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Primary Magazine - Issue 17: A little bit of history


This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 30 October 2009 by ncetm_administrator
Updated on 17 November 2009 by ncetm_administrator

Primary Magazine Issue 17Blaise Pascal
 

A little bit of history - Blaise Pascal

Pascal is a significant figure in mathematics and science. He was a mathematician, physicist, and religious philosopher. This article gives a small insight into this Frenchman, highlighting some of his achievements and parts of his life.

Pascal was considered to be a child prodigy. His father was a judge, who was also interested in mathematics and science. Pascal’s mother died when he was three and shortly after this his father moved the family to Paris where he decided to educate his children at home. All three of his children (he also had two daughters) showed extraordinary intellectual ability, particularly Blaise. He had an amazing aptitude for mathematics and science, so much so that at 11 his father forbade him to do any more until he was 15, so that he could concentrate on his studies of Latin and Greek. However, when he was 12, his father found him writing on a wall with coal, an independent proof that the sum of the angles of a triangle is equal to a straight line! From then on he was allowed to study Euclid and also to sit in on the meetings of some of the greatest mathematicians and scientists of the time. As a result of listening to one of these he produced the proof for what is known as the Mystic Hexagram. It is still known today as Pascal’s Theorem, which states that if a hexagon is inscribed in a circle, then the three intersection points of opposite sides lie on a line (called Pascal’s Line).

 
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You could try the angles in a triangle investigation with your children. It’s good fun and helps to demonstrate this theory (although not strictly in Pascal fashion) in a different, much more basic way. It is also a good example of finding proof at a primary level.
 

investigating triangles

 
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He also helped to create two major areas of research which were new at the time. One, which he did when he was 16, was related to geometry, and the other related to probability and strongly influenced the development of modern economics and social science. This began in about 1654 when he started to investigate the chances of getting different values for rolls of the dice. His discussions with Pierre de Fermat are said to have laid the foundation for the theory of probability. His father, who died in 1651, seemed to spend most of his later life calculating and recalculating the taxes he owed and it was this that led Pascal to construct the first type of mechanical calculator capable of adding and subtracting – and he wasn’t yet 19! Originally he made one to help his father and then some to sell, but they were too expensive for most people to buy and became status symbols for the rich. He improved and refined his first design and over the next 10 years built and sold 50.

Pascal is probably best known in primary school circles for his famous triangle.

Each number is the sum of the numbers directly above it. Have you given this to your class to try? If not, why not give it a go?
 

each number is the sum of the numbers directly above


Pascal's Triangle is more than just a big triangle of numbers. It is used in algebra and probability.
 

Pascal's triangle

In 1646, Pascal joined a religious movement within Catholicism, and five years later had a ‘second conversion’ which caused him to mostly abandon his other work and devote himself to philosophy and theology.

He suffered from a nervous illness throughout his life which gave him constant pain. In 1647 he had a paralytic attack that so disabled him he could not move without crutches. In 1662 Pascal's illness became more violent. His health was fading quickly and he wanted to be moved to the hospital for incurable diseases, but his doctors said that he was too unstable to be carried. In Paris, on 18 August 1662, Pascal went into convulsions. He died the next morning. The last words he said were, 'May God never abandon me.'

 
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Fascinating things to investigate in Pascal’s Triangle

Hockey stick pattern
If a diagonal of numbers of any length is selected starting at any of the one's at the sides of the triangle and ending on any number inside the triangle on that diagonal, the sum of the numbers inside the selection is equal to the number below the end of the selection that is not on the same diagonal itself, eg:

1 + 6 + 21 + 56 = 84
1 + 7 + 28 + 84 + 210 + 462 + 924 = 1716
1 + 12 = 13  
 

Pascal's triangle - hockey stick pattern

 
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Magic 11s
If a row is made into a single number by using each as a digit of the number (adding the tens number of a two-digit numbers to the previous one) it is a multiple of 11. Check it out!  

Row Multi-Digit number Actual Row
Row 0 1 1
Row 1 11 1 1
Row 2 121 1 2 1
Row 3 1331 1 3 3 1
Row 4 14641 1 4 6 4 1
Row 5 161051 1 5 10 10 5 1
Row 6 1771561 1 6 15 20 15 6 1
Row 7 19487171 1 7 21 35 35 21 7 1
Row 8 214358881 1 8 28 56 70 56 28 8 1


Triangular Numbers
They can be found in the diagonal starting at row 3 as shown in the diagram. The first triangular number is 1, the second is 3, the third is 6, the fourth is 10, and so on.
 

Pascal's triangle - triangular numbers

 
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Fibonacci numbers
To find these you need to go up at an angle: you're looking for 1, 1, 1 + 1, 1 + 2, 1 + 3 + 1, 1 + 4 + 3, 1 + 5 + 6 + 1.
 

Pascal's triangle - Fibonacci numbers

 
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Square Numbers
These are found in the same diagonal as the triangular numbers. A Square Number is the sum of the two numbers in any circled area in the diagram. The very first square number is 02. The second is 12, the third is 22 (4), the fourth is 32 (9), and so on.
 

Pascal's triangle - square numbers

To find out more about these investigations and others visit these websites:

Wikipedia has more information about Pascal.
 

 
 
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