Irrational Number A to Z

Did you realise Book Creator could be used in the mathematics classroom? It can! Here are some great examples.

Math books

Jaesung This is my very first experience making an e-book but I will try my best to make the best e-book.

1. Historical Background -- 2. Mathematicians -------- 3. Myth &#38; Fact ------------ 4. Cases ------------------- 5. Book Review ------------ 6. Math around us -------- 7. What we learned -------

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Bomin I'm not good at math at all but I'll work together with my team as much as I can.

Jaewoong I think solving math questions for my math class is not fun but creating e-book about math is so fun and I hope I can get to like Math after this activity.

https://api.bookcreator.com/users/v2/FI1jKsDURCdlFqjcModc0FUoN8U2/books/lSC14a_aSBmEjcGE0irwdA/assets/DAP4HMhQSu-nNsVv3JDjdw.jpg?width=301&height=391

Yoonjea Due to Covid19, I didn't see my friends often but I think this project will help me to work with friends. I will work hard on this.

Rainbow paper border

You may have seen the photos of celebrities in social media like the ones below. In the description of this photo, the overall proportions of the face, such as the eyes, nose, lips, and forehead, are made as close to the golden ratio and considered to be good looking. 

What we see the most on the way to and from school, is a sidewalk block. There are also irrational numbers hidden in this sidewalk block that we passed by without knowing anything. Assuming that the sidewalk block is a square with one side, the length of the diagonal is &#8730;2 according to the Pythagorean theorem. Isn't it fascinating?

4. Credit cards, Post cards &#38; Business cards

If the width and height of A4 paper are set to 1 and x, the width and height of half of the paper are x/2 and 1/2. And if we set up and rearrange the proportional expression, we get x square = 2. And when a number x is squared to become a, we write x square = a, and we call x the square root of a. Therefore, the number that becomes 2 when squared is root2 and -root2, and since the length of the paper is positive, x is root2. Therefore, the ratio of the width and length of the paper is 1: root 2.

If you measure the width and length of the credit card, you can see that it is 8.56 cm and 5.398 cm, respectively, and the width is about 1:1.586. This rain is very close to the golden ratio. Nowadays, because the golden ratio is a beautiful ratio that makes people's eyes comfortable, the horizontal and vertical ratios of books, computer monitors, TV screens, and movie theater screens are deliberately made close to the golden ratio.

Parchment

The Geumgang ratio is the ratio of 1: (Route 2) found in Seokguram Grotto, Gyeongju. It is said that the name Geumgangbi was given because it was as beautiful as Mt. Compared with the Western golden ratio, if the standard of the golden ratio is a pentagram, the standard of the golden ratio is a square. Plant growth and human body also choose Geumgangbi. This is because Geumgangbi includes stability and practicality. Geumgangbi can also be found in various buildings and household items because it contains aesthetic beauty. Therefore, we can find the Geumgangbi in cultural relics such as Seokguram Grotto, Muryangsujeon Hall of Buseoksa Temple, and Cheomseongdae.

Seokguram Grotto, Korea

황금비율을 품다, 부석사 무량수전

지진에도 무너지지 않는 첨성대 축조의 비밀 / YTN 사이언스

Around the 6th century BC, the Pythagorean school, centered on Pythagoras, believed that numbers could only be expressed as integers or ratios of integers (fractions). One day, when one of the Pythagoreans, Hippasos, was looking for the ratio of each line segment that makes up a figure, he drew a square of length 1 and drew a diagonal. there was no So, solving with the Pythagorean theorem, a&#178;=2, 1.414213&#8230; &#8230; A prime number that does not cycle with . So this fact is reported to the people of the school, but the people of the school don't believe it and instead kill Hippasus.

