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Good morning/afternoon/evening! It is a great honor for me to stand here before you today to deliver a speech on the importance and impact of mathematics in our lives. Mathematics, as one of the oldest sciences, has played a crucial role in the development of human civilization. It has not only shaped our understanding of the world but also revolutionized various fields such as technology, economics, and even art. In this speech, I will discuss the significance of mathematics, its historical development, and its applications in different domains.Firstly, let us delve into the significance of mathematics. Mathematics is the language of science, the tool of engineers, and the basis of economics. It enables us to make sense of the world around us and to solve real-world problems. Without mathematics, our lives would be chaotic and unpredictable. Here are some key reasons why mathematics is so important:1. Logic and reasoning: Mathematics teaches us to think logically and critically. It helps us develop problem-solving skills and analytical abilities, which are essential in various aspects of life.2. Predicting and modeling: Mathematics allows us to predict future events and model complex systems. This is crucial in fields such as climate science, finance, and medicine.3. Technology: The development of technology has been heavily dependent on mathematics. From smartphones to airplanes, mathematics has played a vital role in the creation of innovative technologies.4. Economics: Mathematics is the backbone of economics. It helps us understand market trends, make informed decisions, and analyze financial data.5. Art and aesthetics: Mathematics has influenced the development of art and aesthetics throughout history. The Fibonacci sequence, for instance, is a classic example of how mathematics can inspire creativity.Now, let's take a brief look at the historical development of mathematics. The origins of mathematics can be traced back to ancient civilizations such as the Egyptians, Babylonians, and Greeks. Theseearly mathematicians developed basic arithmetic, geometry, and algebraic concepts, which laid the foundation for future advancements.1. Ancient Egypt and Mesopotamia: The ancient Egyptians and Babylonians were the first to develop a numeral system and solve practical problems related to agriculture, construction, and trade.2. Ancient Greece: The Greeks made significant contributions to mathematics, with figures like Pythagoras, Euclid, and Archimedes. They developed theorems, algorithms, and a more rigorous approach to mathematical reasoning.3. Islamic Golden Age: During the Islamic Golden Age, scholars from the Islamic world translated and expanded upon the works of ancient Greek mathematicians. They introduced algebra, trigonometry, and the concept of zero.4. European Renaissance: The Renaissance in Europe marked a revival of interest in mathematics, with figures like Leonardo da Vinci and Galileo Galilei contributing to the field.5. Modern mathematics: In the 19th and 20th centuries, mathematics experienced a period of rapid growth and diversification. New branches of mathematics, such as abstract algebra, topology, and number theory, were developed, leading to significant advancements in various fields.Moving on to the applications of mathematics in different domains, we can see its pervasive influence:1. Technology: Mathematics is the backbone of technology, from the development of computer algorithms to the design of microchips. It plays a crucial role in fields like artificial intelligence, cryptography, and robotics.2. Economics: Mathematics is essential in economics for analyzing market trends, predicting economic behavior, and making informed decisions. It helps us understand the complexities of financial systems and optimize economic policies.3. Medicine: Mathematics is used in medicine to analyze data, develop treatment plans, and create medical devices. It plays a crucial role in fields like biostatistics, epidemiology, and medical imaging.4. Art and design: Mathematics has inspired countless artists and designers throughout history. From the golden ratio in Renaissance artto the fractal patterns in modern architecture, mathematics has influenced the aesthetics of countless works.5. Environmental science: Mathematics helps us understand environmental systems, predict climate change, and develop sustainable solutions. It plays a crucial role in fields like ecology, geology, and meteorology.In conclusion, mathematics is a fundamental discipline that has shaped human civilization and continues to play a vital role in our lives. Its significance cannot be overstated, as it enables us to make sense of the world, solve real-world problems, and drive innovation in various fields. As we move forward, it is essential that we continue to value and promote the study of mathematics, as it will undoubtedly continue to contribute to the advancement of our society.Thank you for your attention, and I hope this speech has highlighted the importance of mathematics in our lives.。
英语作文关于数学公式的应用Mathematical formulas are used in many different fields for various applications. From physics to engineering, from finance to computer science, mathematical formulas play a crucial role in solving complex problems and making important decisions.In physics, mathematical formulas are used to describe the behavior of particles and waves, the motion of objects, and the interaction of forces. For example, the famous formula E=mc^2, proposed by Albert Einstein, describes the relationship between energy (E), mass (m), and the speed of light (c). This formula has had a profound impact on our understanding of the universe and has led to the development of nuclear energy and weapons.In engineering, mathematical formulas are used to design structures, analyze materials, and optimize processes. For instance, the formula F=ma, which describes therelationship between force (F), mass (m), and acceleration (a), is used to calculate the strength and stability of buildings, bridges, and machines. Engineers also use mathematical formulas to model fluid dynamics, heattransfer, and electrical circuits, enabling them to create innovative solutions for real-world problems.In finance, mathematical formulas are used to evaluate risk, estimate returns, and price financial instruments. For example, the Black-Scholes formula, developed by economists Fischer Black and Myron Scholes, is used to calculate the value of stock options and other derivatives. This formula has revolutionized the field of finance and has enabled investors to make more informed decisions about their investments.In computer science, mathematical formulas are used to develop algorithms, analyze data, and optimize performance. For instance, the formula for calculating the Fibonacci sequence is used in computer programs to generate a series of numbers that are widely used in various applications, such as in modeling population growth, predicting stock prices, and creating visually appealing designs.Overall, mathematical formulas are essential tools for solving problems, making predictions, and creating new technologies. They provide a universal language for expressing relationships and patterns, and they enable usto understand the world around us in a precise and systematic way.数学公式在许多不同领域中被用于各种应用。
