...space exploration and deep impact navigation. One of the very early pioneers in astronomy and mathematics was Aryabhatta. Whatever this origins, it cannot be disputed that he lived in Patliputra where he wrote his famous thesis called the “Aryabhatta-Siddanta” more commonly known as the “Aryabhatiya”. This is the only works to have survived to the present day. It contains mathematical and astronomical hypothesis that have been discovered to be quite accurate in contemporary mathematics. For example, he wrote that if 4 is added to 100 and then multiplied by 8 then added to 62,000 then divided by 20,000 the answer will be equal to the circumference of a circle of diameter twenty thousand. This calculates to 3.1416 close to actual value Pi (3.14159). But his greatest donation has to be zero, known as “Shunya” in his times. His other works include theorems on trigonometry, arithmetic, algebra, quadratic equations and the sine table. He also wrote essays on astronomy. For example he was aware that the earth spins on its axis, and that it moves round the sun and the moon rotates round the earth. He discusses about the locations of the planets in relation to its movement around the sun. Aryabhatta refers to the light of the planets and the moon as reflections from the sun. He goes as far as to explain the eclipse of the moon and the sun, day and...
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...Summation notation Summation notation is a convenient way to represent the sum of many terms. (It iis also called Sigma notation, since it uses the Greek letter Σ (Sigma)). If f is a function, then b f (k) = f (a) + f (a + 1) + f (a + 2) + . . . + f (b). k=a . Here are some examples: • • • • 5 k=1 k 3 = 13 + 23 + 33 + 43 + 53 = 225. = 1/1 + 1/2 + 1/3 + . . . + 1/100 ≈ 5.1873. = 1/1 + 1/2 + 1/3 + . . . + 1/n. 100 1 k=1 k n 1 t=1 t 15 x=1 2 = 2 + 2 + . . . + 2 = 15 · 2 = 30 15 times • 8 x=2 sin(π/x) = sin(π/2) + sin(π/3) + . . . + sin(π/8) ≈ 4.4885 Here are some handy rules for working with sums, where c is some constant: • • • n x=m cf (x) = c n x=m f (x). n x=m n x=m n x=m (f (x) n x=m (f (x) − g(x)) = + g(x)) = f (x) − f (x) + n x=m n x=m g(x). g(x). Here are some classical formulas that might come in handy: • • • • • n i=1 n i=1 n i=1 1 = n. c = cn. i= n(n+1) . 2 n(n+1)(2n+1) . 6 n(n+1) 2 2 n 2 i=1 i n 3 i=1 i = = . For details, see Appendix E in the text. Most of the second set of formulas can be proved using mathematical induction. 256 2π Calculator sums. It is often useful to use the calculator to find sums such as k=1 cos2 π + k 256 . This would be tedious to calculate by hand. The two commands to use are sum and seq, which on the TI-83, can be found in the LIST menu, under the OPS and MATH sub-menus, respectively. The command seq has the format: seq(expression,variable, lower, upper) This returns...
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...you must some up the sides and the widths times 2. Prect = 2l +2w. L is 2.5m and width is 10m. (5.0*2)= 10.0m (20*2) = 40m So Prect = 50.0m 3.) You use the area formula of a circle. Acir = ( pi)r^2. D = 4M to find radius you take the diameter and divide it by 2 so radius is 2. Pi is 3.141 or 22/7. Acir = (3.141)2^2 = Area. (3.141)4 = area. Area= 12.564. or 12.571 if you use 22/7 4.) We need to find the volume of a sphere radius is 5 m. your formula is Vsph =(4/3)(pi)r^3. So you take (4/3)(3.141)5^3 or (4/3)(22/7)5^3. V = 420.59 or V = 420.839 5.) We need to find the length of the rectangular prism. Volume is 144 m^3. Width is 2 m. Height is 6 m. your formula would be L = V/H*W So L = 144/2*6 L= 144/12 So L = 12 6.) Find area of triangle. A = 3M B = 4M. Your formula is Atri = (1/2)bh. So Atri = (1/2)4*3. Atri = 2*3. Area =6 7.) You need to find the length of the hypotenuse. Y = 6m and the area is 36m^2. Your formula is 8.) Find the volume of a cylinder your radius is 3m height is 10m. the formula is Vcyl = (pi)r^2h. So it’s (3.141)3^2*10 or (22/7)3^2*10. First one is 28.269*10 V = 282.69 or 28.28*10 = V = 282.85 9.) A sphere would have more 4 units and the radius is 3 the formula for volume in a cube is A^3=V so you take 4^3 = 80.342. a sphere is (4/3)(pi)r^3. So you have (4/3)(22/7)3^3 = 4.190etc *3^3 V= 252.50etc...
