Real Mathematics – Numbers #2

Lottery Bust

No matter how hard you try to avoid them; one day you may need negative numbers. There are a respectable number of people from Britain who would agree with me on this matter. Back in 2007, a British lottery company released a new lottery game named Cool Cash. On every Cool Cash card, there were five boxes that you had to scratch. The box which was on the left bottom corner was called the “temperature of the day”.

Other four boxes were assigned to a prize. In order to win at Cool Cash, your box or boxes should have had lower temperature than the temperature of the day.


In this example our temperature of the day is given as -8 Celsius degrees and our four temperatures are -4, -6, -7 and -7 Celsius degrees. All four degrees are higher than the temperature of the day which meant that this Cool Cash card gives away no prize whatsoever.

Unfortunately this game caused huge problems for the lottery company. It turned out that British people had a poor understanding on negative number notion. Majority of the people called the company and claimed that they won even though they really did not. Ironically, some people thought they lost and threw away their cards which in fact had won a prize. After increasing number of complaints executives of the lottery company realized that they had two choices: Teach mathematics or stop producing Cool Cash cards. They had chosen the second option which was clearly the easiest of the options.

Smallest Number

Negative numbers had challenged mathematicians throughout the history. That is why it is not surprising to see kids having trouble when they try to make calculations with them. Although in most mathematics curricula negative numbers are ought to be thought for 11-12 years old students, some high school students have problems with negative number notion. Even in the 17th century Europe where modern science flourished, many scientists claimed that there can’t be a number that is less than zero.

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Chinese rod numerals.

When one searches for a counter example, it is possible to see the negative number concept in the early civilizations. For instance in ancient China there was a system for distinguishing positive and negative numbers. In ancient China, numbers were represented by rods in two different colors: Red rods were used for positive numbers as black rods for negative numbers. (Today we use red and black to show if an account is in plus or in minus.) Nevertheless Chinese mathematicians believed that a problem or an equation can never have a negative answer.

Adding Words

Ancient China and Egypt was using this technique quite often: Addition to the statement. People were adding extra words in their statements such that these words would change the whole set of the problem. In the end these extra words prevented people from the need of negative numbers. Let me explain this with an example:

Example 1-a: If one has 50 euro in his/her account and shops for 70 euro, that person should end up with -20 euro in total.

Example 1-b: If one has 50 euro in his/her account and shops for 70 euro, that person should end up with 20 euro debt in total.

Addition of the word “debt” changes a negative number into a positive one. This technique is still used today in our daily lives.


People also developed an opposition technique in their language to prevent the need of negative numbers: Choose the opposite of the term.

Example 2-a: A submarine is compared to sea level at -2400 meters height.

Example 2-b: A submarine is compared to sea level at 2400 meters depth.

Oppositions such as up-down, forward-backward, profit-loss, more-loss … are useful for turning a number from negative to positive.

Real Issue

The real issue for understanding negative number notion is that there are no negative numbered things in nature. This is really crucial as we don’t let students think over number concept. Actually whole number notion is not natural; we create numbers in our minds. You can’t go out to nature and run into something (living or not) that is in the shape of number 3. The number 3 makes sense to us when we see 3 cows in a cattle farm.

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Brahmagupta (598-668)

3 apples, 3 trees, 3 cats… Numbers make sense to us whenever you bring them with another concept. However, if someone asks you “what makes 2+1?” you would answer “3”. Here, “3” has no concept attached to itself, yet you used it perfectly.

Almost 1400 years ago, a brilliant Indian scientist named Brahmagupta used negative numbers as we use it today. He explained the negative number notion with another concept attached to it: “Negative numbers stay in the debt side of the number zero.” Brahmagupta also gave rules to negative numbers in calculations. He is the first known person who claimed that a negative multiplying another negative makes a positive.

One wonders…


This is the picture of +2 apples. How can you photograph -2 apples?

M. Serkan Kalaycıoğlu


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