Introduction:Mathematics, as an essential discipline, plays a vital role in our daily lives and academic pursuits. Reflecting on my mathematical journey, I have come to realize the significance of problem-solving, critical thinking, and the beauty of mathematical concepts. This essay aims to explore the insights gained from my experiences with mathematics and the impact it has had on my personal and academic growth.Body:1. The Evolution of Mathematical UnderstandingFrom early arithmetic to complex algebraic equations, my journey in mathematics has been a continuous process of learning and unlearning. I remember the excitement of solving simple addition and subtraction problems as a child, which gradually led to a deeper understanding of more advanced mathematical concepts. Reflecting on this evolution, I appreciate the gradual development of my mathematical skills and the importance of building a strong foundation.2. Problem-Solving SkillsOne of the most significant benefits of studying mathematics is the enhancement of problem-solving skills. Mathematics trains our minds to approach challenges systematically, breaking them down into smaller, manageable parts. Through the process of problem-solving, I have learned to analyze situations, think critically, and develop logical reasoning. These skills have proven to be invaluable not only in mathematics but also in various aspects of life.3. The Beauty of Mathematical ConceptsMathematics is not just about numbers and formulas; it is a languagethat reveals the beauty of patterns, symmetry, and order. From the Fibonacci sequence to the Pythagorean theorem, mathematical concepts have fascinated me with their elegance and simplicity. Reflecting on these ideas, I have come to appreciate the harmony that exists within the realm of mathematics, which inspires awe and wonder.4. The Role of Mathematics in Academic PursuitsMathematics has played a crucial role in my academic journey. It has equipped me with the necessary tools to excel in various subjects, such as physics, engineering, and computer science. Moreover, mathematics has helped me develop a mindset that encourages continuous learning and exploration. Reflecting on this aspect, I acknowledge the importance of mathematics in fostering a well-rounded education.5. The Challenges and Rewards of MathematicsWhile mathematics is a subject that brings immense satisfaction, it also presents its fair share of challenges. The complexity of certain mathematical concepts, the pressure to perform, and the need for perseverance can be daunting. However, overcoming these challenges has taught me resilience and the value of hard work. The rewards of mathematical success, such as the satisfaction of solving a difficult problem or the appreciation of a beautiful mathematical theorem, are worth the effort.Conclusion:In conclusion, reflecting on my experiences with mathematics has allowed me to appreciate the importance of this discipline in my personal and academic growth. Mathematics has not only enhanced my problem-solving skills and critical thinking but also exposed me to the beauty of mathematical concepts. As I continue my journey in mathematics, I am determined to embrace the challenges and rewards that lie ahead, and to further develop my understanding of this fascinating subject.。
介绍一种自己喜欢的植物英语作文My Favorite Plant: The SunflowerHello, my name is Claude and I'm going to tell you all about my favorite plant - the sunflower! Sunflowers are so bright, cheerful and sunny. Just looking at them makes me happy. Let me share why I love sunflowers so much.What Are Sunflowers?Sunflowers are big, tall flowers that can grow over 3 meters (10 feet) high! Their flowers are huge too, with wide circular heads full of tiny flowers called florets. The centers are made up of lots and lots of small florets clustered together in a spiral pattern. Surrounding the centers are big yellow petals.When the sunflower is just a bud, the back of the head tracks the sun's movement across the sky from east to west over the course of the day. This is called heliotropism and helps the flower get as much sunlight as possible. Once the flower blooms fully, it faces east permanently.Sunflowers are members of the daisy family. The name "sunflower" comes from the flower's huge, round bloom that reminds people of the sun. Some types of sunflowers can havereddish petals, but the most common variety has bright yellow petals.Fun Sunflower FactsHere are some cool facts about sunflowers that I think are awesome:• The world's tallest sunflower grew 9.17 meters (30 feet and 1 inch) tall!• Sunflowers can be used to make sunflower oil for cooking and soap.• The leaves, seeds and flowering heads of the sunflower can all be eaten! The seeds are really healthy snacks.• Sunflowers bloom during the summer and early fall. They love soaking up that warm sunshine.• Sunflowers are the national flower of Russia. The country of Ukraine is one of the biggest producers of sunflower seeds.• Some sunflower seeds were even taken into space by NASA astronauts!Why I Love SunflowersThere are so many reasons why sunflowers are my favorite flowers. First of all, they are just so wonderfully sunny and bright! Seeing a big field of sunflowers in full bloom is one of the most cheerful, joyful sights. All those vibrant yellow petals stretching out as far as the eye can see is amazing.I love how sunflowers are smart enough to track and follow the sun's path across the sky when the buds are growing. It's like they are little scientists! Their spiral seed patterns are phenomenal examples of sacred geometry found in nature, just like pinecones, seashells and galaxies. The patterns follow the Fibonacci sequence. Isn't that incredible?Also, sunflowers are very useful plants. We can eat the seeds, use the oil for cooking, and some farm animals also enjoy munching on the plants and seeds too. If humans ever travel to live on other planets, we may be able to grow sunflowers to produce food, since NASA has already tested growing them in space!I find sunflowers to be the most jovial, optimistic flowers. Whenever I'm having a bad day, I just need to look at a photo of a sunflower field and my mood instantly brightens. Sunflowers seem to radiate happiness, warmth and positive energy. Their tall stalks stretch up proudly towards the sky. To me, they symbolizejoy, vibrancy, nourishment and resurrection. Just like the sun's rays allow the flower to bloom, I feel like the sunflower fills me with sunshine and light.My Sunflower GardenBecause I love sunflowers so much, I've started a little sunflower garden in my backyard. Every spring, I till the soil and plant a bunch of sunflower seeds in one corner. I put a small fence around them at first to protect the new sprouts from any curious critters.It's always so exciting to see those first little green heads poking up through the dirt! Over the next few months, I water them faithfully and watch in delight as the stalks grow taller and taller. Once the flower buds appear, I can't wait for them to finally open into big, gorgeous blooms.I love going out in my garden to admire the sunflowers. I'll pick a few for a cheerful bouquet to put in a vase for our kitchen table. Sometimes I like to just sit outside and stare at the sunflowers while listening to the bees buzzing around them. Their bright yellow faces are constantly pointed towards thelife-giving sun, almost like they are worshipping it.When autumn arrives, the sunflowers' petals start to droop down sadly. All the seeds in the big centers get plump and heavy.I always let some sunflowers go to seed, then I harvest the seed heads so I can roast and eat the yummy seeds as snacks. I also leave some seed heads attached for the birds, squirrels and chipmunks to feast on. They seem to love snacking on sunflower seeds for energy in the winter.Did you know that a"sunflower" is also a person who travels from place to place to follow a certain activity, person or show? I like that idea of being a "sunflower" myself - always searching out the light, the sun, the joy and beauty in this world!In conclusion, I hope you can see why sunflowers are so special to me. They are like portable bursts of sunshine that fill me with warmth, energy and optimism. I recommend planting some sunflower seeds yourself this spring. Watching their whole growth cycle from seedlings to towering beauties is such a rewarding experience. And you can enjoy the benefits of fresh sunflower seeds and sunny blooms all summer long! What's not to love about these。