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...Photographic Memory It is not possible to have photographic memory, according to growing evidence it is impossible to recall images with perfect accuracy. There are some people that do have extraordinary memories like for example; Akira Haraguchi, at age 59, recited from memory the first 83,431 decimal places of pi. He is currently the record holder for the most remembered pi digits in the least amount of time studied. There are people that have Herculean memories, this means that they have skills memorizing a specific task, but can be incompetent recognizing someone’s face for example. Alan Searleman, a professor of psychology at St. Lawrence University in New York, says that the closest to having a photographic memory is having an eidetic memory. Eidetikers are persons who can vividly describe an image, for example they can scan a picture of a garden for 30 seconds and say how many petals are in the garden, but sometimes they forget after just a few minutes or they are not accurate enough. With this Searleman says that if they have photographic memory, they would not have any errors at all. Studies have also found that eidetikers are born, not made. You can improve your memory by practicing or learning tricks to remember things. And Searleman also found out that children tend to have more eidetic memory than adults, but once they start growing up they begin losing that ability, most of it at age six when they learn to process information more abstractly. Reference: ...
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...Measuring Earth with a Stick Have you ever heard of the Greek mathematician and astronomer Eratosthenes? His name is probably best known among astronomers. Why do they think so highly of him? Eratosthenes was born about 276 B.C.E. and received some of his education in Athens, Greece. He spent a good part of his life, however, in Alexandria, Egypt, which at that time was under Greek rule. In about 200 B.C.E., Eratosthenes set out to determine the dimensions of the earth by using a simple stick. “Impossible!” you may say. How did he do it? In the city of Syene (now called Aswan), Eratosthenes observed that at noon on the first day of summer, the sun was directly overhead. He knew this because there was no shadow cast when the sunlight reached the bottom wells. However, at noon on the same day in the city of Alexandria, which was located 5,000 stadia (stadia were Greek units of length. Through the exact value varied locally, one stadium is believed to have been 530 to 600 feet) to the north of Syene, a shadow could be observed. That gave Eratosthenes an idea. Eratosthenes set up a gnomon, a simple upright stick. When the sun was overhead at noon, he measured the angle of the shadow that the stick cast in Alexandria. He determined the angle to be 7.2 degrees from vertical. Now, Eratosthenes believed the earth to be spherical, and he knew that there are 360 degrees in a circle. So he divided 360 by the angle he had measured, 7.2. The result? His angle was one...
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...Essay on the movie "Pi" (1997) written and directed by Darren Aronofsky. Pi is a black and white movie that fits into Classicism form of film. Mathematician Maximillian Cohen (played by Sean Gulette) is a genius who leads a strange and lonely life. He shares a small apartment with Euklid, his homemade supercomputer. He's a mathematical genius who suffers from head-splitting migraine attacks, hallucinations, extreme paranoia, and some form of social anxiety disorder . After many unsuccessful treatments, he has become addicted to several painkillers . He is obsessed to find a pattern within the number pi. He thinks Mathematics is the language of nature, and believes everything around us can be represented and understood through numbers and that if you graph the numbers properly patterns will emerge. Max meets Lenny Meyer (played by Ben Shenkman), a Hasidic Jew who does mathematical research on the Torah. Lenny explains to Max how some people believe that the Torah is a string of numbers that form a code sent by God. Lenny's research is similar to real mathematical theories, which intrigues Max. Lenny also mentions that he and his fellow researchers are searching for a 216-digit number that is repeated throughout the text of the Torah. Max finds the 216-digit number code that he thinks it will unlock patterns in the stock market to predict future changes of how stock will increase or decrease. A Wall Street businesswoman Marcy Dawson (played by Pamela Hart) and a group of...
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...always comes at a cost. In Ang Lee’s film, Life of Pi, the protagonist embarks on a journey of self discovery that results in great personal growth. Tim Winton’s short story Big World is also a story of a boy’s rite of passage. Both texts explore the personal cost of their protagonists’ discoveries as they must endure great suffering, isolation and the loss of innocence in order to discover themselves and their place in the big picture of life. Lee’s film is structured to portray an ordeal that fluctuates between great suffering and great joy. Pi cannot be free to discover himself while he exists within the comfort zone of his family. The sinking of the Tsimtsum casts him into the ‘ocean of life’ where he must fend for himself. The aerial shot of his head, dwarfed by the vast, dark ocean emphasises his vulnerability as he is cast adrift. His intense suffering is revealed most powerfully in the storm scene where wide shots again portray his vulnerability in the wild sea and Christ-like imagery shows him screaming at his ‘God’, arms spread in supplication: “Why are you scaring him? I’ve lost my family. I’ve lost everything. I surrender. What more do you want?” Pi’s life raft is swept away, symbolising the loss of his haven and material possessions and a high angle shot of Richard Parker shows that even the tiger – a symbol of Pi’s braver and more primitive self – is afraid. A sustained scene that cuts between shots of Pi bracing himself against the sides of the boat...
