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Irrational Numbers

Math Project: Quarter 2
by

Kaya Nighthood

on 15 February 2013

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Transcript of Irrational Numbers

Irrational Numbers Melani Zuckerman and Cara Krupnikoff-Salkin An irrational number is a number that cannot be expressed as a fraction. There is an unlimited amount of irrational numbers, and many of them have become famous. Some of these numbers are: pi-the ratio of the circumference to the diameter, tau- the ration of the circumference to the radius(two pi), phi-the golden ratio, e- the base of the natural logarithm, and the square root of two. Any non-repeating, non-terminating decimal is an irrational number. This means that the number has no pattern to it, and has endless possibilities of digits. Before a non terminating decimal can
be officially called a irrational number,
it must pass through many mathematical formulas. Firstly, and most importantly, it must be affirmed that it is not the solution to any two numbers. In other words, integers multiplied together should equal the possible irrational number. Phi The Golden Ratio Phi, also know as the golden ratio, the golden section, the divine proportion, the golden number... is a number that appears in almost everything.

Phi can be found in roses, mollusk shells, pyramids, and many famous paintings.

It is indeed, as Mario Livio puts it, "The world's most astonishing number." The first 100 digits of phi are:
1.61803398874989484820458683436563811772030917980576286213544862270526046281890244970720720418939113748475408807538689175212663386222353693179318006076672635 44333890865959395829056383226613199282902... Another thing that must be proven is that when mathematicians put the number in continued fraction form, it still has an infinite number of decimal places. In other words, it is fulfilling the definition of an irrational number; it cannot be put into a simple fraction form. How to find phi: There are many ways to find this magnificent number.

Phi is closely associated with Fibonacci numbers(a pattern of digits adding the two before to get the next one... 0,1,1,2,3,5,8,13,21,34,55,89...). Actually, one of the ways to find phi is to divide one Fibonacci number by it's previous one. The further in the sequence you go, the closer the number is to phi.

Another way to find phi is by finding the quantity of the square root of 5 minus 1, and then dividing that by 2. This is the simplest way to prove that phi is an irrational number.

Of course, there are many other ways to find phi, but these are the two well known ones. The History of Irrational Numbers: In around the 5th century(the time of the Greeks) Pythagoras and the Pythagoreans only believed in counting numbers and fractions. For them, there were no negative or irrational numbers, and everything was easy. Then, Hippasus, a Greek mathematician discovered that there were indeed irrational numbers.

The first irrational number discovered was the square root of 2. This was found when using the Pythagorean Theorem, both legs were made 1. If both legs were 1, then the hypotenuse's length would be the square root of 2. This was where they discovered irrationality.

They didn't have algebra back then, but they used an equation muck like the algebraic formula shown here. ... Math Project Quarter Two Math Project Quarter Two By, Math Project Quarter Two 3.1415926535897932384626433832795028841971693993551058209749445923078164062862089986280348 Fibonacci and Lucas Numbers The Fibonacci sequence- 1,1,2,3,5,8,13,21,34,55,89...
and the Lucas sequence- 1,3,4,7,11,18,29,47,76,123