一串数字的作文英语英文回答：As a collection of numerical digits, my essence lies in the realm of mathematics, where I serve as a versatile tool for quantifying, measuring, and representing various concepts and phenomena. I am an integral part of the human experience, facilitating communication, problem-solving, and the advancement of knowledge across diverse fields.Take, for instance, the number "7." It holds cultural significance in many parts of the world, often associated with good luck, perfection, or completion. In Christianity, it symbolizes the seven days of creation and the seven deadly sins. In ancient Greek mythology, it represents the seven wonders of the ancient world.Conversely, the number "13" carries a negative connotation in some cultures, particularly in Western society, where it is associated with superstition andmisfortune. This stems from the belief that Judas Iscariot, the betrayer of Jesus, was the 13th guest at the Last Supper.Another example is the number "pi" (π), an irrational number representing the ratio of a circle's circumference to its diameter. It has fascinated mathematicians for centuries, with its endless, non-repeating decimal expansion. Pi plays a crucial role in geometry, trigonometry, and physics, enabling us to calculate the area, volume, and other properties of circular objects.The Fibonacci sequence, a series of numbers where each subsequent number is the sum of the two preceding ones, embodies the concept of the golden ratio, a harmonious proportion found in nature and art. It is evident in the spiral patterns of seashells, the arrangement of leaves on a stem, and the proportions of the human body.In the realm of computer science, binary numbers (consisting of 0s and 1s) form the foundation of digital technology. They allow computers to store and processinformation efficiently, enabling the development of software, hardware, and the internet.Numbers also serve as a powerful means of communication. We use them to convey quantities, dates, measurements, and even emotions. The number "5" can express approval or agreement, while the number "10" might indicate perfectionor completeness.Moreover, numbers empower us to make informed decisions. Statistical data, financial figures, and scientific measurements provide valuable insights into the worldaround us, helping us understand trends, patterns, and risks. They enable us to make rational choices, allocate resources effectively, and solve complex problems.In essence, the series of numerical digits that defines me is not merely a collection of symbols but a versatileand indispensable tool that permeates every aspect of human life. From unlocking the mysteries of the universe to facilitating everyday communications, numbers shape our perceptions, guide our decisions, and empower us to explorethe world around us.中文回答：作为一串数字，我的本质在于数学领域，在那里我充当一个多功能工具，用于量化、测量和表示各种概念与现象。
USA AMC 10 20001In the year , the United States will host the International Mathematical Olympiad. Let , , and be distinct positive integers such that the product . What's the largest possible value of the sum ?SolutionThe sum is the highest if two factors are the lowest.So, and .2Solution.3Each day, Jenny ate of the jellybeans that were in her jar at the beginning of the day. At the end of the second day, remained. How many jellybeans were in the jar originally?Solution4Chandra pays an online service provider a fixed monthly fee plus an hourly charge for connect time. Her December bill was , but in January her bill was because she used twice as much connect time as in December. What is the fixxed monthly fee?SolutionLet be the fixed fee, and be the amount she pays for the minutes she used in the first month.We want the fixed fee, which is5Points and are the midpoints of sides and of . As moves along a line that is parallel to side , how many of the four quantities listed below change?(a) the length of the segment(b) the perimeter of(c) the area of(d) the area of trapezoidSolution(a) Clearly does not change, and , so doesn't change either.(b) Obviously, the perimeter changes.(c) The area clearly doesn't change, as both the base and its corresponding height remain the same.(d) The bases and do not change, and neither does the height, so the trapezoid remains the same.Only quantity changes, so the correct answer is .6The Fibonacci Sequence starts with two 1s and each term afterwards is the sum of its predecessors. Which one of the ten digits is the last to appear in thet units position of a number in the Fibonacci Sequence?SolutionThe pattern of the units digits areIn order of appearance:.is the last.7In rectangle , , is on , and and trisect . What is the perimeter of ?Solution.Since is trisected, .Thus,.Adding, .8At Olympic High School, of the freshmen and of the sophomores took the AMC-10. Given that the number of freshmen and sophomore contestants was the same, which of the following must be true?There are five times as many sophomores as freshmen.There are twice as many sophomores as freshmen.There are as many freshmen as sophomores.There are twice as many freshmen as sophomores.There are five times as many freshmen as sophomores.SolutionLet be the number of freshman and be the number of sophomores.There are twice as many freshmen as sophomores.9If , where , thenSolution, so ...10The sides of a triangle with positive area have lengths , , and . The sides of a second triangle with positive area have lengths , , and . What is the smallest positive number that is not a possible value of ?SolutionFrom the triangle inequality, and . The smallest positive number not possible is , which is .11Two different prime numbers between and are chosen. When their sum is subtracted from their product, which of the following numbers could be obtained?SolutionTwo prime numbers between and are both odd.Thus, we can discard the even choices.Both and are even, so one more than is a multiple of four.is the only possible choice.satisfy this, .12Figures , , , and consist of , , , and nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be in figure 100?SolutionSolution 1We have a recursion:.I.E. we add increasing multiples of each time we go up a figure. So, to go from Figure 0 to 100, we add.Solution 2We can divide up figure to get the sum of the sum of the first odd numbers and the sum of the first odd numbers. If you do not see this, here is the example for :The sum of the first odd numbers is , so for figure , there are unit squares. We plug in to get , which is choice13There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color?SolutionIn each column there must be one yellow peg. In particular, in the rightmost column, there is only one peg spot, therefore a yellow peg must go there.In the second column from the right, there