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Transcript of Pascal's Triangle
Infinite (Tanton) Most Basic Patterns of Pascal's Triangle 1s Counting #s Triangle #s Triangle Numbers 1 1+2 1 3 1+2+3 6 1+2+3+4 10 ... ... -Row Sums Row 0 Row 1 Row 2 Works Cited "Binomial theorem." Encyclopædia Britannica. Encyclopædia Britannica Online. Encyclopædia Britannica, 2011. Web. 26 Apr. 2011. <http:/www.brittanica.com/ EBchecked/topic/65749/binomial-theorem>
"Blaise Pascal." Encyclopædia Britannica. Encyclopædia Britannica Online. Encyclopædia Britannica, 2011. Web. 16 May. 2011. <http://www.britannica.com/EBchecked/ topic/445406/Blaise-Pascal>.
Daintith, John, and Richard Rennie. "binomial coefficient." Science Online. Facts On File, Inc. Web. 26 Apr. 2011. <http://www.fofweb.com/activelink2.asp?ItemID=WE40&SID =5&iPin=DMATH0149&SingleRecord=True>.
"Pascal’s triangle." Encyclopædia Britannica. Encyclopædia Britannica Online. Encyclopædia Britannica, 2011. Web. 26 Apr. 2011. <http://www.britannica.com/ EBchecked/topic/445453/Pascals-triangle>.
"Pascal's Triangle." World of Mathematics. Ed. Brigham Narins. Vol. 2. 2001. 473-4. Print.
Tanton, James. "Pascal's triangle." Science Online. Facts On File, Inc. Web. 26 Apr. 2011. <http://www.fofweb.com/activelink2.asp?ItemID=WE40&SID=5&iPin=EMATH0622&Si>
Weisstein, Eric W. "Pascal's Triangle." MathWorld. http://mathworld.wolfram.com/ PascalsTriangle.html
"Yang Hui." Encyclopædia Britannica. Encyclopædia Britannica Online. Encyclopædia Britannica, 2011. Web. 14 May. 2011. <http://www.britannica.com/EBchecked/ topic/1073781/Yang-Hui>. 1 3 3 1 1 (2 ) 0 2 (2 ) 1 4 (2 ) 2 8 (2 ) 3 16 (2 ) 4 32 (2 ) 5 32 (2 ) 6 x2 x2 x2 x2 x2 x2 Row 3 Row 4 Row 5 Row 6 Row n 2 n 1 digit 2 digits 3 digits 4 digits 5 digits 6 digits 7 digits n+1 digits http://www.fofweb.com/Science/MainDetailPrint.asp?iPin=SciIllus01608&LrgImg=1&WinType=Free http://sites.google.com/site/malachid/pascal-base.gif Blaise Pascal Chinese History More Early History Combination Problems How many ways can you pick items from a set of ? k n How many ways can you pick your two socks from the thirteen socks in your drawer? (Tanton) Examples How many ways can you pick a team of four students out of a class of 20? How many ways can you pick three hats out of five? k- 2
n- 13 k- 4
n- 20 k- 3
n- 5 Solving
Combination Problems The normal way to solve combination problems is to use the formula n!
(n-k)! k! Order doesn't matter.
Items can only be picked once. So, using the example of the hats, where k is 3 and n is 5, this formula would be (Tanton) 5!
