. It is noted that there is nothing special about the zero in the sequence of multiplication.
  • #1
rcgldr
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Since the other thread is closed, I thought I create one with a simple example of water in a tank that most people will understand.

Imagine a tank designed to hold water with markers every inch from bottom to top. The marker halfway up the tank is chosen to be labeled zero. The next marker above is +1, and the next marker above that is +2. The first marker below the zero marker is labled -1, and the marker below that is -2, and so on.

The tank was and is being filled at a rate of 2 markers per minute. Currently, the water level in the tank has just reached the zero marker. At what marker will the water level be in the tank 3 minutes from now: answer +3 minutes times +2 markers per minute = +6 markers. Where was the water level at 3 minutes before the water reached the zero marker: answer -3 minutes times +2 makers per minute = -6 markers.

In the mean time there is a second tank, marked in the same way with the zero marker at the halfway point, that started off filled. It was and is being drained so that that water level is decreasing by 2 markers per minute. The water level is currently at the zero marker. At what marker will the water level be in the tank 3 minutes from now: answer +3 minutes times -2 markers per minute = -6 markers. Where was the water level at 3 minutes before the water reached the zero marker: answer -3 minutes times -2 makers per minute = +6 markers.

As another example, sand flowing through an marked hour glass. The upper half is being drained, while the bottom half is being filled, same questions and answers as above.

These are just two examples of common real world objects that are easily witnessed. A video tape could be used to record and play time "forwards" (positively) or "backwards" (negatively) to help in understanding.

Anything involving easily viewed constant motion and time are good candidates for demonstrating multiplication of signed numbers.
 
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  • #2
This sounds very similar to the idea of just continuing the multiplication tables:
-3*2=-6
-3*1=-3
-3*0=0
-3*-1=3
...
There is nothing that special about the zero in that sequence.
 

1. Why does a negative multiplied by a negative equal a positive?

When two negative numbers are multiplied together, the resulting product is always a positive number. This is because multiplying a number by a negative essentially means flipping the sign of that number. So when two negative numbers are multiplied, the negative signs cancel out, resulting in a positive number.

2. Is there a mathematical explanation for this rule?

Yes, there is a mathematical explanation for why a negative times a negative is positive. It is based on the concept of the distributive property, which states that a(b+c) = ab + ac. When applied to negative numbers, this property can be written as (-a)(-b) = (-1)(a)(-1)(b) = (-1)(-1)(ab). Since multiplying a number by -1 is the same as changing its sign, (-1)(-1) equals 1. Therefore, (-a)(-b) = ab, resulting in a positive product.

3. Can you give an example to illustrate this rule?

Sure. Let's take the expression (-2)(-3). Using the distributive property, we can rewrite this as (-1)(2)(-1)(3). Since (-1)(2) = -2 and (-1)(3) = -3, the expression becomes (-2)(-3) = (-1)(2)(-1)(3) = (-1)(-1)(2)(3). As mentioned before, (-1)(-1) equals 1, so (-2)(-3) = 1(2)(3) = 6, which is a positive number.

4. Does this rule apply to all numbers, including fractions and decimals?

Yes, this rule applies to all numbers, including fractions and decimals. The concept of changing the sign when multiplying by a negative remains the same, so the rule holds true for all types of numbers.

5. How is this rule used in real-world applications?

This rule is used in various real-world applications, such as calculating temperature changes, determining profit/loss in finance, and solving equations in physics and chemistry. It is an essential concept in mathematics and has practical uses in many fields.

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