What is happening to the probability as the number of coins gets larger? Draw a histogram of the 10 th row of Pascal's triangle, that is, a bar chart, where each column on the row numbered 10 is shown as a bar whose height is the Pascal's triangle number.
Try it again for tow 20 if you can or use a Spreadsheet on your computer. The shape that you get as the row increases is called a Bell curve since it looks like a bell cut in half.
It has many uses in Statistics and is a very important shape. Make a Galton Quincunx. This is a device with lots of nails put in a regular hexagon arrangement. Its name derives from the Latin word quincunx for the X-like shape of the spots on the 5-face of a dice: Hopper for balls balls fall onto nails with an equal chance of bouncing to left or right each time balls collect in hoppers The whole board is tilted forward slightly so that the top is raised off the table a little.
When small balls are poured onto the network of nails at the top, they fall through, bouncing either to the right or to the left and so hit another nail on the row below. Eventually they fall off the bottom row of nails and are caught in containers. If you have a lot of nails and a lot of little balls good sources for these are small steel ball-bearings from a bicycle shop or ping-pong balls for a large version or even dried peas or other cheap round seeds from the supermarket then they end up forming a shape in the containers that is very much like the Bell curve of the previous exploration.
You will need to space the nails so they are as far apart as about one and a half times the width of the balls you are using. Let's see how the curve of the last two explorations, the Bell curve might actually occur in some real data sets. Measure the height of each person in your class and plot a graph similar to the containers above, labelled with heights to the nearest centimetre, each container containing one ball for each person with that height.
What shape do you get? Try adding in the results from other classes to get one big graph. This makes a good practical demonstration for a Science Fair or Parents' Exhibition or Open Day at your school or college.
Measure the height of each person who passes your display and "add a ball" to the container which represents their height. What shape do you get at the end of the day? What else could you measure? The weight of each person to the nearest pound or nearest grams; their age last birthday; but remember some people do not like disclosing their age or knowing too accurately their own weight! In the USA this might be up to several thousands! Do all of these give the Bell curve for large samples?
If not, why do you think some do and some don't? Can you decide beforehand which will give the Bell curve and which won't? If a distribution is not a Bell curve, what shape do you think it will be? How can mathematics help? Write out the first few powers of Do they remind you of Pascal's triangle? Why does the Pascal's triangle pattern break down after the first few powers? To finish, let's return to a human family tree. Suppose that the probability of each child being male is exactly 0.
If a couple have 2 children, what are the four possible sequences of children they can have? What is it if they have 3 children? In what proportion of the couples that have 3 children will all 3 children be girls?
Suppose a couple have 4 children, what is the probability now that all 4 will be girls? A Generating Function for the Fibonacci Numbers To find the value of the decimal fraction we look at a generalisation, replacing 10 by x. The decimal fraction 0. Now here is the interesting part of the technique! We have done this so that each Fibonacci number in P x is aligned with the two previous Fibonacci numbers.
Since the sum of the two previous numbers always equals the next in the Fibonacci series, then, when we take them away, the result will be zero - the terms will vanish! Finding such a polynomial for other series of numbers is an important part of modern mathematics and has many applications. He used it exactly as we have done here for the Fibonacci numbers!
An exact Fractions Calculator The decimal expansions of fractions here in the Calculator are produced to any number of decimal places , unlike an ordinary calculator which only gives perhaps a maximum of 15 decimal places.
Even numbers 2 digits each 00 02 04 Even numbers 3 digits each Odd numbers 2 digits each 01 03 Odd numbers 3 digits each Powers of 2 2 digits each 01 02 04 Powers of 2 3 digits each Powers of 3 2 digits each 01 03 09 Powers of 3 3 digits each Squares 2 digits each 00 01 04 09 Triangular 2 digits each 01 03 06 19 Fibonacci single digits 0 1 1 2 3 Fibonacci 2 digits each 00 01 01 02 03 Fibonacci 3 digits each Fibonacci backwards single digits Can you find exact fractions for the following where all continue with the Fibonacci series terms?
Calvin Long solves the general problem for all Fibonacci-type sequences i. This article examines all the decimal fractions where the terms are F a , F 2a , F 3a taken k digits at a time in the decimal fraction. The whole collection of articles is now available in book form by mail order from The Fibonacci Association. It is readable and not too technical.
There is also a list of formulae for all kinds of generating functions, which, if we substitute a power of 10 for x, will give a large collection of fractions whose decimal expansion is , for example: the Lucas Numbers see this page at this site e. Scott's Fibonacci Scrapbook , Allan Scott in Fibonacci Quarterly vol 6 number 2, April , page is a follow-up article to the one above, extending the generating functions to Lucas cubes and Fibonacci fourth and fifth powers.
Note there are several corrections to these equations on page 70 of vol 6 number 3 June Alice : Choose any two numbers you like, Bill, but not too big as you're going to have to do some adding yourself. Write them as if you are going to add them up and I'll, of course, be looking the other way! Bill : OK, I've done that.
Bill chooses 16 and 21 and writes them one under the other: 16 21 Alice : Now add the first to the second and write the sum underneath to make the third entry in the column.
Bill : I don't think I'll need my calculator just yet Ok, I've done that. Keep on doing this, adding the number you have just written to the number before it and putting the new sum underneath. Stop when you have 10 numbers written down and draw a line under the tenth. There is a sound of lots of buttons being tapped on Bill's calculator! Bill : OK, the ten numbers are ready. Bills column now looks like this: 16 21 37 58 95 Alice : Now I'll turn round and look at your numbers and write the sum of all ten numbers straight away!