&#60;Image Source: http://www.kdnews.co.kr/20347&#62;

Pre-Algebra 32 - Irrational Numbers

CAPTION Half page image nullam nunc eros, vehicula feugiat

After that, in the 7th century India, the irrational number found by Hippasos was finally recognized, and at first, irrational numbers such as ka15 and ka10 were expressed instead of the root symbols we know. After some time, the symbol &#8730; was created by taking the r from the Greek word radix, which means root. Then, around 1500, French mathematician Descartes began to think about how to change &#8730;. The reason is that it is difficult to see at a glance how far &#8730; is applied. So, Descartes added horizontal bars until he wanted to express the square root. By creating this symbol, the scope of application of the route has been increased.

https://api.bookcreator.com/users/v2/FI1jKsDURCdlFqjcModc0FUoN8U2/books/lSC14a_aSBmEjcGE0irwdA/assets/pUvO5w68Qs2WhTXEzBrgAA.jpg?width=1228&height=810

By Jaesung, Bomin, Jaewoong, &#38; Yoonjea &#55356;&#57096;&#55357;&#56397;from Deogweon Middle School

Bunting border

Bomin I was shocked when I first learned that Hippasus was killed because he found irrational numbers. And I learned a lot about Math while making this e-book.

Yoonjae It was interesting to know that the golden ratio is hidden in the Fibonacci sequence. This project was useful because I found the interesting side of irrational numbers that I thought it was just boring in school class. It was even more fun to work with my friends.

Jaewoong I made my first e-book with my friends, it was fun and it was a good opportunity to know more about irrational numbers and Hippasus Pythagoras. I don't hate math any more.

Jaesung My teammates and I struggled with how to make an e-book at first, but the more we added, the more we were able to familiarize with the Book Creator. Making an e-book was fun, and I thank my math teacher for giving me this opportunity.

4. Cases in Our Daily life

Circumference is a mathematical constant representing the ratio of circumference to diameter. It is written with the Greek letter &#960; and read as pi (&#960;). Because Archimedes' calculation is widely known, it is also called the Archimedes constant, and in Germany, it is also called the Rudolph number after Rudolf van Kohlen calculated the circumference to 35 decimal places in the 1600s. The value of the circumference is 3.1415926535897932384626433832795028841.... Since it is a non-recurring infinite decimal, 3.14 is used as an approximation value or used as the symbol &#960;.

The golden ratio or golden division is a ratio such that the ratio of any two numbers is equal to the ratio of the sum to the greater of the two numbers, and is an irrational number with an approximate value of about 1.618.  The expression for the golden ratio is as follows.

Since Euclid studied its properties, many mathematicians have studied the golden ratio found in nature.

Dmitri Kessel - Google Arts & Culture

&#55356;&#57263;Where it is used: statues, credit cards, transportation cards, etc.

Stars pattern

1) The Number of Spirals of Pine Cones and Sunflowers

Monster Stroll Fractions Game in Math Playground

1. It is easy to observe the spiral curve structure in nature. If you count the number of spirals in the pine cone, it has the number of spirals such as 1,2,3,5,8. , or the number of spirals of small and large pine cones) 2. In the arrangement of seeds on the bud of sunflower, it can be seen that spiral curves appear in the right and left directions, and the number of these spirals is also 1, 1, 2, 3, 5, 8, 13, 21... .. that is, it can be seen that it follows the Fibonacci number.

We couldn't find any problems with irrational numbers in the Math Playground website. So, while looking for a fraction game, we found a game called, "Monster Stroll Fractions." This game is related to fractions but there are examples of fractions such as 2/3 and 1/6 that produce irrational numbers when converted to decimals.

Fibonacci Numbers – Sequences and Patterns – Mathigon

Monster Stroll Fractions | Math Playground

Irrational numbers in the Fibonacci sequence

<a href="https://en.wikipedia.org/wiki/Fibonacci">Leonardo Fibonacci</a> studied this number, referring to the breeding of rabbits in 1202. It was Fibonacci who made an achievement to study and systematize this sequence in Western Europe, hence the name <a href="https://en.wikipedia.org/wiki/Fibonacci_number">Fibonacci sequence</a>. The Fibonacci sequence includes 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, etc. At this time, when I find the ratio between these numbers, it gradually becomes an irrational number and eventually comes out as an irrational number as the golden ratio.