are two spaces for pegs. One of them is in the same row as the corner peg, so there is only one remaining choice left for the yellow peg in this column.By similar logic, we can fill in the yellow pegs as shown:After this we can proceed to fill in the whole pegboard, so there is only arrangement of the pegs. The answer is14Mrs. Walter gave an exam in a mathematics class of five students. She entered the scores in random order into a spreadsheet, which recalculated the class average after each score was entered. Mrs. Walter noticed that after each score was entered, the average was always an integer. The scores (listed in ascending order) were , , , , and . What was the last score Mrs. Walter entered?SolutionThe sum of the first scores must be even, so we must choose evens or the odds to be the first two scores.Let us look at the numbers in mod .If we choose the two odds, the next number must be a multiple of , of which there is none.Similarly, if we choose or , the next number must be a multiple of , of which there is none.So we choose first.The next number must be 1 in mod 3, of which only remains.The sum of the first three scores is . This is equivalent to in mod .Thus, we need to choose one number that is in mod . is the only one that works.Thus, is the last score entered.15Two non-zero real numbers, and , satisfy . Which of the following is a possible value of ?SolutionSubstituting , we get16The diagram shows lattice points, each one unit from its nearest neighbors. Segment meets segment at . Find the length of segment .SolutionSolution 1Let be the line containing and and let be the line containing and . If we set the bottom left point at , then , , , and .The line is given by the equation . The -intercept is , so . We are given two points on , hence we cancompute the slope, to be , so is the lineSimilarly, is given by . The slope in this case is , so . Plugging in the point gives us , so is the line .At , the intersection point, both of the equations must be true, soWe have the coordinates of and , so we can use the distance formula here:which is answer choiceSolution 2Draw the perpendiculars from and to , respectively. As it turns out, . Let be the point on for which ., and , so by AA similarity,By the Pythagorean Theorem, we have ,, and . Let , so , thenThis is answer choiceAlso, you could extend CD to the end of the box and create two similar triangles. Then use ratios and find that the distance is 5/9 of the diagonal AB. Thus, the answer is B.17Boris has an incredible coin changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?SolutionConsider what happens each time he puts a coin in. If he puts in a quarter, he gets five nickels back, so the amount of money he has doesn't change. Similarly, if he puts a nickel in the machine, he gets five pennies back and the money value doesn't change. However, if he puts a penny in, he gets five quarters back, increasing the amount of money he has by cents.This implies that the only possible values, in cents, he can have are the ones one more than a multiple of . Of the choices given, the only one is18Charlyn walks completely around the boundary of a square whose sides are each km long. From any point on her path she can see exactly km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?SolutionThe area she sees looks at follows:The part inside the walk has area . The part outside the walk consists of four rectangles, and four arcs. Each of the rectangles has area . The four arcs together form a circle with radius . Therefore the total area she can see is, which rounded to the nearest integer is .19Through a point on the hypotenuse of a right triangle, lines are drawn parallel to the legs of the triangle so that the trangle is divided into a square and two smaller right triangles. The area of one of the two small right triangles is times the area of the square. The ratio of the area of the other small right triangle to the area of the square isSolutionLet the square have area , then it follows that the altitude of one of the triangles is . The area of the other triangle is .By similar triangles, we haveThis is choice(Note that this approach is enough to get the correct answer in the contest. However, if we wanted a completely correct solution, we should also note that scaling the given triangle times changes each of the areas times, and therefore it does not influence the ratio of any two areas. This is why we can pick the side of the square.)20Let , , and be nonnegative integers such that . What is the maximum value of ?SolutionThe trick is to realize that the sum is similar to the product .If we multiply , we get.We know that , therefore.Therefore the maximum value of is equal to the maximum value of . Now we will find this maximum.Suppose that some two of , , and differ by at least . Then this triple is surely not optimal.Proof: WLOG let . We can then increase the value ofby changing and .Therefore the maximum is achieved in the cases where is a rotation of . The value of in this case is . And thus the maximum of is.21If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) must be true?I. All alligators are creepy crawlers.II. Some ferocious creatures are creepy crawlers.III. Some alligators are not creepy crawlers.SolutionWe interpret the problem statement as a query about three abstract concepts denoted as "alligators", "creepy crawlers" and "ferocious creatures". In answering the question, we may NOT refer to reality -- for example to the fact that alligators do exist.To make more clear that we are not using anything outside the problem statement, let's rename the three concepts as , , and .We got the following information:▪If is an , then is an .▪There is some that is a and at the same time an .We CAN NOT conclude that the first statement is true. For example, the situation "Johnny and Freddy are s, but only Johnny is a "meets both conditions, but the first statement is false.We CAN conclude that the second statement is true. We know that there is some that is a and at the same time an . Pick one such and call it Bobby. Additionally, we know that if is an , then is an. Bobby is an , therefore Bobby is an . And this is enough to prove the second statement -- Bobby is an that is also a .We CAN NOT conclude that the third statement is true. For example, consider the situation when , and are equivalent (represent the same set of objects). In such case both conditions are satisfied, but the third statement is false.Therefore the answer is .22One morning each member of Angela's family drank an -ounce mixture of coffee with milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Angela drank a quarter of the total amount of milk and a sixth of the total amount of coffee. How many people are in the family?SolutionThe exact value "8 ounces" is not important. We will only use the fact that each member of the family drank the same amount.Let be the total number of ounces of milk drank by the family and the total number of ounces of coffee. Thus the whole family drank a total of ounces of fluids.Let be the number of family members. Then each family member drank ounces of fluids.We know that Angela drank ounces of fluids.As Angela is a family member, we have .Multiply both sides by to get .If , we have .If , we have .Therefore the only remaining option is .23When the mean, median, and mode of the list are arranged in increasing order, they form a non-constant arithmetic progression. What is the sum of all possible real values of ?SolutionAs occurs three times and each of the three other values just