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...Algorithm Workbench 1. Design an algorithm that prompts the user to enter his or her height and stores the users input in a variable named height. display “enter height” input variable= height 2. Design an algorithm that prompts the user to enter his or her favorite color and stores the users input in a variable named color. Display “enter favorite color” Input variable=color 3. Write an assignment statements that perform the following operations with the variables a, b, and c. a. ADDS 2 to a and stores the result in b a+2=b b. Multiplies b times 4 and stores the result in a b*4=a c. Divides a by 3.14 and stores the result in b a/3.14=b d. Subtracts 8 from b and stores the result in a 8-b=a 4. Assume the variables result w, x, y, and z are all integers, and that w=5, x=4, y=8, and z=2. What value will be stored in result in each of the following statements? a. Set result = x+y 12 b. Set result = z*2 4 c. Set result = y/x 4 d. Set result = y-2 6 5. Write a pseudocode statement that declares the variable COST so it can hold real numbers. Declare real variable cost 6. Write a pseudocode statement that declares the variable TOTAL so it can hold integers. Initialize the variable with the value 0. Declare Real Price=0 7. Write a pseudocode statement that assigns the value 27 to the variable content. Count=27 8. Write a pseudocode statement that assigns the sum of 10 and 14 to...
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...In this work, we present an extremely efficient approach for a fast numerical evaluation of highly oscillatory spherical Bessel integrals occurring in the analytic expressions of the so-called molecular multi-center integrals over exponential type functions. The approach is based on the Slevinsky-Safouhi formulae for higher derivatives applied to spherical Bessel functions and on extrapolation methods combined with practical properties of sine and cosine functions. Recurrence relations are used for computing the approximations of the spherical Bessel integrals, allowing for a control of accuracy and the stability of the algorithm. The computer algebra system Maple was used in our development, mainly to prove the applicability of the extrapolation methods. Among molecular multi-center integrals, the three-center nuclear attraction and four-center two-electron Coulomb and exchange integrals are undoubtedly the most difficult ones to evaluate rapidly to a high pre-determined accuracy. These integrals are required for both density functional and ab initio calculations. Already for small molecules, many millions of them have to be computed. As the molecular system gets larger, the computation of these integrals becomes one of the most laborious and time consuming steps in molecular electronic structure calculation. Improvement of the computational methods of molecular integrals would be indispensable to a further development in computational studies of large molecular systems. Convergence...
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...A. In order to complete this task, I first had to locate 3 circular items for which to measure the circumference and diameter in both traditional and metric units. The items used in the task were a cork drink coaster, a compact disk and a round candle jar. In order to obtain the most accurate measurements, I used a roll of yarn, permanent marker and a ruler with both inch and centimeter measurements. To calculate the circumference, I used the yarn and made a complete revolution around the outer edge of each item and using the marker, made a dot on the yarn at the point where it crossed its beginning end. By laying that piece of yarn flat, it is possible to gather an accurate measurement with the ruler. To collect the measurements for the diameter of each item was very similar with the exception of the yarn did not go around the item but rather straight across it. The results are found below. All measurements obtained, as well as all measurements in general, are considered to be approximate. There is always room for an inherent amount of error, or variation, in every measurement. Measuring cannot be compared to such as counting. I can count the number of apples on a tree and know that it is an exact number. An exact measurement can never be known but by using the best measuring tools, the variation can be reduced. An example of this is a person’s height. I am 5’4” but depending on the person measuring me or the form of measurement being used, the measurement could...
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...more threads? Can you tie any into the Quest pattern? Father’s admonitions to stay away from the tiger in the zoo In chapter 8 Ravi, Pi and their parents are at the zoo and Pi’s father is teaching his sons a lesson on the dangerousness of a tiger by letting a tiger brutally kill a goat right in front of him. He is trying to teach Pi how incredibly dangerous the Tiger is and that he should avoid them entirely; this is made evident by his accompanying speech: “Tigers are very dangerous. I want you to understand that you are never—under any circumstances—to touch a tiger, to pet a tiger, to put your hands through the bars of a cage, even to get close to a cage. Is that clear?” This thread, initiated at the very beginning of the novel, remains through to the ending moments of Pi’s trip across the sea. The thread like Pi’s experience develops and changes throughout. Initially Pi is warned of the dangers of a tiger and made to see it firsthand, this event foreshadows the mid-story where Pi is stuck on a lifeboat with the tiger he was made to fear, at this point he is forced to decide: obey his father and flee from the ship or, against everything he has been taught, stay with this ferocious feline. By challenging his father’s views he is able to develop his own, this theme—growth through trial—is very well supported in this thread; though initially Pi was afraid of the tiger as he...
