Real Mathematics – Numbers #5

The word fraction has a Latin root “fractio” which means “to break”. Let’s take a slightly different approach for fractions. Instead of breaking, we will use “folding”.

Fractions with Papers

In order to understand the four arithmetical operations in fractions, we can use a standard A4 paper. First we will fold the paper into two halves. Every half represents the fraction ½. We can continue folding one of the pieces in two halves. In the end we will have the following papers which we can use for four mathematical operations:

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a. 1
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b. 1/2+1/2
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c. 1/2+1/4+1/4
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d. 1/2+1/4+1/8+1/8
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e. 1/2+1/4+1/8+1/16+1/16

Addition

We are aware of the fact that all five of the versions are equal to one another and they each add up to 1. If we take a look at versions b and c we can conclude that ½ is equal to ¼ + ¼. If we substitute 1/2s in the version b we can find the following result:

1 = ½ + ½ = ¼ + ¼ + ¼ + ¼.

Subtraction

Let’s continue from the previous. If we subtract ¼ from both sides of ½ = ¼ + ¼, then we get:

½ – ¼ = ¼ + ¼ – ¼

½ – ¼ = ¼

Multiplication

Let’s start with another A4 paper and take its whole as 1. This time we will fold the paper and use the marks on it.

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Assume that we are trying to find ½ * ¾.

Check the first fraction and fold the paper into two halves since it is ½.

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Then check the second fraction and fold the paper into four equal parts.

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Since second fraction is ¾, mark 3 parts of the paper.

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Now unfold the A4 paper completely. It shows that there are 8 equal parts and 3 of them are marked. Hence solution is 3/8.

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One wonders…

  1. Using the paper method show the connection(s) between 1/8 and 1/32.
  2. Find 1/8 – 1/32 with the paper method.

Game

Imagine a gambling game in which you don’t have to gamble your money. In the worst case scenario, you will win nothing. A dream for gamblers, isn’t it?

Let’s say you get 128.000 dollars and 6 red and black cards (3 for each). Here is how the game goes: In every hand you must bet half of your total money. When you pick a red card, you will get twice what you played. When you pick a black card, you will lose all the money you bet in that hand.

When the game is finished (after all 6 cards are played) if you end up with more than 128.000 dollars you will get that surplus.

What is your strategy to win? Explain your answer.

M. Serkan Kalaycıoğlu

Real Mathematics – Numbers #4

In schools we start learning mathematics with learning what numbers are. Unfortunately numbers are taken for granted and being overlooked just because it starts in the elementary school. The truth is this part of mathematics is a joint work of countless civilizations that lasted thousands of years. Although, categorizing and defining all those information were done only in the near past. This means that things we learn in the first few years of school have so much more depth than we think they have.

Especially fractions (or rational numbers) weren’t used in Europe in the sense we understand them today until the 17th century. In fact for a long time people thought of fractions not as numbers but as two numbers being divided to one another.

Rhind

Ancient Egyptians were one of the first known civilizations that used fractions. They created one of the most important and oldest documents in the history of civilizations using papyrus trees. Around 4000 years ago they started writing valuable information on papyrus leafs. Rhind papyrus is one of those documents. It is believed to be written around 1800 BC. Thanks to Rhind, we can understand how ancient Egyptians used fractions.

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It is uncanny how commonly they used fractions in Rhind. Although they were obsessed with unit fractions as they found ways to describe every fraction with them.

Unit Fraction: Fractions that have 1 on their numerators.

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In the ancient Egypt they used a shaped that looks like an open mouth (or an eye). This shape was the notation for the unit fraction. Denominator of the fraction would be placed under the mouth.

Table of 2/n

Inside Rhind there is a method for describing fractions in the form of 2/n (when n is odd) with two unit fractions. Table starts with 2/3 and ends with 2/101. In the papyrus it says that 2/3 is equal to ½ + 1/6. For the rest of the papyrus a formula was given in order to describe fractions in the form of 2/3k: It is 1/2k + 1/6k.

Let’s try it for 2/9. 9 is 3k, hence k=3. This gives 2/9 = 1/6 + 1/18. Ingenious, isn’t it?

Next number on the table is 2/5 = 1/3 + 1/15. This is also a general formula just like the previous one. Any fraction in the form of 2/5k can be shown as 1/3k + 1/15k.

Fraction Line

Besides ancient Egyptians, Babylonians used fractions too. But their choice of symbols was so confusing, it is impossible to understand which number is written. You could only check the rest of the calculation (if there is any) and guess which number is being used.