once, regardless of what we choose the mode will always be .The sum of all numbers is , therefore the mean is .The six known values, in sorted order, are . From this sequence we conclude: If , the median will be . If , the median will be . Finally, if , the median will be .We will now examine each of these three cases separately.In the case , both the median and the mode are 2, therefore we can not get any non-constant arithmetic progression.In the case we have , because. Therefore our three values inorder are . We want this to be an arithmetic progression. From the first two terms the difference must be . Therefore thethird term must be .Solving we get the only solution for this case: . The case remains. Once again, we have ,therefore the order is . The only solution is when , i. e., .The sum of all solutions is therefore .24Let be a function for which . Find the sum of all values of for which .SolutionIn the definition of , let . We get: . Aswe have , we must have , in other words .One can now either explicitly compute the roots, or use Vieta's formulas. According to them, the sum of the roots of is . In our case this is .(Note that for the above approach to be completely correct, we should additionally verify that there actually are two distinct real roots. This is, for example, obvious from the facts that and .)25In year , the day of the year is a Tuesday. In year , the day is also a Tuesday. On what day of the week did the of year occur?SolutionClearly, identifying what of these years may/must/may not be a leap year will be key in solving the problem.Let be the day of year , the day of year and the day of year .If year is not a leap year, the day will bedays after . As , that would be a Monday.Therefore year must be a leap year. (Then is days after .) As there can not be two leap years after each other, is not a leap year. Therefore day is days after . We have . Therefore is weekdays before , i.e., is a.(Note that the situation described by the problem statement indeed occurs in our calendar. For example, for we have =Tuesday, October 26th 2004, =Tuesday, July 19th, 2005 and =Thursday, April 10th 2003.)。
Ladies and Gentlemen,Good evening. It is my great honor to stand before you today and embark on a journey through the fascinating realm of mathematics, which we often refer to as the language of the universe. Mathematics is not just a subject we study in schools; it is the very fabric that intertwines the cosmos and shapes our understanding of the world around us. Today, I invite you to join me as we explore the wonders of the mathematical universe.The universe itself is a grand work of art, and mathematics is its blueprint. From the smallest subatomic particles to the vast expanse of galaxies, the universe is governed by patterns and structures that can be described and understood through mathematical formulas. Let us delve into the depths of this magnificent universe and discover the role of mathematics in shaping its very essence.I. The Foundations of MathematicsThe journey of mathematics begins with the very foundations upon whichit is built. Mathematics is a discipline that relies on logic, rigor, and precision. It is a language that transcends cultures and languages, and it has been the driving force behind scientific and technological advancements throughout history.1. Numbers: The Universal LanguageNumbers are the building blocks of mathematics. They are the essence of quantity and magnitude, and they have been a part of human existence since the dawn of time. From counting objects to measuring distances, numbers have allowed us to make sense of the world and communicate our understanding of it.2. Arithmetic: The Basic OperationsArithmetic, the study of numbers and their properties, forms the cornerstone of mathematics. The four basic operations—addition, subtraction, multiplication, and division—are the foundation upon which more complex mathematical concepts are built.3. Geometry: The Language of Shape and SpaceGeometry is the branch of mathematics that deals with the properties, relations, and measurements of points, lines, surfaces, and solids. It is the language of shape and space, and it has played a crucial role in understanding the physical world.II. The Mathematical Universe: A Journey Through PatternsThe universe is filled with patterns and regularities that can be discovered and understood through mathematics. Let us explore some of these patterns and their significance.1. The Fibonacci Sequence: Nature's CodeThe Fibonacci sequence, a series of numbers in which each number is the sum of the two preceding ones, is a classic example of a mathematical pattern found in nature. This sequence can be observed in the arrangement of leaves on a plant, the spiral patterns of seashells, and even the branching of trees. The Fibonacci sequence reveals the underlying order and beauty of the natural world.2. The Golden Ratio: The Divine ProportionThe golden ratio, also known as the golden mean, is an irrational number approximately equal to 1.618. It is a ratio that has been found to be aesthetically pleasing and has been used in art, architecture, and nature. The golden ratio can be found in the proportions of the human body, the design of famous buildings like the Parthenon, and even in the Fibonacci sequence.3. Fractals: The Geometry of NatureFractals are complex patterns that are self-similar across different scales. They are found in nature in the form of snowflakes, coastlines, and even the patterns of lightning. Fractals demonstrate the intricate and beautiful structures that can arise from simple mathematical rules.III. Mathematics in the CosmosMathematics plays a crucial role in understanding the cosmos. From the smallest particles to the largest structures, the universe is governed by mathematical laws and principles.1. The Big Bang Theory: Mathematics and the Origin of the UniverseThe Big Bang theory, which describes the origin and evolution of the universe, is based on mathematical equations and observations. The Friedmann equations, which describe the expansion of the universe, are a testament to the power of mathematics in unraveling the mysteries of the cosmos.2. General Relativity: The Geometry of Space-TimeAlbert Einstein's theory of general relativity, which describes gravity as the curvature of space-time, is a profound example of how mathematics can be used to understand the fundamental forces that govern the universe.3. Quantum Mechanics: The Mathematics of the MicrocosmQuantum mechanics, the branch of physics that deals with the behavior of matter and energy at the smallest scales, relies heavily on mathematics. The Schrödinger equation, which describes the behavior of quantum particles, is a complex mathematical expression that has revolutionized our understanding of the universe.IV. The Impact of Mathematics on SocietyMathematics has had a profound impact on society, shaping the way welive and work. From the development of technology to the advancement of medicine, mathematics has been a driving force behind human progress.1. Technology: The Pillar of Modern SocietyTechnology has become an integral part of our lives, and mathematics is the backbone of its development. From computers to smartphones, from the internet to artificial intelligence, mathematics has enabled the creation of countless technological marvels that have transformed the world.2. Medicine: Healing Through NumbersMedicine, one of the most critical fields of human endeavor, has been revolutionized by mathematics. Mathematical models and algorithms are used to predict disease outbreaks, analyze genetic information, and develop new drugs and treatments.V. ConclusionIn conclusion, the mathematical universe is a wondrous and intricate tapestry of patterns and structures that governs the cosmos. Mathematics is not just a subject to be studied; it is the very language that describes the universe and its wonders. As we journey through the mathematical universe, we come to appreciate the beauty and elegance of numbers, shapes, and patterns that shape our world.Ladies and gentlemen, let us embrace the power of mathematics and continue to explore the mysteries of the universe. For in the words of the great mathematician Pythagoras, "All is number." Thank you.。
