Real Mathematics – Geometry #2

Mathematics without numbers

Around 2700 years ago ancient Greeks were in total control of every part of science (Philosophy, geometry and mathematics in particular.). For centuries Greek hundreds of historically important figures like Thales, Pythagoras, Eudoxus and Euclid dominated mathematics.

Ancient Greek mathematicians had a significant difference. Unlike their colleagues from other parts of the world, they choose not to use number symbols. According to them, geometry was the foundation of mathematics, and like everything in mathematics numbers arose from geometry as well.

Even though they created respectable number systems and symbols, comparing to their advanced knowledge in other branches of mathematics (particularly geometry) they were behind with numbers. It was like as if they didn’t care about number systems and symbols as much as they cared geometry and other parts of science.

It is mesmerizing to hear that founders of geometry didn’t need numbers in their works.

Ruler, Compass and Unit

In ancient Greece, philosophers (meaning scientists) used magnitude instead of numbers. They were drawing straight line segments to show a magnitude. In other words, ancient Greek mathematicians were drawing lines instead of writing number symbols. Moreover, they used unmarked ruler and compass as their only tools. (I’ll be explaining the use of them in the upcoming articles.)

Q: How did Greeks manage to make mathematics without numbers?

Assume that we have positive integers a and b.

Addition

Their addition makes a+b. Using straight line segments we can show a+b as follows:

a+b

Extraction

If a is greater than b, extraction can be written as a-b. This can be shown with line segments like the following:

a-b

Multiplication

Multiplication of them gives a.b. We can use properties of triangles in order to explain multiplication with lines. Assume that we have a triangle with side a and 1:

Now we will extend the sides of this triangle so that, the side a will become a.b while the side 1 becomes b.

Division

Let’s say that we want to find a/b with lines. This time we can use a similar approach that we used in multiplication. First we construct a triangle with sides a and b:

Then we shorten the sides so that length of the side a will become a/b while length of the side b becomes 1.

Taking the Square Root

To take the square root of the number a, first we should draw a straight line segment that has length a+1. Then we mark the segment such that left side of the mark will have length a, and right will have length 1. Finally draw a semicircle that has diameter a+1. Now draw a perpendicular from the circle’s boundary to the marked point. That perpendicular line will have length √a.

One wonders…

If ancient Greeks knew how to make calculations, does it mean that they were involved with algebra and number theory too? (Check out the name Diophantus.)

M. Serkan Kalaycıoğlu

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