Pascal s Triangle Introduction!

Transcription

1 Math 0 Section 2A! Page! 209 Eitel Section 2A Lecture Pascal s Triangle Introduction! A Rich Source of Number Patterns Many interesting number patterns can be found in Pascal's Triangle. This pattern was named after the French mathematician Blaise Pascal (2-2) who brought the triangle to the attention of Western mathematicians. it was known as early as 00 in China, where it was known as the "Chinese Triangle". The first rows of Pascal s Triangle To build the triangle, start with a "" at the top as the first row, The second row has the number written twice centered below the first. The rd row starts and ends with a. The middle number is found by adding the 2 numbers directly above it. The 4th row starts and ends with a. The two middle numbers are found by adding the 2 numbers directly above them. 2 2 Row Row 4

2 Math 0 Section 2A! Page 2! 209 Eitel Each row starts and ends with a. Each number between the ends is found by adding the two numbers directly above them. 2 4 = + 4 Row Row = 4 + The first rows of Pascal s Triangle

3 Math 0 Section 2A! Page! 209 Eitel The first rows of Pascal s Triangle written without the squares Pascal s Triangle is symmetric about the center The right and left sides of the line of symmetry are reflections of each other

4 Figurate Numbers A Figurate Number is a number that can be represented by a regular geometrical arrangement of equally spaced points. Figurate numbers are most commonly expressed in the form of regular triangles, squares, pentagons, hexagons, etc. One of the fist persons to develop this topic was Pythagoras. Pythagoras was a Greek mathematician (c 0-49 bc) and is often even titled as the father of numbers. Aristotle, another famous mathematician, wrote that Pythagoras thought that all things were numbers. He believed that everything was related to mathematics and that numbers were the ultimate reality and, through mathematics, everything could be predicted and measured in rhythmic patterns. Many of his first discoveries about number patterns were based on the shapes of figures he could make with pebbles. By recording the number of sides a figure had and the number of pebbles along a given side, he found patterns for the figures. In this manner, the figurate numbers were discovered. The first Counting Numbers 2 4 The first Triangular Numbers 0 The first Square Numbers The first 4 Tetrahedral Numbers Math 0 Section 2A! Page 4! 209 Eitel

5 Math 0 Section 2A! Page! 209 Eitel Patterns in Pascal s Triangle Each Diagonal of Pascal s Triangle is a figurate number. Each Diagonal of Pascal s Triangle is represents by a polynomial function Diagonal Constant f(n) = Diagonal 2 Counting Numbers f(n) = n + Diagonal Triangular Numbers f(n) = n (n + ) / 2 Diagonal Tetrahedral Numbers f(n) = n (n+) (n+2) / Diagonal Pentatope Numbers f(n) = n (n+) (n+2) (n + ) / 24 Diagonal Hexateron Number f(n) = n (n+) (n+2) (n + ) ( n +4) ) / 20

6 Math 0 Section 2A! Page! 209 Eitel Downward and Inward Sums Start with any of the cells on the outside edge of the triangle, a cell with a, and then travel down the diagonal toward the interior of the triangle as far as you like, At some cell stop going down that diagonal and change the direction for the last cell. The sum of the numbers in the diagonal will be the number in the offset cell

7 Math 0 Section 2A! Page! 209 Eitel 2 Days of Christmas Gifts In the Christmas song, "The 2 Days of Christmas", The gifts are: Day : A partridge in a pear tree, Day 2: Two turtle doves, Day : Three french hens, Day 4: Four calling birds, Day : Five gold rings, Day : Six geese a-laying Day : Seven swans a-swimming, Day : Eight maids a-milking, Day 9: Nine ladies dancing, Day 0: Ten lords a-leaping, Day : Eleven pipers piping, Day 2: Twelve drummers drumming. First we need to figure out how many gifts we receive on any given day. It is a cumulative song, so you receive one partridge in a pear tree on the first day, then on the second day you receive two turtle doves and another partridge in a pear tree, and so on for twelve days. So the sequence for each day is: + 2 = = = =... The number of gifts we receive on any given day,,,, 0,,... are the numbers in the rd diagonal in Pascal's triangle. The rd diagonal contains the "triangular numbers" which are each the ( ) sum of the first N integers. The expression for the is T n = n n + 2 We now need to find the total number of gifts after n days. This is the sum of the numbers of gifts given each day = = = = We now need to find the total number of gifts after n days are the number sin the 4th diagonal. The 4th diagonal contains the "tetrahedral numbers"which are each the sum of the first N triangular numbers. The expression for the is T n = n ( n + )(n + 2) The grand total gift count is the 2th tetrahedral number: 4.