fortran习题答案Fortran习题答案Fortran是一种古老而又强大的编程语言，广泛应用于科学计算和工程领域。
对于初学者来说，掌握Fortran的基本语法和解决问题的能力是非常重要的。
在学习过程中，练习习题是提高编程能力的有效途径。
本文将为读者提供一些Fortran习题的答案，帮助他们更好地理解和掌握这门编程语言。
1. 编写一个Fortran程序，计算并输出1到100之间所有偶数的和。
```fortranprogram sum_of_evensimplicit noneinteger :: i, sumsum = 0do i = 2, 100, 2sum = sum + iend doprint *, "The sum of even numbers from 1 to 100 is:", sumend program sum_of_evens```2. 编写一个Fortran程序，计算并输出1到n之间所有整数的平方。
```fortranprogram squaresimplicit noneinteger :: i, nprint *, "Enter a positive integer:"read *, nprint *, "The squares of integers from 1 to", n, "are:"do i = 1, nprint *, i**2end doend program squares```3. 编写一个Fortran程序，计算并输出斐波那契数列的前n个数。
```fortranprogram fibonacciimplicit noneinteger :: i, n, fib(100)print *, "Enter the number of terms in the Fibonacci sequence:" read *, nfib(1) = 0fib(2) = 1do i = 3, nfib(i) = fib(i-1) + fib(i-2)end doprint *, "The first", n, "terms of the Fibonacci sequence are:"do i = 1, nprint *, fib(i)end doend program fibonacci```4. 编写一个Fortran程序，计算并输出一个矩阵的转置矩阵。
用英语写喜欢数学这个科目的作文全文共3篇示例，供读者参考篇1I Love MathematicsMathematics, the universal language of numbers, shapes, and patterns, has always been my favorite subject. I have a deep passion for solving complex equations, finding the patterns in numbers, and exploring the relationships between different mathematical concepts. The beauty of mathematics lies in its precision and logic, and the satisfaction of finding the right answer after a challenging problem is unparalleled.One of the reasons I love mathematics is its applicability in everyday life. From calculating grocery bills to measuring ingredients for a recipe, mathematics is always present in our daily activities. Understanding mathematical concepts not only helps us make better decisions but also sharpens our analytical and problem-solving skills. Moreover, mathematics is the foundation of many other fields, such as engineering, computer science, and physics, making it an essential subject for anyone pursuing a career in STEM fields.Another reason why I love mathematics is the sense of accomplishment it gives me. Solving a difficult problem or proving a theorem through a series of logical steps is incredibly rewarding. Mathematics challenges my mind and forces me to think critically and creatively, which helps improve my cognitive abilities and enhances my overall mental acuity. The feeling of mastering a difficult mathematical concept is truly gratifying and fuels my passion for the subject even more.Furthermore, mathematics is a subject that transcends boundaries and connects people from different cultures and backgrounds. The language of mathematics is universal, and mathematical concepts are the same no matter where you go in the world. This universal language of numbers and symbols allows us to communicate ideas and collaborate with others, fostering a sense of unity and camaraderie among mathematicians worldwide.In addition, mathematics is a subject that constantly evolves and challenges us to think outside the box. New mathematical theories and concepts are being developed all the time, pushing the boundaries of our understanding and opening up new possibilities for exploration and discovery. The ever-changing nature of mathematics keeps me engaged and eager to learnmore, as there is always something new and exciting to discover in the world of numbers and shapes.In conclusion, mathematics is not just a subject to me; it is a way of thinking, a way of problem-solving, and a way of understanding the world around us. I love mathematics for its precision, logic, and beauty, as well as its practical applications and universal appeal. Mathematics has enriched my life in countless ways and continues to inspire me to think critically, explore new ideas, and never stop learning. I am grateful for the endless opportunities that mathematics has provided me and look forward to continuing my journey of mathematical discovery for years to come. Mathematics will always hold a special place in my heart, and I will forever cherish the joy and excitement it brings me every day.篇2I like mathematics. It's not just a subject or a discipline for me, it's a way of thinking and problem-solving that I find incredibly satisfying and fulfilling. From a young age, I have always been drawn to numbers and patterns, and as I have grown older, my love for mathematics has only deepened.One of the things I love most about mathematics is its universality. Mathematics is the language of the universe, and its principles can be applied to any area of study or profession. Whether you are a scientist, an engineer, a programmer, or a businessperson, a solid understanding of mathematics is essential. This universality of mathematics means that the skills and knowledge I acquire through studying mathematics are incredibly valuable and transferrable.Another aspect of mathematics that I find fascinating is its inherent beauty. Mathematics is full of elegant and intricate patterns and structures that are a joy to discover and explore. I love the way that different branches of mathematics, such as algebra, geometry, calculus, and statistics, all interconnect and complement each other, creating a rich tapestry of knowledge and understanding.But perhaps the thing I love most about mathematics is the way it challenges and stretches my mind. Solving a difficult mathematical problem requires patience, creativity, and persistence, and the satisfaction that comes from finally cracking a tough nut is unparalleled. Mathematics forces me to think in new and innovative ways, pushing me to explore and develop my analytical and problem-solving skills.In conclusion, mathematics is more than just a subject to me—it's a passion and a way of life. I am grateful for the opportunities I have had to study mathematics and for the doors that it has opened for me. I look forward to continuing my exploration of this beautiful and challenging discipline and to discovering all that it has to offer.