Number History and TheoryReviewing the contribution made by Blaise Pascal

tonumber theory, and the underlying concepts

discovered.

Presentation by Stephen Hughes.MATI003AZ2010/1 - Number Theory, History And Modelling (A) (2010/1)

“Since we cannot know all that there is to be known about anything, we ought to know a little about everything.”

“We are generally the better persuaded by the reasons we discover ourselves than by those given to us by others.”

Who is Blaise Pascal?

Blaise Pascal was a French mathematician bornon June 19th 1623 in Claremont-Ferrand.

He was interested in many different areas;

• Mathematics• Physics• Theology• Philosophy

And is most famously known for ‘Pascal’sWager’.

“If you gain, you gain all. If you lose, you lose nothing. Wager then, without hesitation, that He exists. ”

Pascal The Mathematician

• Traité du triangle arithmétique ("Treatise on the Arithmetical Triangle“) explored binomial coefficients.

• This resulted in the creation of Pascal’s Triangle.

• Worked on probability and game theory, collaborating with Fermat.

• Later applied probability to theology (Pascal’s Gambit).

• Famously offered a prize for the quadrature of a cycloid. He eventually published his own proof under a fake nameI will be exploring the features of Pascal’s

Triangle“All of our reasoning ends in surrender to feeling.” 

Pascal’s TriangleBelow is a visual representation of some of the

interesting facts about Pascal’s Triangle

“If all men knew what others say of them, there would not be four friends in the world.”

How to Construct the Triangle

The picture below shows how the triangle

is constructed;

1+1 = 2

10+5 = 15

7+21 = 28

9+1 0= 19

“It is the fight alone that pleases us, not the victory.”

Basic Properties

The diagonals (Dx) form a sequence of numbers;

D1 is always 1

D2 is the natural number series

D3 is the triangular number series

D4 is the tetrahedral number series

D5 is the pentalope number seriesThe second number in the diagonal determines the Figurate

numbers based on (n-1)-dimensional regular simplex. With the triangular numbers n-1 = 2 which implies a 2 dimensional nature.

“Even those who write against fame wish for the fame of having written well, and those who read their works desire the fame of having read them.”

Lets Talk about rows

These patterns apply to all subsequent rows;

The red lines represent the Fibonacci number sequence.

This is the 6th row of the sequence. The sum of all the green values is 2^(n-1) where n = row number.

If the first non 1 number of a row is a prime, then all the numbers in the row are divisible by that number. Example is in blue.

There is one more trick which is involving the number 11…

“Imagination disposes of everything; it creates beauty, justice, and happiness, which are everything in this world.” 

The magic 11Incidentally there is a clever way of instantlyMultiplying anything by 11;

21x11

Add up the digits and place it in the middle

2+1 = 3 21x11 = 231

The digits on any given row reflect

11^(n-1). Be careful beyond 11^4

though!The red shaded hexagons indicate where digits are carried over.

A value of 10 means we add a 1 onto the previous hexagon.11^5 = 161051 and not 15101051

“Imagination decides everything.” 

A pretty cool Hockey game…We can form a Hockey Stick shape (red) whereby the

sum of the diagonal equals the number positioned to the converse diagonal.

The petals

The product of the green petals is;21x8x84 = 14,112

The product for the yellow petals is;7x56x36 = 14,112

Let us break them into prime factors;Yellow = (7)x(2x2x2x7)x(2x2x3x3)Green = (7x3)x(2x2x2)x(3x2x2x7)

Their prime factors are equal

“Small minds are concerned with the extraordinary, great minds with the ordinary.”

RecapRows

• The sum of row n is equal to 2^(n-1).

• If value 2 of row n is prime, then it divides the other numbers perfectly.

• The digits on row n form the digits of 11^(n-1).

• The fibonacci sequence is contained in a skewed row style.

Diagonals

• D1 =1.• D2 = N.• D3 = triangular numbers.• Dn = (n-1)-dimensional

regular simplex.

Hockey Stick

• The sum of any diagonal is found by taking the next value on the converse side of the diagonal.

Petals

• The petal patterns shows that nearby numbers have the same prime factors.

Pick one of these bullet points and we

will look closely into the mathematics

behind this…“Nothing fortifies scepticism more than the fact that there are some who are not sceptics; if all were so, they would be wrong.”

Underlying mathematicsThe mathematics behind these facts is extremely

interesting.In the space below I will go through one such proof;

“Nature is an infinite sphere of which the center is everywhere and the circumference nowhere.”

Probability• You can use pascal’s triangle to find probabilities.

• Laid the foundation for subsequent probability theory.

• Based around the binomial distribution theory.

You have sticks labelled A,B,C. You want to know how many different ways you can arrange these into distinct groups of two sticks.

• The problem reduces to ‘pick 2 from 3’.

• We can write it in the form nCr (which means pick r from n)

• Therefore our problem is 3C2 (pick 2 from 3)

“Chance gives rise to thoughts, and chance removes them; no art can keep or acquire them.”

nCr in the triangleRecall that nCr means ‘pick r from n’

We will solve the previous problem using Pascal’s

Triangle; 3C2

n = row numberr = place number

Note that the leftmost 1 is r=0

So we need to find row 3 and place 2

This diagram shows that 3C2 equals 3

The general formula is on the right;

Example: 4C2 should equal 6; (4x3x2x1)/[(2x1)(2x1)] = 6

“Earnestness is enthusiasm tempered by reason.”

Fractals - Sierpinski's Triangle

• A fractal is a pattern formed after a specific iteration

• Discovered by Sierpinski, if you shade in all the odd numbers of the triangle you get a pattern.

• This pattern repeats indefinitely.

• The black numbers are odd

• The white numbers are even

“We view things not only from different sides, but with different eyes; we have no wish to find them alike.”

Conclusion

• Explored binomial distributions.

• Contributed towards probability theory.

• Pascal’s triangle has fundamental consequences for number theory.

• Jack of all trades – Pascal enjoyed exploring alternate subject areas

“Through space the universe encompasses and swallows me up like an atom; through thought I comprehend the world.”

References

Bajaj, G (2000). Free powerpoint presentation template. [photograph]. Available at:http://www.indezine.com/powerpoint/freetemplate/1615.html [Accessed 4th Nov 2010]

Blaise Pascal (n.d). Quotations from Famous people. [online] Available at:http://www.brainyquote.com/quotes/quotes/b/blaisepasc390555.html [Accessed 7th Nov

2010]

Malcfifty (2010). Pascal’s Triangle. [video online] Available at:http://www.youtube.com/watch?v=YUqHdxxdbyM [Accessed 4th Nov 2010]

MathForum (2008). Exploring Pascal’s Triangle. [image]. Available at:http://mathforum.org/workshops/usi/pascal/petals_pascal.html [Accessed 4th Nov 2010]

Peter Fox (1998). "Cambridge University Library: the great collections". Cambridge University Press.

p.13. ISBN 0521626471

Pierce, Rod. "The Sierpinksi Triangle" Math Is Fun. Ed. Rod Pierce. [online] Available at:http://www.mathsisfun.com/sierpinski-triangle.html [Accessed 8th Nov 2010]

Rogers, B (2003). Pascal’s life and times, "The Cambridge Companion to Pascal", Eds. Nicholas

Hammond, Cambridge University Press.

“Man's greatness lies in his power of thought.”