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...This program will calculate the area and circumference of a circle. */ #include <iostream> #include <cmath> using namespace std; int main () { float radius; float circumference; float area; cout << "Please enter the radius of a circle: "; cin >> radius; cout << "\n"; circumference = 2 * 3.1416 * radius; area = 3.1416 * radius * radius; cout << "************************************" << "\n" << "*Area and Circumference of A Circle*" << "\n" << "************************************" << "\n" << "\tRadius= " << radius << "\n" << "\tArea= " << area << "\n" << "\tCircumference= " << circumference << "\n"; cin.get(); return 0; } //end main ------------------------------------------------- C program to Calculate the...
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...Self-check – Pg. 85 1) 22/7 = 3 7/22 = 0 22%7 = 1 7%22 = 0 b. 16/15 = 1 15/16 = 0 16%15 = 1 15%16 = 0 3 -a. 3%4= undefined – this is invalid because 3 divided by 4 in .75, which is zero. b. (989 – 1000 = -11 / 3 = -3.6 – invalid, this is not an integer. c. 4 %3 = 1 – Valid d. 3.14 * -1 = -3.14 – valid e. 3/-4 = -.75 – invalid, this is not an integer f. ¾ = .75 – valid g. 3 % .75 = 0 – valid 4= a. 7 % 3 = 1 b. (989 – 1000) / 7 = -1 c. 3%7 = -4 d. PI * 2.0 = 6.2 e. 7/-3= -2 f. 7/3 = 2 g. 7%(a7/3) = 1 Page 102 3. It’s missing the return (0) 4. Max_speed , Sphere size 2. if ((num V=I*R<0) Printf(“number = %lf”); If (area=length*width) Printf(“number = %d”); If (Rseries=R1+R2+R3) Printf(“number = %lf”); If Rparallel = R1*R2 / (R1+R2) Printf(“number = %lf”); 3. a. they considered this variable as a double, but I could pass as a integer. Also the printf statement has “%f” representing a double variable. b. This makes more sense considering 15 * 0.5 equals a double variable: hence the “x” representing the double. c. This program has 4 variables, the first two are double’s so they are labeled with “x,y”. the next two are int so they are labeled “I,j”. for each of the variable the printf statement show’s 2 “%f” and 2 “%d” confirming the variables type. At the end of printf, it shows the variables in order to keep track of them. d. “printf(“%d”, A); is merely showing that the variable...
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...the Law of Indices. The Law of Indices are related to exponents and how exponents are added, subtracted, multiplied and divided. Next in the Arithmetica Infinitorum, he find the area of a curve that uses the equation y=xm. He then used this curve to prove that the ratio of the curve to the parallelogram that has the same base and altitude as the curve, has a ratio of one to m plus one. John Wallis also proved that y=axm, where a is constant and m is any number that is either positive or negative. In his treatise, he proves how this works when a is equal to two and m is equal to negative one. In this treatise, he also talks about the pi. In the treatise, he finds the value of pi as . He did this by first determining the value of pi as four root ⅔ . After this, he determined that the value of pi is equal to 1, , , . He then noticed that the value of pi is interpolated between 1 and ⅙ . Using this, he was able to create the formula . He continued he love of math in 1657 when he published the Mathesis Universalis, that talks about algebra and geometry. In the Mathesis Universalis, he talks about infinity. John Wallis invented infinity and then introduced it in his book. In this book, he show how the symbol a3 is used. John wallis was the first to explain how exponent like a3, is used, however, he was not the first to introduce exponents. He also explained how negative and fractional exponents are used. ...
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...kill a goat right in front of him. He is trying to teach Pi to not go near an animal such as tigers because they are incredibly dangerous. I believe this thread represents the foreshadowing of what is to come for Pi. When Pi and Ravi’s father was telling them about staying away from tigers he said “Tigers are very dangerous. I want you to understand that you are never—under any circumstances—to touch a tiger, to pet a tiger, to put your hands through the bars of a cage, even to get close to a cage. Is that clear?” (37) This evidence indicates that tigers are extremely dangerous and any contact with them can cause serious injury. This thread develops and changes throughout the novel because later on Pi is stuck on a lifeboat with a tiger and doesn’t really have a choice whether or not he has to go near it. Pi needs to decide whether he is going to listen to his father’s lesson or fight for his life. Pi spends most of his days on the lifeboat training Richard Parker to let him on the lifeboat more often and establish dominance over him. Pi soon learns that Richard Parker is not as big of a threat as his father made him out to be. 2) The name ‘Pi’ As a child, one of Pi’s biggest concerns was defending his name because all of his classmates always teased him about his name, calling him “Pissing”. The name Pi is symbolic too who he really is. His name makes reference to math, the number of pi, 3.14. The number of pi is theoretically a never ending number, although is shortened...
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