She turns round and almost immediately writes underneath: Bill taps away again on his calculator and is amazed that Alice got it right in so short a time [gasp! The sum of all ten numbers is just eleven times the fourth number from the bottom.
Also, Alice knows the quick method of multiplying a number by eleven. The fourth number from the bottom is , and there is the quick and easy method of multiplying numbers by 11 that you can easily do in your head: Starting at the right, just copy the last digit of the number as the last digit of your product.
Here the last digit of is 8 so the product also ends with 8 which Alice writes down If ever you get a sum bigger than 10, then write down the units digit of the sum and remember to carry anything over into your next pair to add. Alice sees that the left digit is 2 which, because there is nothing being carried from the previous pair, becomes the left-hand digit of the sum. You can see how it works using algebra and by starting with A and B as the two numbers that Bill chooses.
What does he write next? Now look at the fourth number up from the bottom. What is it? So the trick works by a special property of adding up exactly ten numbers from a Fibonacci-like sequence and will work for any two starting values A and B! Perhaps you noticed that the multiples of A and B were the Fibonacci numbers? This is part of a more general pattern which is the first investigation of several to spot new patterns in the Fibonacci sequence in the next section.
This article introduces the above trick and generalises it. Practice here with "Bill" Here is your very own "Bill" to practice on. Click on the "Show Bill's list" button and he will think of two numbers and show you his list. Enter your answer in the Sum: box Click on Am I right? Click on Show Bill's list as often as you like to get a new list. This article introduces the above trick and generalises it to sums of more numbers.
Another generalisation. Another Number Pattern Dave Wood has found another number pattern that we can prove using the same method. It looks like the differences seem to be 'copying' the Fibonacci series in the tens and in the units columns. If we continue the investigation we have: f 13 -f 8 is - 21 which is or 21 tens and 2; f 14 -f 9 is - 34 which is or 34 tens and 3; f 15 -f 10 is - 55 which is or 55 tens and 5; f 16 -f 11 is - 89 which is or 89 tens and 8; f 17 -f 12 is - which is or tens and 13; From this point on, we have to borrow a ten in order to make the 'units' have the 2 digits needed for the next Fibonacci number.
Later we shall have to 'borrow' more, but the pattern still seems to hold. You will see that some are just magnifications of smaller ones where all the sides have been doubled, or trebled for example.
The others are "new" and are usually called primitive Pythagorean triangles. Any Pythagorean triangle is either primitive or a multiple of a primitive and this is shown in the table above. Using the Fibonacci Numbers to make Pythagorean Triangles There is an easy way to generate Pythagorean triangles using 4 Fibonacci numbers. Take, for example, the 4 Fibonacci numbers: 1, 2, 3, 5 Let's call the first two a and b.
This is one side, s, of the Pythagorean Triangle. Multiply together the two outer numbers here 1 and 5 giving 5. This is the second side, t, of the Pythagorean triangle.
This is the third side, h, of the Pythagorean triangle. We have generated the 12, 5,13 Pythagorean triangle, or, putting the sides in order, the 5, 12, 13 triangle this time. Try it with 2, 3, 5 and 8 and check that you get the Pythagorean triangle: 30, 16, Is this one primitive? In fact, this process works for any two numbers a and b , not just Fibonacci numbers.
The third and fourth numbers are found using the Fibonacci rule : add the latest two values to get the next. Four such numbers are part of a generalised Fibonacci series which we could continue for as long as we liked, just as we did for the real Fibonacci series.
All primitive Pythagorean triangles can be generated in this way by choosing suitable starting numbers a and b but not all non-primitive ones are!
For the reason for this and lots more on Pythagorean triangles see my Pythagorean Triangles page. This article explores many ways of introducing the Fibonacci numbers in class starting from the Pythagorean triples, with an extensive Appendix of references useful for the teacher and comparing different approaches.
Highly recommended! Fibonacci Numbers as the sides of Pythagorean Triangles Can we form a triangle not necessarily right-angled from three distinct Fibonacci numbers? No, because of the following condition that must be true for any and all triangles: in any triangle the longest side must be shorter than the sum of the other two sides This is called the Triangle Inequality.
Since three consecutive Fibonacci numbers already have the third number equal to the sum of the other two, then the Triangle Inequality fails. Or, if you prefer, the two shorter sides collapse onto the third to form a straight line when you try to construct a triangle from these numbers. If the smallest side is smaller, that makes it worse, as it does if the longer side gets longer!
So we have No triangle has sides which are three distinct Fibonacci numbers So can we have a triangle with three Fibonacci numbers as sides, but with two sides equal? Yes: 3,3,5 will do. No Pythagorean triangle has two equal sides. If we ask for just two sides which are Fibonacci numbers, the third being any whole number, then there are at least two Pythagorean triangles with Fibonacci numbers on two sides: 3 , 4, 5 and 5 , 12, 13 It is still an unsolved problem as to whether there are any more right-angled Pythagorean triangles with just two Fibonacci numbers as sides.
Can we have any other Pythagorean triangles with a Fibonacci number as the hypotenuse the longest side? Weisstein, Eric W. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Walk through homework problems step-by-step from beginning to end.
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Update all multiples of p. Now traverse through the range. Sqrt n ;. Pow y, 2. Previous Program to find Nth term in the series 0, 2, 1, 3, 1, 5, 2, 7, 3,…. Recommended Articles. Count numbers in a given range having prime and non-prime digits at prime and non-prime positions respectively.
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