Blaise Pascal and Pascal’s Triangle

PASCAL’S TRIANGLE

* ABOUT THE MAN * CONSTRUCTING THE TRIANGLE * PATTERNS IN THE TRIANGLE * PROBABILITY AND THE TRIANGLE

Blaise PascalJUNE 19,1623-AUGUST 19, 1662

French religious philosopher, physicist, and mathematician.

Mother died when he was 3 years old.He was home schooled.

His father, Étienne Pascal, decided that Blaise was not to study mathematics before the age of 15 and all mathematics texts were removed from their house.

Blaise however, his curiosity raised by this, started to work on geometry himself at the age of 12. He discovered that the sum of the angles of a triangle are two right angles and, when his father found out, he relented and allowed Blaise a copy of Euclid.

Étienne had been appointed as a tax collector for Upper Normandy.

Pascal invented the first digital calculator to help his father with his work collecting taxes.

He worked on it for three years between 1642 and 1645.

The device, called the Pascaline, resembled a mechanical calculator of the 1940s. This, almost certainly, makes Pascal the second person to invent a mechanical calculator for Schickard had manufactured one in 1624.

The Pascaline

Pascal was not the first to study the Pascal Triangle

As of today, the triangle appears to have been discovered independently by both the Persians and the Chinese during the eleventh century.  Although no longer in existence, the work of Chinese mathematician Chia Hsien (ca. 1050).

Khu Shijiei triangle, depth 8,

1303.

In correspondence with Pierre de Fermat, Pascal laid the foundation for the theory of probability.

This correspondence consisted of five letters and occurred in the summer of 1654. They considered the dice problem and a point problem.

The dice problem asks how many times one must throw a pair of dice before one expects a double six while the problem of points asks how to divide the stakes if a game of dice is incomplete.

Sometime around then he nearly lost his life in an accident.

The horses pulling his carriage bolted and the carriage was left hanging over a bridge above the river Seine.

Although he was rescued without any physical injury, it does appear that he was much affected psychologically.

Not long after he underwent another religious experience, on 23 November 1654, and he pledged his life to Christianity and stopped doing mathematics.

In 1658 Pascal started to think about mathematical problems again as he lay awake at night unable to sleep for pain.

To forget about the pain he wrote his last work on the cycloid, the curve traced by a point on the circumference of a rolling circle.

Pascal the philosopher

Pascal's most famous work in philosophy is Pensées, a collection of personal thoughts on human suffering and faith in God which he began in late 1656 and continued to work on during 1657 and 1658.

This work contains 'Pascal's wager' which claims to prove that belief in God is rational with the following argument.

If God does not exist, one will lose nothing by believing in him, while if he does exist, one will lose everything by not believing.

Pascal died at the age of 39 in intense pain after a malignant growth in his stomach spread to the brain.

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*Pascal was first to discover the importance of the patterns of Pascal’s triangle.

CONSTRUCTING THE TRIANGLE

* START AT THE TOP OF THE TRIANGLE WITH THE NUMBER 1; THIS IS THE ZERO ROW.* NEXT, INSERT TWO 1s. THIS IS ROW 1.* TO CONSTRUCT EACH ENTRY ON THE

NEXT ROW, INSERT 1s ON EACH END,THEN ADD THE TWO ENTRIES ABOVE IT TO THE LEFT AND RIGHT (DIAGONAL TO IT).

* CONTINUE IN THIS FASHION INDEFINITELY.

1

CONSTRUCTING THE TRIANGLE

1 ROW 0 1 1 ROW 1 1 2 1 ROW 2 1 3 3 1 ROW 3 1 4 6 4 1 R0W 4 1 5 10 10 5 1 ROW 5 1 6 15 20 15 6 1 ROW 6 1 7 21 35 35 21 7 1 ROW 7 1 8 28 56 70 56 28 8 1 ROW 8 1 9 36 84 126 126 84 36 9 1 ROW

9

2

PALINDROMES

EACH ROW OF NUMBERS PRODUCES A PALINDROME.