8 Math 0 Section 2A! Page! 209 Eitel 2 Days of Christmas Gifts 2 Diagonal Total gifts each day Diagonal 4 Total number of Gifts after n days Total number of Gifts after 2 days The 2th tetrahedral number The expression for the 2th tetrahedral number is T n = n ( n + )(n + 2) The 2th tetrahedral number is 2( 2 +)(2 + 2) T 2 = 2 T 2 = ( )(4) T 2 = 4

9 Math 0 Section 2A! Page 9! 209 Eitel Square Numbers The sum of any 2 consecutive numbers in Diagonal are square numbers Diagonal 2 + = 4 = = = = = = 4 = Diagonal 2 + = 9 = = 2 = = 49 = = =

10 Math 0 Section 2A! Page 0! 209 Eitel The Sums of the Squares in any Row Let the number of a row be n. The sums of the squares of all the numbers in row n will be found in the center of the (2n )th row An alternate way to see this is that if you drew the diagonals from the s at both ends of the row downward and inward they would meet at a center number that would be the sums of the squares of the row you stared with. 2 Row Row Row Row Row If n = The sums of the squares of the numbers in row is = = If n = than the total will be the center number in the (2n )th row 2 The number in the center of the th row is Row n = The sums of the squares of the numbers in row is = = 22 ( ) = = Row If n = than the total will be the center number in the (2n )th row 2( ) = 2 = Row The number in the center of the th row is 22

11 Powers of 2 The sum of all the numbers in any row is a power of 2. For any row n the sum of the number in row n is 2 n Row Row 2 Row Row 4 Row Row Row Row Row Sum = 2 = 2 0 = Sum = 2 2 = 2 = 2 Sum = 2 = 2 2 = 4 Sum = 2 4 = 2 = Sum = 2 = 2 4 = Sum = 2 = 2 = 2 Sum = 2 = 2 = 4 Sum = 2 = 2 = Sum = 2 9 = 2 = 2 Math 0 Section 2A! Page! 209 Eitel

12 Math 0 Section 2A! Page 2! 209 Eitel Mersenne Numbers A Mersenne Number is any number the can be expressed in the form that falls short of being 2 to the power of n. That is for some values of 2 n Example! Example 2! Example If n = than 2 = 2 =! If n = 2 than 2 2 = 4 = Mersenne Numbers and Pascal s Triangle! If n = than 2 = If you add up every single number in the first n rows, you ll get the nth Mersenne Number Example! Example 2! Example Sum of the first rows! Sum of the first 2 rows`! Sum of the first rows = 2! + 2 = = 2 2! = = 2 Example 4! Example! Example Sum of the first 4 rows! Sum of the first rows`! Sum of the first rows = = 2 4! = = 2! = = 2 Mersenne Prime Numbers A prime number is any number that can only be divided by and itself. can only be divided by and so prime. can be divided by,2, and so is not prime. Some of the Mersenne Numbers are prime. The search for Mersenne Prime Numbers is one of the most computationally intensive and actively pursued areas of advanced and distributed computing. = Blaise Pascal, Marin Mersenne and Pierre de Fermat Pierre Fermat, was one of the truly great figures in the history of mathematics, often stating his results without proof in his letters. Pascal, Mersenne and Fermat all lived in Europe in the 00 s. They often shared thoughts with each other through letters, many of which have survived and give us insight into the thinking of these great mathematicians. In, Fermat wrote to Mersenne implying that he had solved what is perhaps the most beautiful problem of all arithmetic, finding the precise sum of powers in an arithmetic progression, no matter what the power. His proof was based on the patterns he had found in Pascal s Triangle.

13 Math 0 Section 2A! Page! 209 Eitel Fibonacci's Sequence If you take the sum of the numbers in each shallow diagonal the sums represent the numbers in the Fibonacci sequence. = 2 = = 2 = = = = = 2 = = =

14 Math 0 Section 2A! Page 4! 209 Eitel Fibonacci's Sequence If you take the sum of the numbers in each shallow diagonal the sums represent the numbers in the Fibonacci sequence. This example uses hexagons to outline the numbers in Pascal s Triangle. It makes it easy to follow the diagonals used to find the numbers used as the sum in each diagonal Leonardo Fibonacci (c. c. 20) was an Italian mathematician considered by many to be the most talented western mathematician of the Middle Ages. He introduced and popularized the Hindu- Arabic number system (also called the decimal system) to Europe. He contributed greatly to number theory, and during his life published many important texts. He is also known for the Fibonacci Series.