If you count the petals around you, almost all the petals are 3, 5, 8, 13... is made of Lilies and irises have 3 petals, chrysanthemum dianthus camellia has 5 petals, peony cosmos has 8 petals, and marigolds and marigolds 13. Aster and chicory have 21 sheets, plantain and daisy have 34 sheets, and wormwood has 55 and 89 sheets depending on the type. Also, the leaves of plants grow in the same order as 1, 1, 2, 3, 5, 8, 13, 21....

Jaesung For me, the part that impressed me the most was about Geumgangbi that can be found even in the remains of prehistoric huts in Korea. The reason is that people of the past felt the Geumgangbi as beauty just like us.

Bomin The phrase that moved me is "The paper we used carelessly hides a special irrational number of people's efforts and wisdom to reduce waste". Because it is strange that the idea of irrational numbers is added to the campaign to reduce paper waste.

Jaewoong I was impressed with the words, 'I am not a betrayer! I have just made known to the world a surprising fact' because Hippasus only found irrational numbers and was killed by Pythagoras.

Yoonjae I said, 'After much deliberation, Descartes decided to add a horizontal bar to the point where he wanted to express it as a square root.' The sentence that impressed me the most was because I could feel Descartes' sincerity to make people feel comfortable.

<a href="https://en.wikipedia.org/wiki/Ren%C3%A9_Descartes">Descartes</a> was a mathematician of the 17th century who accomplished many great achievements in science and philosophy as well as essential in mathematics, such as the use of the coordinate plane and the unknown x. It is also famous for simply solving the inconvenience of the old &#8730;. &#160;As his representative story, Descartes, who was thinking while lying in bed, looked at the fly on the ceiling and wondered if there was any way to indicate the location. So what he thought was the coordinate plane. He drew two lines that meet each other on the ceiling, and called these two lines the horizontal and vertical axes, respectively. The position of the fly could be indicated by the values of the horizontal and vertical axes.

Let us find the timeline of the mathematicians we researched so far.

Timeline of Mathematics – Mathigon

The infinite life of pi - Reynaldo Lopes

When the value of F is obtained in this way, the number of F looks irregular, but 1.4 is an approximation of &#8730;2, the second 2 is the value multiplied by &#8730;2 twice, and the third 2.8 is the approximate value of multiplying &#8730;2 three times. To explain, the iris of the camera controls the amount of light by changing the width of the light entering part by covering the lens. If the F-number is increased by one level, the amount of light entering through the round lens is reduced by half. At this time, since it increases to a value of &#8730;2, an irrational number can be found at the camera aperture.

(This is where we can find the value of the aperture in the camera.)

F1.4 , F2 , F2.8 , F4 , F5.6 , F8 , F11 , F16 are engraved numbers on the camera lens. These numbers are called F numbers. The F-values formula is as follows.

MATH behind F STOP numbers (aperture sizes) - how to memorize aperture numbers

<a href="https://en.wikipedia.org/wiki/Archimedes">Archimedes</a> invented the concept of infinite primes in an era when there was no modern integral. An approximation was found using the elimination method, and was calculated using a regular decagon. According to a famous story, when there were rumors that the king's crown had been mixed with other substances, he summoned Archimedes to find out. When he entered the bathroom, he saw the water overflowing from his body and accidentally discovered the principle of buoyancy, and exclaimed, "I found it, Eureka!

By applying the Pythagorean theorem, the theorem named after his teacher, to a right-angled triangle with two sides of length 1, <a href="https://en.wikipedia.org/wiki/Hippasus">Hippasus</a> discovered the number &#8730;2, proving that it is a number that cannot be expressed in a perfect integer ratio. At that time, the Pythagoreans believed that the origin of all things was a natural number, and taught that all numbers can be expressed as ratios of natural numbers. In other words, the existence of irrational numbers implied the Pythagorean error. So he was killed by the Pythagoreans.

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