Real Mathematics: Numbers #3

Negative integers are a “game changer” when they sneak into our lives. Everything we learn about calculation gets a little bit more complex with the introduction of negative integers. Now we have rules that say crazy things such as “multiplication of two negative integers gives you a positive one”.  How come two negatives make a positive? Is there a sensible explanation?

Q: (-2).(-3) = ?

Algebraic Method: It is being taught that when two negative integers are being multiplied, absolute values 2 and 3 are being done and a plus sign comes to the top of the calculation. This gives +6 or 6 as a result. Almost all of the students learn how to do this calculation but most of them have no idea why it is being done like this.

There are numerous examples to explain the reason. I prefer giving a specific geometric explanation as an example.

Geometric Method: We teach that integers can be shown on an infinite straight line what we call “number line”.  Middle of this line is assigned to the number zero. Left of zero is for negative integers as right side is for positives.

While teaching multiplication with negative integers, you could imagine a line that is perpendicular to the original number line. Upwards would be assigned to the positive integers as negative integers go to downwards.

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Rules

Assume that we have to multiply two numbers.

  1. Sign of the first number tells us which way we are facing: Upwards or downwards. Positive would mean “look up” as negative means “look down”.
  2. Sign of the second number tells us the way we are taking our steps: We could either go forward (which means the number is positive) or go backwards (meaning that the number is negative). You could imagine that backward steps are like how Michael Jackson was “moonwalking”.
  3. Value of the first number tells us how many steps are taken in each time.
  4. Value of the second number tells us how many times those steps are going to be taken.

Let’s solve our original problem with the geometric method: What is (-2).(-3) ?

  • First number is negative: We are facing downwards.
  • Second number is negative: We are taking our steps backwards.
  • Value of first number is 2: We are taking 2 steps in each time.
  • Value of the second number is 3: We are taking those steps 3 times.

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In the end, we are facing down, moving backwards (like Michael Jackson’ moonwalk), 2 steps at a time and 3 times. Our arrival would be +6 as shown on the graphic.

One wonders…

  1. Try to prove that the algebraic method is true. (Hint: Start with assuming that the multiplying two negatives won’t make a positive.)
  2. Try to find another example for geometric method from real life.

M. Serkan Kalaycıoğlu

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