15 The numbers each row represents the digits in the power of. Row Row 2 Row Row 4 Row Row Row Row Number = = 0 Number = = Number = 2 = 2 Number = = Number = 44 = 4 Number =,0 = Number =,, = Number = 9,4, = Row 2 is = Row is 2 = 2 Row 4 is = Row is 44 = 4 If a number in a cell with more than digit you cary the left most digit to the next cell on the left. Row The st number from the right is. The 2nd number from the right is. The rd number from the right is 0. Keep the 0 and carry the to the 0 on left. 0 The 4th number from the right is 0 + =. Keep the and carry the. 0 The th number from the right is + = with the carry. 0 The th number from the right is. 0. Row is,0 = Row The st number from the right is. The 2nd number from the right is. The rd number from the right is. Keep the and carry the. The 4th number from the right is 20 + = 2. Keep the and carry the 2. The th number from the right is + 2 =. Keep the and carry the. The th number from the right is + =. The th number from the right is.. Row is,, = Math 0 Section 2A! Page! 209 Eitel

16 Math 0 Section 2A! Page! 209 Eitel The prime numbers greater than in diagonal divide evenly into all the numbers in that row that are greater than. Diagonal 2 Row Row Row Row Row 2 The first prime number greater than in Diagonal is in row. The prime number 2 divides 2. The next prime number greater than in Diagonal is in row 4. The prime number divides. The next prime number greater than in Diagonal is in row. The prime number divides and 0. The next prime number greater than in Diagonal is in row. The prime number divides, 2 and The next prime number greater than in Diagonal is in row 2. The prime number divides,,, 0 and 4 The next prime number greater than in Diagonal will be in row 4. The prime number will be and will divide the other numbers >

17 Fractals If you shade all the even numbers in one color and the odd numbers in another color you will get a fractal design. This is the recursive of Sierpinski's Triangle. A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Fractals are images of dynamic systems the pictures of Chaos. Fractal patterns are extremely familiar, since nature is full of fractals. For instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc. Abstract fractals can be generated by a computer calculating a simple equation over and over. Math 0 Section 2A! Page! 209 Eitel

18 Math 0 Section 2A! Page! 209 Eitel The Chinese used the Triangle long before Pascal did. This drawing is entitled "The Old Method Chart of the Seven Multiplying Squares". It is from the front of Chu Shi-Chieh's book "Ssu Yuan Yü Chien" (Precious Mirror of the Four Elements), written in AD 0. That was more than 00 years before Pascal., In the book it says the triangle was known about more than two centuries before that.

19 Math 0 Section 2A! Page 9! 209 Eitel Binomial Expansion When expanding a binomial expression of the form (x + y) n the coefficients of each term in the polynomial can be found in Pascal's triangle. If you are expanding (x + y) 4 you would look at the th row which is 4 4. These are the coefficients of the terms in the expansion. (x + y) 4 = x 4 + 4x y + x 2 y 2 + 4x y +y 4 The coefficients of each term in the expansion (x + y) n will be found in the n+ th row. (x + y) 0 = (x + y) = x +y 2 (x + y) 2 = x 2 + 2xy + y 2 (x + y) = x + x 2 y + x y 2 + y (x + y) 4 = x 4 + 4x y + x 2 y 2 + 4x y + y 4 (x + y) 4 = x + x 4 y +0x y 2 +0x 2 y + x y 4 + y ( x + y) 0 = ( x + y) = x+y 2 ( x + y) 2 = x 2 + 2xy +y 2 ( x + y) = x + x 2 y + x y 2 +y 4 4 ( x + y) 4 = x 4 + 4x y + x 2 y 2 + 4x y +y 4

20 The Quincunx or Galton Board and Pascal s Triangle The Galton Board consists of a vertical board with offset rows of pegs. Marbles are dropped from the top,and bounce either left or right as they hit the pegs. Eventually, they are land in slots at the bottom. The height of the beads accumulated in each of the slots will eventually approximate a bell shape curve and forms what is known as the Gaussian distribution (Carl Friedrich Gauss, -). This bell shaped curve is called a Normal Curve in Statistics. he bell curve, is important in statistics and probability theory. It is used in the natural and social sciences to represent random variables. Overlaying Pascal s Triangle onto the pins shows the number of different paths that can be taken to land in each slot Pascal's Marble Run - this example has rows of pins and collecting slots Overlaying Pascal s Triangle onto the pins shows the number of different paths that can be taken to land in each slot At first it looks like the balls will end up at the bottom land in completely random slots and you would be correct. But after a large number of balls are dropped they pile up into a nice pattern. This pattern is called the Normal Distribution and is an important tool in the study of Statistics and Probability NOTE: A link on the website will lead to a video about this device and its application to Pascals Triangle Math 0 Section 2A! Page 20! 209 Eitel

21 Math 0 Section 2A! Page 2! 209 Eitel BAD That s not a good sign 0 penny packing puzzle d print