Transcript PASCAL’S TRIANGLE

PASCAL’S TRIANGLE
PASCAL’S TRIANGLE
* ABOUT THE MAN
* CONSTRUCTING THE TRIANGLE
* PATTERNS IN THE TRIANGLE
* PROBABILITY AND THE TRIANGLE
Blaise Pascal
JUNE 19,1623-AUGUST 19, 1662
*French religious philosopher, physicist, and
mathematician.
*“Thoughts on Religion”. (1655)
*Syringe, and Pascal’s Law. (1647-1654)
*First Digital Calculator. (1642-1644)
*Modern Theory of Probability/Pierre de Fermat. (1654)
*Chinese mathematician Yanghui, 500 years
before Pascal; Eleventh century Persian
mathematician and poet Omar Khayam.
*Pascal was first to discover the importance of the
patterns.
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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.
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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
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
ONE OF THE POLYGONAL NUMBERS
FOUND IN THE SECOND DIAGONAL
BEGINNING AT THE SECOND ROW.
1
2
THE TRIANGULAR NUMBERS
1
1 1
{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}
* * *
*
{15}
* *
* * *
* * * *
* * * * *
THE SQUARE NUMBERS
ONE OF THE POLYGONAL NUMBERS
FOUND IN THE SECOND DIAGONAL
BEGINNING AT THE SECOND ROW.
THIS NUMBER IS THE SUM OF THE
SUCCESSIVE NUMBERS IN THE
DIAGONAL.
1
2
THE SQUARE NUMBERS
1
1 1
(1) 2 1
* *
1 (3) 3 1 * *
14 6 4 1
1 5 10 10 5 1
1 6 15 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 *
1 615 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 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
* * * *
* * * *
* * * *
* * * *
* * * *
*
*
*
*
*
1
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 ADJECENT
DIAGONAL BELOW IT.
2
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
THE SUM OF THE ROWS
1
THE SUM OF THE NUMBERS IN ANY
ROW IS EQUAL TO 2 TO THE “Nth”
POWER ( “N” IS THE NUMBER OF THE
ROW).
2
THE SUM OF THE ROWS
2 TO THE 0TH POWER=1
2 TO THE 1ST POWER=2
2 TO THE 2ND POWER=4
2 TO THE 3RD POWER=8
2 TO THE 4TH POWER=16
2 TO THE 5TH POWER=32
2 TO THE 6TH POWER=64
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
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.
2
PRIME NUMBERS
I
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 84126126 84 36 9 1
THE FIRST ELEMENTS IN ROWS
THREE, FIVE, AND SEVEN
ARE PRIME NUMBERS.
NOTICE THAT THE OTHER
NUMBERS ON THESE ROWS,
EXCEPT THE ONES, ARE
DIVISIBLE BY THE FIRST
ELEMENT.
1
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. IT DOESN’T MATTER TO YOU
WHICH HAT IS ON TOP. IT JUST MATTERS WHICH
TWO HATS YOU PICK. 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.
2
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 1
ROW 7
1 7 21 35 35 21 7 1
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. THE ANSWER IS 56. THIS IS
THE THIRD NUMBER IN THE EIGHTH
ROW.
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
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PASCAL’S TRIANGLE