篇3I Love MathematicsMathematics is often considered a difficult and intimidating subject by many students, but for me, it has always been a source of joy and fascination. Ever since I was a young child, I have had a natural affinity for numbers and patterns, and as I grew older, my love for mathematics only deepened.One of the main reasons I love mathematics is because of its universal language. Mathematics is a language that transcends cultural and linguistic barriers, allowing people from all over the world to communicate and understand complex concepts through symbols and equations. This universal nature of mathematics has always fascinated me, as it shows the power of logic and reasoning to connect people and ideas across the globe.Another reason I love mathematics is because of its practical applications in everyday life. From calculating the total bill at a restaurant to designing complex algorithms for artificial intelligence, mathematics plays a crucial role in almost every aspect of our modern world. Being able to apply mathematical concepts to real-life problems gives me a sense of satisfaction and fulfillment, knowing that I am using my knowledge to make a positive impact on the world around me.Furthermore, mathematics has a beauty and elegance that is unmatched by any other subject. The symmetrical patterns of fractals, the harmonious proportions of the Fibonacci sequence, and the elegant simplicity of Euler's identity are just a few examples of the breathtaking beauty that can be found in mathematics. Studying these patterns and structures not only deepens my appreciation for the subject but also inspires me to explore new mathematical ideas and concepts.In addition to its practical applications and beauty, mathematics also provides a mental challenge that I find incredibly rewarding. Solving complex problems and proving theorems require a high level of concentration, logical reasoning, and creativity, pushing me to think outside the box and approach problems from different angles. The feeling ofsatisfaction that comes from successfully solving a difficult mathematical problem is unparalleled, and it motivates me to continue pushing myself to learn and grow as a mathematician.In conclusion, mathematics is not just a subject for me – it is a passion, a way of thinking, and a source of endless fascination. Its universal language, practical applications, beauty, and mental challenge have captivated me since childhood, and I continue to be inspired by the infinite possibilities and complexities that mathematics has to offer. I am grateful for the opportunity to study and explore this amazing subject, and I look forward to delving even deeper into the world of mathematics in the years to come. Mathematics will always hold a special place in my heart, and I will continue to pursue my love for this incredible subject for as long as I live.。
数学专有名词英文词典Mathematics Glossary: A Comprehensive English Dictionary of Mathematical TermsIntroduction:Mathematics is a language of numbers, shapes, patterns, and relationships. It plays a crucial role in various fields, including science, engineering, economics, and finance. To effectively communicate and understand mathematical concepts, it is essential to have a solid grasp of mathematical vocabulary. This article aims to provide a comprehensive English dictionary of mathematical terms, allowing readers to enhance their mathematical knowledge and fluency.A1. Abacus: A counting device that uses beads or pebbles on rods to represent numbers.2. Absolute Value: The distance of a number from zero on a number line, always expressed as a positive value.3. Algorithm: A set of step-by-step instructions used to solve a particular problem or complete a specific task.4. Angle: The measure of the separation between two lines or surfaces, usually measured in degrees.5. Area: The measure of the amount of space inside a two-dimensional figure, expressed in square units.B1. Base: The number used as a repeated factor in exponential notation.2. Binomial: An algebraic expression with two unlike terms connected by an addition or subtraction sign.3. Boundary: The edge or perimeter of a geometric shape.4. Cartesian Coordinates: A system that uses two number lines, the x-axis and y-axis, to represent the position of a point in a plane.5. Commutative Property: The property that states the order of the terms does not affect the result of addition or multiplication.C1. Circle: A closed curve with all points equidistant from a fixed center point.2. Congruent: Two figures that have the same shape and size.3. Cube: A three-dimensional solid shape with six square faces of equal size.4. Cylinder: A three-dimensional figure with two circular bases and a curved surface connecting them.5. Decimal: A number written in the base-10 system, with a decimal point separating the whole number part from the fractional part.D1. Denominator: The bottom part of a fraction that represents the number of equal parts into which a whole is divided.2. Diameter: The distance across a circle, passing through the center, and equal to twice the radius.3. Differential Equation: An equation involving derivatives that describes the relationship between a function and its derivatives.4. Dividend: The number that is divided in a division operation.5. Domain: The set of all possible input values of a function.E1. Equation: A mathematical statement that asserts the equality of two expressions, usually containing an equal sign.2. Exponent: A number that indicates how many times a base number should be multiplied by itself.3. Expression: A mathematical phrase that combines numbers, variables, and mathematical operations.4. Exponential Growth: A pattern of growth where the quantity increases exponentially over time.5. Exterior Angle: The angle formed when a line intersects two parallel lines.F1. Factor: A number or expression that divides another number or expression without leaving a remainder.2. Fraction: A number that represents part of a whole, consisting of a numerator anda denominator.3. Function: A relation that assigns each element from one set (the domain) to a unique element in another set (the range).4. Fibonacci Sequence: A sequence of numbers where each number is the sum of the two preceding ones.5. Frustum: A three-dimensional solid shape obtained by slicing the top of a cone or pyramid.G1. Geometric Sequence: A sequence of numbers where each term is obtained by multiplying the previous term by a common ratio.2. Gradient: A measure of the steepness of a line or a function at a particular point.3. Greatest Common Divisor (GCD): The largest number that divides two or more numbers without leaving a remainder.4. Graph: A visual representation of a set of values, typically using axes and points or lines.5. Group: A set of elements with a binary operation that satisfies closure, associativity, identity, and inverse properties.H1. Hyperbola: A conic section curve with two branches, symmetric to each other, and asymptotic to two intersecting lines.2. Hypotenuse: The side opposite the right angle in a right triangle, always the longest side.3. Histogram: A graphical representation of data where the data is divided into intervals and the frequency of each interval is shown as a bar.4. Hexagon: A polygon with six sides and six angles.5. Hypothesis: A proposed explanation for a phenomenon, which is then tested through experimentation and analysis.I1. Identity: A mathematical statement that is always true, regardless of the values of the variables.2. Inequality: A mathematical statement that asserts a relationship between two expressions, using symbols such as < (less than) or > (greater than).3. Integer: A whole number, either positive, negative, or zero, without any fractional or decimal part.4. Intersect: The point or set of points where two or more lines, curves, or surfaces meet.5. Irrational Number: A real number that cannot be expressed as a fraction or a terminating or repeating decimal.J1. Joint Variation: A type of variation where a variable is directly or inversely proportional to the product of two or more other variables.2. Justify: To provide a logical or mathematical reason or explanation for a statement or conclusion.K1. Kernel: The set of all inputs that map to the zero element of a function, often used in linear algebra and abstract algebra.L1. Line Segment: A part of a line bounded by two distinct