Welcome to the Workshop Presentation Robert Noyce Teacher Scholarship Program Conference Renaissance Washington DC Hotel July 8, 2011 Workshop Session IV Meeting Room # 6 10:35 am to 11:50 am Exploring the World of Irrational Numbers Dr. Viji K. Sundar Professor of Mathematics California State University, Stanislaus [email protected] God made numbers; the rest is the work of man. Mathematics is Beautiful. Mathematics is Beautiful? Mathematics is Beautiful!! Pre-requisites Basic Set Theory Equal sets and Equivalent Sets One to one correspondence Finite and Infinite Sets Countable and Uncountable Sets A set is infinite it it is NOT finite. Infinite set equivalent to one of its subsets. More Ideas to Grow on … Let us recall what we know Let N be the set of Natural Numbers Let E be the set of Even Numbers Let O be the set of odd Numbers N = E U O Which set is ‘bigger?’ N or E N or O … Justify. Let Z be the set of Integers Let Q be the set of Rational Numbers Let Ir be the set of Irrational Numbers Let R be the set of Real Numbers Which is ’ bigger or larger’ has ‘more’ members ? 6 Natural Numbers Whole Numbers Integers Rational Numbers Irrational Numbers Real Numbers Cardinality of Numbers Cardinality - the number of members in a set. Cardinality can be finite or infinite. Cardinality of ‘number of heads in this room - 30’? Cardinality of Z is denumerably infinite. Z ~ Q Cardinality of Q is denumerably infinite. Z ~ Q Cardinality of total grains of sand in beaches is ? Z equivalent to Q Z ~ Q Density of Rational Numbers that between any two rational numbers we can insert a rational number. Can you find a rational number between 2 and 3? Can you find 5 rational numbers between 2 and 3? This means, between any two rational numbers there exist infinite number of rational numbers. Isn't it amazing? can also be written as 0.5 can also be written as 0.666666666... can also be written as 0.38181818... can also be written as 0.62855421687… the decimals will repeat after 41 digits Be careful when using your calculator to determine if a decimal number is irrational. The calculator may not be displaying enough digits to show you the repeating decimals, as was seen in the last example above. What is an irrational number? What is a rational number? A number that can be expressed as a ratio of two integers where the denominator Is not zero… Can be expressed as a fraction a/b where b is not zero. I want to know… What is an irrational number? A number that can not be expressed as a/b. A Real Number That is not a rational number is an irrational number !!!!! Irrational Numbers Every Real Number has a decimal representation For Rational numbers this will Either terminate or Non-terminating repeating When it is Non-terminating Non repeating … it is Irrational Where ARE All of Those Irrationals? Pi is defined as the (constant) ratio of the circumference to the diameter in any circle. the circumference and diameter of every circle are known to be related by …. is bigger than 3 but less than 4 Hoax or Truth - Do your own search π = 3 Indiana Bill is the popular name for bill #246 of the 1897 π = 3.2 Oklahoma legislators had passed a law making pi equal to 3.0. Congresswoman Martha Roby (R-Ala.) is sponsoring HR 205, The Geometric Simplification Act, declaring the Euclidean mathematical constant of pi to be precisely 3. Motivation …. Squaring a Circle? Historically speaking .. Check this out The Bible says Pi = 3. A little known verse of the Bible reads: " And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about.” (I Kings 7, 23) C = 30 cubits height = radius = 5 30 cubits Lives intimately with the circle In recent years, the computation of the expansion of pi has assumed the role of a standard test of computer integrity. The number has been the subject of a great deal of mathematical (and popular) folklore. It's been worshipped, maligned, and misunderstood. Overestimated, underestimated, and legislated. The Pi video How To Transform The Number Pi Into A Song by Michael John Blake To celebrate Pi Day — the 14th day of the third month — Canadian musician Michael Blake has set the mathematical constant to a tune." The idea just hit me one day — what would happen if I found some kind of equation or formula that I could transfer to music?" he told news.com.au."I thought about the Fibonacci sequence, because I liked the sound of that, but it didn't really work in my mind so the next one I tried was Pi."Blake's music video on YouTube explains how each of the eight notes in any major scale can be assigned a numerical value — C being one, D two and so on. Euler's number e The number e is irrational. Furthermore, it is transcendental (All rational numbers are algebraic) Euler's number That is it is not a root of a polynomial with rational coefficients . All transcendental numbers are irrational. Only few are known as it is difficult to prove that a number is transcendental. So you have a lot of problems to solve! e … base of the Natural Logarithms 10 … base of common algorithm e Pi is a famous irrational number. People have calculated Pi to over one million decimal places … no pattern. 3.1415926535897932384626433832795 (and more ...) The number e (Euler's Number) is another famous irrational number. The first few digits look like this: 2.7182818284590452353602874713527 (and more ...) Phi the Golden Ratio is an irrational number. 1.61803398874989484820... (and more ...) Many square roots, cube roots, etc are also irrational numbers. √3 1.7320508075688772935274463415…( more ...) Euler's number e e … base of the Natural Logarithms 10 … base of common algorithm The irrational e is : a number for algebra 2 students defines trig functions for Pre Calculus students a limit by calculus students But ‘e’ has a role in financial calculus and population growth. The compound-interest problem Jacob Bernoulli discovered this constant by studying a question about compound interest. Ref: khanacademy.com Play this video http://www.khanacademy.org/video/compound-interest-and-e-part--3?playlist=Finance The number e and The Rule of 70 The Rule of 70 is useful for financial as well as demographic analysis. It states that to find the doubling time of a quantity growing at a given annual percentage rate, divide the percentage number into 70 to obtain the approximate number of years required to double. For example, at a 10% annual growth rate, doubling time is 70 / 10 = 7 years. The number e and The Rule of 70 Similarly, to get the annual growth rate, divide 70 by the doubling time. For example, 70 / 14 years doubling time = 5, or a 5% annual growth rate. Where is the Math? Mathematics Behind the Rule of 70 The use of natural logs arises from integrating the basic differential equation for exponential growth: dN/dt = rN, over the period from t=0 to t = the time period in question, where N is the quantity growing and r is the growth rate. The integral of that equation is: The integral of that equation is: N(t) = N(0) x ert where N(t) quantity after t intervals have elapsed, N(0) is the initial value of the quantity, r is the average growth rate over the interval in question, t is the number of intervals. The golden section is a line segment divided according to the golden ratio: The total length a + b is to the length of the longer segment a as the length of a is to the length of the shorter segment b. Video for the Golden Ratio Movie Time http://www.youtube.com/watch?v=-6w9VYbptPQf Pi is a famous irrational number. People have calculated Pi to over one million decimal places and still there is no pattern. 3.1415926535897932384626433832795 (and more ...) The number e (Euler's Number) is another famous irrational number. The first few digits look like this: 2.7182818284590452353602874713527 (and more ...) Phi the Golden Ratio is an irrational number. 1.61803398874989484820... (and more ...) Many square roots, cube roots, etc are also irrational numbers. √3 1.7320508075688772935274463415059 (etc) √99 9.9498743710661995473447982100121 (etc) That is all folks! That is the end of the tour for now!! This is just the beginning of your journey into the wondrous world of Irrational Numbers! There are innumerable Irrational Numbers waiting to be found and classified! May your future be rich with discoveries!!!