(5-3)! 3! or 10 Using Pascal's Triangle We can find the number of combinations in the triangle. The number of combinations is in the place in the row. k n So, going back to the example of the hats, where k is 3 and n is 5, The answer should be in the 3rd place in row 5 Row 0 Row 1 Row 2 Row 3 Row 4 Row 5 Row 6 So, we can find the number of cominations by: Counting down down the total number of objects to choose from
Counting to the right the number of items that we want to choose. -Symmetrical -Three Diagonals What are binomial coefficients? The coefficients of the variables in a binomial expansion. A binomial expansion is the non-simplified form of writing the sum of two variables raised to a power. ex: The binomial expansion of (x+y) is
and the binomial expansion of (x+y) is 2 9 x +2xy+y 2 2 (Wolfram Alpha) The numbers multiplying the variables in these expressions are the binomial coefficients. x +2xy+y 2 2 1, 9, 36, 84, 126, 126, 84, 36, 9, 1 1, 2, 1 Binomial Coefficients in Pascal's Triangle 2 1, 9, 36, 84, 126, 126, 84, 36, 9, 1 1, 2, 1 x +2xy +y 2 (x+y) = 2 (x+y) = 9 Row 0- (x+y) Row 1- (x+y) Row 3- (x+y) Row 2- (x+y) Row 7- (x+y) Row 6- (x+y) Row 5- (x+y) Row 4- (x+y) Row 8- (x+y) Row 9- (x+y) 0 1 2 3 4 5 6 7 8 9 All of the binomial coefficients are found in Pascal's Triangle ... The earliest known representation of Pascal's Triangle was drawn in the 12th century in China by Yang Hui, a Chinese mathematician. The triangle was included in his book along with a geometric method for solving quadratic equations and many magic squares. Yang wrote in his book that he copied it from an even older book written by Jia Xian around 1050 AD. Jia Xian is thought to be the original discoverer of Pascal's Triangle, but Yang Hui popularized it. By the 13th century, using the triangle to find the binomial coefficients was already known as the "Old Method" in China In China, Pascal's Triangle is called the Yanghui triangle. ("Yang Hui") Yang Hui's representations of the triangle www.britannica.com/EBchecked/topic/445453/Pascals-triangle (The - symbol is the character for 1) There is also evidence of Persian astronomer Omar Khayyam studying the patterns of the binomial coefficients in the 11th century Also in the 11th century, Indian mathmaticians realized that the triangle represented their patterns of long and short sounds in poetry. ("Pascal's Triangle") ("Pascal's Triangle") Michael Stifel, a German mathmatician published this version of what was then known as the "Figurate Triangle" in 1544. http://milan.milanovic.org/math/english/fibo/fibo0.html Hockey Stick Pattern Multiples of Two in Pascal's Triangle Other Multiples Fibonacci Numbers Magic 11s Virtual Pascal's Triangle http://nlvm.usu.edu/en/nav/frames_asid_181_g_4_t_1.html?open=activities&from=search.html?qt=pascal%27s+triangle Going diagonally inwards from any edge of the triangle for any distance, the sum of the numbers that you crossed will equal the outer number below it. http://www.learner.org/courses/mathilluminated/images/units/2/hockeysticks.gif The sum of the numbers on the shallow diagonals form the fibonacci sequence. http://www.britannica.com/EBchecked/topic/445453/Pascals-triangle Coloring in the even numbers in Pascal's triangle reveals a pattern of smaller triangles. If we could extend Pascal's Triangle and this pattern infinitely, the shaded region would exactly form Serpinski's Triange, a fractal. http://library.thinkquest.org/TQ0312134/pascalproblems.html http://zeuscat.com/andrew/chaos/sierpinski.html Pascal was a French mathematition, scholar, physicist, religious philosopher, and writer in the 1600s. Pascal rediscovered the triangle and used it in his work on probability theory. He is the namesake of the triangle in the Western world because he was he popularized the triangle in Europe. http://sites.google.com/site/malachid/pascal-overlay-mod3.gif Highlighting the multiples of any number will result in triangles forming. The multiples of 4 http://sites.google.com/site/malachid/pascal-overlay-mod3.gif The multiples of 3 http://en.wikipedia.org/wiki/File:TrianguloPascal.jpg Pascal's drawing of the triangle. Each row is also a power of 11. 1 = 11 11 = 11 121 = 11 1331 = 11 14641 = 11 161051 = 11 1771561 = 11 0 1 2 3 4 5 6