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In the Babylon civilization number system has base 60. The one on the left is 12, other one is 15. But they can also mean 12+(15/60) too. Lack of symbol for fractions caused a lot of problems in this civilization.

Around 1500 years ago Indian mathematicians were shining. They found the number system we use today and even the number zero was invented (or discovered). Their brilliance was key for fractions too as they showed fractions one under the other. Muslims were the ones who thought of putting the fraction line between numbers.

In the end we owe our modern notation to Indian and Muslim mathematicians.

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The way 7/15 was shown in the old Indian symbols.

One wonders…

Try to find answers to following questions about the Rhind:

  1. Why did they only consider odd numbers in the denominator of the fractions?
  2. Find 2/7 and 2/11 using their methods.
  3. What happens after 11?
  4. Try to come up with a general formula for 3, 5, 7 and 11.

M. Serkan Kalaycıoğlu

Real Mathematics: Numbers #6

Achtung! Following article contains advertisement for a chocolate company even though I won’t get paid.

How to get the most Milkinis?

Assume that

                – Class has 8 students.

                – All the students adore Milkinis and would do anything to have some.

                – There are 6 Milkinis bars in total.

Rules of the game:

                – 3 tables in the whole classroom named A, B and C.

                – Tables get 1,2 and 3 Milkinis bars in order.

                – On every table each student will get equal share of Milkinis bars. (eg. If there are 4 students on table A, those 4 students will get ¼=0,25 Milkinis bars each.)

                – Ambition is to select the table that has more Milkinis outcome for the student. (eg. While selecting, if student has a chance to get more chocolate from table A, he/she will choose that table.)

Q: Imagine you are one of the 8 students. In order to get the most Milkinis bars, what kind of strategy should you use? (On which turn you should select your table.)

Playing

As long as we don’t change the rules or anything, we will be starting the game in the same fashion every single time: First student will select table C, because it will give him/her 3 Milkinis bars which is greater than A’s 1 and B’s 2 bars.

Second student will choose B, so he/she will get 2 bars of Milkinis.

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Third one has to choose table C as he/she will get 3/2=1,5 bars of Milkinis. Now we have 1 student sitting in table B, 2 students sitting in table C. A is still empty.

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Forth student has 3 identical choices. In all three tables student will be getting same amount of Milkinis bar. (1 Milkinis bar.) Let’s assume forth student selects table B.

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Fifth one will get;

1/1=1 bar from table A,

2/3=0,66 from B,

3/3=1 from C. Let’s say the student selects table C.

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Sixth student will select table A as it will get him/her a full bar of Milkinis.

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Seventh will be selecting table C. Now table C has 4 students and 3 Milkinis bars. Each student here gets ¾=0,75 Milkinis.

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Eight student must select table B as each student on table B gets 2/3=0,66 Milkinis.

Result

Our game is finished with 1 student sitting at the table A, 3 at B and 4 at C. This would result that students get 1 bar of Milkinis from A, 0,66 from B and 0,75 from C. Student who goes to table A is the clear winners in the situation, who was the sixth choice.

One wonders…

  1. Is there a spot while selecting that guarantees the most chocolate?
  2. Why did I feel the need of using decimal point as using fractions would give me the same amount?
  3. One Milkinis bar has 4 little parts. Which student(s) would get the least Milkinis? How many parts of Milkinis would it be?

History of Decimal Point

In early math education teachers discuss decimal point right after teaching what fractions are. If they both mean the same thing, why do we teach both of them?

At the first sight it looks like a waste of time to show the same thing with two different notations. But in truth fractions and decimal point are both very useful and critical in math. Using decimal point might seem confusing, although when it comes to comparing two or more numbers, decimal point notation is easier to the eye than fractions are. (eg. Comparing 0,66 with 0,60 is easier and faster than comparing 2/3 and 3/5.) Also it takes less time when you write down huge numbers as decimals.

Fractions have a history of at least 4000 years. Decimal point notation is relatively a baby next to fractions. In his book History of Mathematics David E. Smith mentions a priest named Christopher Clavius (1537-1612). According to Smith, Clavius is the first known person who used decimal point systematically. In a book his book Clavius made a table called “Tabula Sinuum” where he wrote down his astronomical calculations in decimal point notation.

pellos

In 1492, Francesco Pellos wrote in his arithmetic book that 1/10th of 5836943 makes 583694.3 as shown. Although Pellos wrote the first known decimal point notation in his book, historians of mathematics claim that Clavius should be considered as the inventor of decimal point notation.

M. Serkan Kalaycıoğlu