Negative * Negative = Positive

Why??

Here, we don't so much "prove," as demonstrate why this is true. For students learning new concepts demonstrations are usually more satisfying than rigorous proofs. Rigorous proofs are for pure mathematicians.

Reformatted: January 2010

 

Method 1: Graphing

On any graph of a straight line, slope is "rise over run," that is change in y divided by change in x. It does not matter what points we use or how we count between them the answer will be the same.

slope

Slope is:

  • M = 2/4 = 1/2
  • M = 5/10 = 1/2
  • M = -3/(-6) = 1/2 (-6 because we went backwards, -3 because we went down)

We find that the same slope can be determined from positive/positive or from negative/negative. Thus negative/negative has the same sign as positive/positive which is positive.

 

For this one the slope is negative (going down). It is shown as:

  • M = 6/(-6) = -1 (negative in the run meaning backwards)
  • M = -6/6 = -1 (negative in the rise meaning down)

Both positive / negative and negative / positive are negative.

slope

Method 2:

We know the formula for area of a rectangle. If we change the length of the sides we can watch how this effects the area and we can draw to show what the parts of the equation are.

rectangle

We start with a rectangle with sides of L and W.

The area is: A=LW. We change the sides to L+c and W+d. This can be written as: A=(W+c)(L+d), or expanded to:

A = LW + Lc + Wd + cd, where each part represents a small rectangle making up the greater rectangle.

What happens if we shorten the sides rather than lengthen them?

We can draw it again. Now the shaded regions represent parts of the rectangle that were removed.

Our equation will be: A=(W-c)(L-d) or:

A = LW - Lc - WD + CD

Why is CD added instead of subtracted? Because the area in rectangle CD was subtracted out once as part of A1=Lc, and then subtracted out again as part of A2=WD Since, A3=CD was subtracted out twice as part of two other rectangles, we now add it back in so that it is only subtracted out once. This shows us that when we foil (W-c)(L-d) to get LW-Lc-Wd+CD the (-c)(-d) must be positive.

rectangle

ractangle Watch how rectangle c*d gets subtracted out twice. So we add it back in so that it is only subtracted out once.

Related Lessons

 

Method 3: Word Problems

As stated in a previous lesson, negative numbers are not used for counting, they are used for measurement. As a result word problems for negatives should involve measurement not counting. This is hard to do without using examples from physics. Also, from that lesson, recall that negatives can represent [1] downward change: ie. direction, [2] below a reference point, and [3] ordered comparisons.

[a] Money Examples

I personally dislike money examples, but people seem to go here first. It is easy to show a negative times a positive, but how do we show a negative times a negative?

Positive * Negative: Last month our CEO signed 54 checks for $100 each:

54*(-100) = $-5,400.

The negative means the company spent -5,400. Notice this is a counting example, count 54 checks, and cannot be used to show negative * negative. How can we fix that problem? By asking a question about change.

 

Negative * Negative: This month our CEO signed 8 less checks for $100. What did our company save over the previous month?

(-8)*(-100) = $800.

The -8 show us that the amount decreased (changed downward) by 8. The positive $800 show us that that our company account changed in the positive direction. We did not determine the actual value of the spending or the account. We determined the change in spending.

[b] Electronic Examples:

Our three primary concerns in any electrical components are voltage(V), current (I), and power (P). They are related by the simple formula: P=VI. With current and voltage a negative refers to a reversed direction. For power, a positive means power being given to (or consumed by) an element, and a negative means power flowing out of an element.

curcuit

Look at the elements in this circuit. For the voltage source S, the current is flowing towards the positive voltage, or by definition, in the opposite direction. This gives us a positive voltage times negative current eg: P=(9)(-2)=-18. The negative in the power tells us that power is being delivered, or leaving the battery.

For the resistor, the current is flowing towards the negative voltage, or by definition, in the same direction. We have a positive voltage times positive current resulting in positive power, P=9*2=18, that is power being driven into the element.

Think of the resistor as being like a light bulb in a flash light. What happens if we turn the battery around? If the battery chamber will fit reversed batteries, the light will still light. Since the voltage source (battery) is reversed, the current and voltage from the battery will also be reversed. At the lamp, we have negative voltage and negative current resulting in positive power: P=(-9)(-2)=+18. What does the positive 18 for power tell us? It tells us that the lamp is receiving power, or lighting. Isn't that what we observe?

[c] Extensions:

Now its your turn. Have students collect simple equations from what ever source they may find them. Ask them to interpret what a negative might mean to the various parts of the equation. Remember how we used a negative to mean "decreases" in the money example. Discuss the equations to be sure the answers make sense.

Examples:

  • F=ma: negative acceleration means slowing down, negative mass must mean change in mass
  • S=rt: negative time could only come from a comparison: where was it before now?
  • PV=nRT: for each of these a negative would only refer to a comparison
  • y=mx+b: this one is easy, show the whole graph to prove that the parts make a straight line
 
 

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