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1

THE TRIANGULAR NUMBERS 1 1 1 * {15} {1} 2 1 * * * 1 {3} 3 1 * * {10} * * * 1 4 {6} 4 1 * * * * * * * 1 5 10 {10} 5 1 * * * * * * * * * 1 6 15 20 {15} 6 1 * * * * {6}

* {1} * * {3} * * *

2

THE SQUARE NUMBERS

1 1 1 (1) 2 1 * * 1 (3) 3 1 * * 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1

2

THE SQUARE NUMBERS

1 1 1 1 2 1 1 (3) 3 1 * * * 1 4 (6) 4 1 * * * 1 5 10 10 5 1 * * * 1 615 20 15 6 1

3

THE SQUARE NUMBERS

1 1 1 1 2 1 1 3 3 1 * * * * 1 4 (6) 4 1 * * * * 1 5 10 (10) 5 1 * * * * 1 6 15 20 15 6 1 * * * *

4

THE SQUARE NUMBERS

1 1 1 1 2 1 * * * * * 1 3 3 1 * * * * * 1 4 6 4 1 * * * * * 1 5 10 (10) 5 1 * * * * * 1 6 15 20 (15) 6 1 * * * * *

5

THE HOCKEY STICK PATTERN

IF A DIAGONAL OF ANY LENTH IS SELECTED AND ENDS ON ANY NUMBER WITHIN THE TRIANGLE, THEN THE SUM OF THE NUMBERS IS EQUAL TO A NUMBER ON AN ADJACENT DIAGONAL BELOW IT.

1

THE HOCKEY STICK PATTERN

1 1 1 1 2 1 (1) 3 3 1 1 (4) 6 4 1 1 5 (10) 10 5 {1} 1 6 15 (20) 15 {6} 1 1 7 21 [35] 35 {21} 7 1 1 8 28 56 70 {56} 28 8 1 1 9 36 84 126 126 [84] 36 9 1

2

THE SUM OF THE ROWS

THE SUM OF THE NUMBERS IN ANY ROW IS EQUAL TO 2 TO THE “Nth” POWER ( “N” IS THE NUMBER OF THE ROW).

1

THE SUM OF THE ROWS

2 TO THE 0TH POWER=1 1 2 TO THE 1ST POWER=2 1 1 2 TO THE 2ND POWER=4 1 2 1 2 TO THE 3RD POWER=8 1 3 3 1 2 TO THE 4TH POWER=16 1 4 6 4 1 2 TO THE 5TH POWER=32 1 5 10 10 5 1 2 TO THE 6TH POWER=64 1 6 15 20 15 6

1

2

PRIME NUMBERS

IF THE 1ST ELEMENT IN A ROW IS A PRIME NUMBER, THEN ALL OF THE NUMBERS IN THAT ROW, EXCLUDING THE 1s, ARE DIVISIBLE BY THAT PRIME NUMBER.

1

PRIME NUMBERS

I 1 1 THE FIRST ELEMENTS IN

ROWS 1 2 1 THREE, FIVE, AND SEVEN 1 *3 3 1 ARE PRIME NUMBERS. 1 4 6 4 1 NOTICE THAT THE OTHER 1 *5 10 10 5 1 NUMBERS ON THESE

ROWS, 1 6 15 20 15 6 1 EXCEPT THE ONES, ARE 1 *7 21 35 35 21 7 1 DIVISIBLE BY THE FIRST 1 8 28 56 70 56 28 8 1 ELEMENT. 1 9 36 84126126 84 36 9 1

2

PROBABILITY/COMBINATIONS

PASCAL’S TRIANGLE CAN BE USED IN PROBABILITY COMBINATIONS.

LET’S SAY THAT YOU HAVE FIVE HATS ON A RACK, AND YOU WANT TO KNOW HOW MANY DIFFERENT WAYS YOU CAN PICK TWO OF THEM TO WEAR.

1

Remember Combinations?

SO THE QUESTION IS “HOW MANY DIFFERENT WAYS CAN YOU PICK TWO OBJECTS FROM A SET OF FIVE OBJECTS….”

THE ANSWER IS 10. THIS IS THE SECOND NUMBER IN THE

FIFTH ROW. IT IS EXPRESSED AS 5:2, OR FIVE CHOOSE TWO.

PROBABILITY/COMBINATIONS

ROW O 1 ROW 1 1 1 ROW 2 1 2 1 ROW 3 1 3 3 1 ROW 4 1 4 6 4 1 ROW 5 ----------- 1 5 (10) 10 5 1 ROW 6 1 6 15 20 15 6 1ROW 7 1 7 21 35 35 21 7 1

2

PROBABILITY/COMBINATIONS

HOW MANY COMBINATIONS OF THREE LETTERS CAN YOU MAKE FROM THE WORD FOOTBALL?

USING THE TRIANGLE YOU WOULD EXPRESS THIS AS 8:3, OR EIGHT CHOOSE THREE.

What is the answer?

3

PROBABILITY/COMBINATIONS

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 ROW 8-----1 8 28 (56) 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1

4