endpoints.2. Logarithm: The exponent or power to which a base number must be raised to obtain a given number.3. Limit: The value that a function or sequence approaches as the input or index approaches a particular value.4. Linear Equation: An equation of the form Ax + By = C, where A, B, and C are constants, and x and y are variables.5. Locus: The set of all points that satisfy a particular condition or criteria.M1. Median: The middle value in a set of data arranged in ascending or descending order.2. Mean: The average of a set of numbers, obtained by summing all the values and dividing by the total count.3. Mode: The value or values that appear most frequently in a data set.4. Matrix: A rectangular array of numbers, symbols, or expressions arranged in rows and columns.5. Midpoint: The point that divides a line segment into two equal halves.N1. Natural Numbers: The set of positive whole numbers, excluding zero.2. Negative: A number less than zero, often represented with a minus sign.3. Nonagon: A polygon with nine sides and nine angles.4. Null Set: A set that contains no elements, often represented by the symbol Ø or { }.5. Numerator: The top part of a fraction that represents the number of equal parts being considered.O1. Obtuse Angle: An angle that measures more than 90 degrees but less than 180 degrees.2. Octagon: A polygon with eight sides and eight angles.3. Origin: The point (0, 0) on a coordinate plane, where the x-axis and y-axis intersect.4. Order of Operations: The set of rules for evaluating mathematical expressions, typically following the sequence of parentheses, exponents, multiplication, division, addition, and subtraction.5. Odd Number: An integer that cannot be divided evenly by 2.P1. Parabola: A conic section curve with a U shape, symmetric about a vertical line called the axis of symmetry.2. Pi (π): A mathematical constant representing the ratio of a circle's circumference to its diameter, approximately equal to3.14159.3. Probability: The measure of the likelihood that a particular event will occur, often expressed as a fraction, decimal, or percentage.4. Prime Number: A natural number greater than 1 that has no positive divisors other than 1 and itself.5. Prism: A three-dimensional figure with two parallel congruent bases and rectangular or triangular sides connecting the bases.Q1. Quadrant: One of the four regions obtained by dividing a coordinate plane into four equal parts.2. Quadrilateral: A polygon with four sides and four angles.3. Quartile: Each of the three values that divide a data set into four equal parts, each containing 25% of the data.4. Quotient: The result obtained from the division of one number by another.5. Quaternion: A four-dimensional extension of complex numbers, often used in advanced mathematics and physics.R1. Radius: The distance from the center of a circle or sphere to any point on its circumference or surface, always half of the diameter.2. Radical: The symbol √ used to represent the square root of a number or the principal root of a higher-order root.3. Ratio: A comparison of two quantities, often expressed as a fraction, using a colon, or as a verbal statement.4. Reflection: A transformation that flips a figure over a line, creating a mirror image.5. Rhombus: A parallelogram with all four sides of equal length.S1. Scalene Triangle: A triangle with no equal sides.2. Sector: The region bounded by two radii of a circle and the arc between them.3. Series: The sum of the terms in a sequence, often represented using sigma notation.4. Sphere: A three-dimensional object in which every point on the surface is equidistant from the center point.5. Square: A polygon with four equal sides and four right angles.T1. Tangent: A trigonometric function that represents the ratio of the length of the side opposite an acute angle to the length of the adjacent side.2. Theorem: A mathematical statement that has been proven to be true based on previously established results.3. Transversal: A line that intersects two or more other lines, typically forming angles at the intersection points.4. Trapezoid: A quadrilateral with one pair of parallel sides.5. Triangle: A polygon with three sides and three angles.U1. Union: The combination of two or more sets to form a new set that contains all the elements of the original sets.2. Unit: A standard quantity used to measure or compare other quantities.3. Unit Circle: A circle with a radius of 1, often used in trigonometry to define trigonometric functions.4. Undefined: A term used to describe a mathematical expression or operation that does not have a meaning or value.5. Variable: A symbol or letter used to represent an unknown or changing quantity in an equation or expression.V1. Vertex: A point where two or more lines, rays, or line segments meet.2. Volume: The measure of the amount of space occupied by a three-dimensional object, often expressed in cubic units.3. Variable: A symbol or letter used to represent an unknown or changing quantity in an equation or expression.4. Vector: A quantity with both magnitude (size) and direction, often represented as an arrow.5. Venn Diagram: A graphical representation of the relationships between different sets using overlapping circles or other shapes.W1. Whole Numbers: The set of non-negative integers, including zero.2. Weighted Average: An average calculated by giving different weights or importance to different values or data points.3. Work: In physics, a measure of the energy transfer that occurs when an object is moved against an external force.4. Wavelength: The distance between two corresponding points on a wave, often represented by the symbol λ.5. Width: The measurement or extent of something from side to side.X1. x-axis: The horizontal number line in a coordinate plane.2. x-intercept: The point where a graph or a curve intersects the x-axis.3. x-coordinate: The horizontal component of a point's location on a coordinate plane.4. xy-plane: A two-dimensional coordinate plane formed by the x-axis and the y-axis.5. x-variable: A variable commonly used to represent the horizontal axis or the input in a mathematical equation or function.Y1. y-axis: The vertical number line in a coordinate plane.2. y-intercept: The point where a graph or a curve intersects the y-axis.3. y-coordinate: The vertical component of a point's location on a coordinate plane.4. y-variable: A variable commonly used to represent the vertical axis or the output in a mathematical equation or function.5. y=mx+b: The equation of a straight line in slope-intercept form, where m represents the slope and b represents the y-intercept.Z1. Zero: The number denoted by 0, often used as a placeholder or a starting point in the number system.2. Zero Pair: A pair of numbers that add up to zero when combined, often used in integer addition and subtraction.3. Zero Product Property: The property that states if the product of two or more factors is zero, then at least one of the factors must be zero.4. Zero Slope: A line that is horizontal and has a slope of 0.5. Zeroth Power: The exponent of 0, which always equals 1.Conclusion:This comprehensive English dictionary of mathematical terms provides an extensive list of vocabulary essential for understanding and communicating mathematical concepts. With the knowledge of these terms, readers can enhance their mathematical fluency and explore various branches of mathematics with greater confidence. Remember, mathematics is not just about numbers, but also about understanding the language that describes the beauty and intricacies of the subject.。