Phi is related to these because the further along in the sequence you fo the reatio between a number and its previous one gets closer and closer to phi. This means that phi is found in a large variety of things, because these sequences do. For example, the reproducing of rabbits is done in these amounts. Also, a pineapple uses these numbers in its spirals. Of course, the Fibonacci numbers are also found in a logarithmic spiral, which shows just how much phi is used. PI The Golden Rectangle (and Other Golden Shapes) Pi is a well known irrational number, equivalent to the ratio of a circles circumference to its diameter. It is often used in mathematical areas such as algebra and advanced geometry. Pi (3.1415926535897932384… The Platonic Solids The Golden Triangle The Golden Rectangle The golden rectangle is a shape where the length divided by the width (or vice versa) makes phi (or a number approximated to phi). When drawing a square in the golden rectangle, the remainder will be a golden rectangle, which can be divided in to squares and golden rectangles infinitely. Similarly, the squares are at the lengths of adjacent Fibonacci numbers, and a golden spiral (and many other amazing patterns) can go through it evenly. The Pentagram The pentagram is a shape very closely related with phi. When a regular pentagon is drawn, and its points are connected, by the way of every other point, this shape will appear, and an unlimited amount of pentagrams can appear. One of the ways the pentagram is associated with phi so much is how if one dimension is divided by any other unequal length (or vice versa, depending on the lengths), it will become this miraculous number. This shape is an isosceles triangle with the proportions of one side to phi. Like the golden rectangle, it can be further divided into a continuous amount of golden triangles, and also follows the Fibonacci sequence (making the Fibonacci spiral). The Platonic Solids are 3-D shapes made of regular shaped faces. They are the:
1. Tetrahedron-A triangular based pyramid composed of equilateral triangle face. It has 6 faces.
2. Cube(Hexahedron)-A 6 faced rectangular prism with equal faces. This solid is the most known of these 5 solids.
3. Octahedron- A figure composed of eight triangular faces.
4. Dodecahedron- This figure is made of 12 octagonal faces.
Icosahedron- 20 triangular faces make the icosahedron. Tau Tau, (pi multiplied by 2,) is very much like the value of a circle's circumference divided by the diameter. That is to say, that tau is very much like pi. They both are very useful in mathematics and help with varying subjects. While pi is the circumference divided by the diameter, tau is the circumference divided by the radius; sothey go pretty much hand in hand. These two numbers are argued about greatly over which one is better, but the truth is that we should acknowledge and use both. The main thing tau should be used for is radians. The problem with using pi in these is that pi of a circle is really only half of a circle. So one fourth of a circle is not pi over 4, but pi over 2. This adds extra steps and makes these calculations much more complex. If tau was used for this instead, 1/4 of a circle would be equal to the value of tau over 4, simplifying things greatly. Other uses of tau:




Gaussian distribution, Fourier transform, Cauchy's integral formula, Riemann zeta function, and Euler's identity* Many famous equations use two pi instead of tau. If we were to replace these with tau, they would still make sense. These include the: * euler's identity uses pi and makes negative one, but if tau is used instead, it makes one. It can go either way for this one. Pi is closely related to the circle, so it is used in many mathematical formulas, specifically concerning circles, spheres, or ellipses. It can also be used in statistics, mechanics, number theories, and other well known mathematical topics. Pi's Uses Pi in Popular Culture:
Outside the Sciences Many people have been very interested with pi.
In fact, it is a popular practice around the world to memorize the digits of pi. This is referred to as piphology. The world record for the most pi digits ever memorized is amazingly, 67,890 digits! Another popular practice is actually writing. Many people use pi as guidelines for their stories. From poems to novels, pi is in many instances present. The numbers of pi are not listed, though. The digits are used to determine how many letters each word is supposed to be. For example, if I was expressing the first three digits of pi (3.14), I would say " 'Dad, a lion!' " Many books have been published in this abstract form. Another way that pi is represented in a, non-mathematical form is pi day. There are a actually two days that are known as Pi day, national and international. National pi day is 3/14, at least here in the US, because the first 3 digits of pi are 3.14. International pi day is on July 22, because pi can also be represented as 22/7, which, switched around is 7/22. Pi's Less Known Uses
and Appearences Believe it or not, Pi doesn't only appear in algebra and geometry textbooks. It actually appears in everyday life and everyday uses.
For example, pi actually appears
many times in fundamental laws of the universe, such as Heisenberg's uncertainty principal. Phi in Architecture and Other Arts It can be argued on both sides about how much the golden ratio is actually involved in ancient architectures. The problem is, famous phi structures, such as the Great Pyramids and the Parthenon, have dimensions that have been guessed and approximated to equal a number close to phi, like 1.72, which is one of the rations approximated for the Parthenon. So it's actually very hard to truly know if ancient civilizations knew about phi or not.
Phi can be seen greatly in the Platonic solids. Different dimensions, surface areas, and volumes all involve phi. The dodecahedron and icosahedron are the most related. The same problem appears in art. Paintings such as Madonna in Glory exhibit ratios that are near the golden ratio, but these tend to be farfetched approximations, with thicker lines for golden rectangles and such, making it harder to tell just how close the number is to phi. However, some paintings such as Sacrament of the Last Supper, by Salvador Dali, show true phi promises. This one has a dodecahedron displayed over everything, representing the universe. Phi can be seen in so much more, and is definitely very mysterious. Tau Day:
Tau day is on June 28, and although the number still isn't known to many, (as it was invented not long ago,)the traditions are like that of pi, except most people who celebrate both have half of a pie on pi day (3-14 or in international writing 22/7) and a full one on tau day. In conclusion, pi is a very integrated
part of the world, from literature to
the laws of the universe.
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