Nature of Mathematics ‘The emergence of digital technology has sparked a monumental rise in the rate at which we consume and produce data. Before the Internet, it could take hours to get several volumes of resources from the library for a research paper. Today, a few minutes (or seconds, depending on the speed of your connection) using your mobile device’s browser could get you the same information, or even more. A few decades ago, it took hours for photographs to be printed and shared. Now, it only takes a matter of seconds for your perfect selfie to be uploaded and viewed by your relatives and friends on the other side of the world. In this fast-paced society, how often have you stopped to appreciate the beauty of the things around you? Have you ever paused and pondered about the underlying principles that govern the universe? How about contemplating about the processes and mechanisms that make our lives easier, if not more comfortable? Most people do the same routine tasks every day and the fundamental concepts that make these activities possible are often overlooked. As rational creatures, we tend to identify and follow patterns, whether consciously or subcor jously. Recognizing patterns feels natural, like our brain is hardwired to recognize them. Early, humans recognized the repeating interval of day and night, the cycle of the moon, the rising and falling of tides, and the changing of the seasons. Awareness of these patterns allowed humans to survive. In a similar fashion, many flora and fauna also follow certain patterns such as the arrangement of leaves and stems in a plant, the shape of a snowflake, the flowers’ petals, or even the shape of a snail's shell. Do you also notice patterns around you? What other examples can you think of? In this chapter, we will be looking at patterns and regularities in the world, and how mathematics comes into play, both in nature and in human endeavors.Gaeeeemes CHAPTER 1 | Nature of Mathematics . ‘SECTION 1.1 Patterns and Numbers in Nature and the World In the general sense of the word, patterns are regular, repeated, or recurring forms or designs. We see patterns every day—from the layout of floor tiles, designs of skyscrapers, tothe way we tie our shoelaces. Studying patterns help students in identifying relationships and finding logical connections to form generalizations and make predictions. Let’s take a look at this pattern: S000 What do you think will be the next face in the sequence? It should be easy enough to note that the pattern is made up of two smiling faces—one without teeth and one with teeth. Beginning with a toothless face, the two faces then alternate. Logically, the face that EZ what comes Next? ‘What is the next figure in the pattern below? [- I = "t 2 F fal Solution Looking at the given figures, the lines seem to rotate at 90-degree intervals in a | counterclockwise direction, always parallel to one side of the square. Hence, either A or | B could be the answer. Checking the other patterns, the length of the lines inside the square follow a decreasing trend. So again, either A or B could be the answer. Finally, looking at the number of the lines inside the box, each succeeding figure has the number of lines increase by 1. This means that the next figure should have five lines inside. This | leads to option A as the correct choice. |SECTION 1.1 | Patterns and Numbers in Nature and the World Ce TR TTT ‘What is the next figure in the given pattern? | ke [STAUEEERA what Number comes Next? What number comes next in 1, 3, 5, 7,9, _? | Solution Looking at the given numbers, the sequence is increasing, with each term being two ‘more than the previous term: 3 = 1 +2; 5=3+2;7=5+2;9=7 +2. Therefore, the next term should be 11 =9 +2. Seas What number comes next in 1, 4, 9, 16, 25, __? Patterns indicate a sense of structure and organization that it seems only humans are capable of producing these intricate, creative, and amazing formations. It is from this perspective that some people see an “intelligent design” in the way that nature forms. Snowflakes and Honeycombs Recall that symmetry indicates that you can draw an imaginary fine across an object and the resulting parts are mirror images of each other. ‘The figure above is symmetric about the axis indicated by the dotted line. Note that the left and right portions are exactly the same. This type of symmetry, known as line or bilateral symmetry, is evident in most animals, including humans. Look in a mirror and see how the left and right sides of your face closely match.eee CHAPTER 1 | Nature of Mathematics ‘©JanakaMaharageDharmasena/iStock/Thinkstock Leonardo da Vinci's Vitruvian Man showing the proportions and symmetry of the human body \ There are other types of symmetry depending on the number of sides or faces that are \ symmetrical, Take a look at these images. i Q i 3 i 3g z 3 Spiderwort with three-fold symmetry ‘This starfish has a five-fold symmetry. @abadonianviStock/ThinkstockNote that if you rotate the spiderwort and starfish above by several degrees, you can still achieve the same appearance as the original position. This is known as rotational symmetry. The smallest angle that a figure can be rotated while still preserving the original formation is called the angle of rotation. For the spiderwort, the angle of rotation is 120° while the angle of rotation for the baby starfish is 72°. ‘A mote common way of describing rotational symmetry is by onder of roti, Order of Rotation A figure has a rotational symmetry of order 1 (n-fold rotational symmetry) if I/n of a complete turn leaves the figure unchanged. To compute for the angle of rotation, we Use the following formula: \ Angle of rotation = 362° Consider this image of a snowflake. ‘©ChaoticMind75/Stock/Thinkstock Itcan be observed that the patterns on a snowflake repeat six times, indicating that there is a six-fold symmetry. To determine the angle of rotation, we simply divide 360° by 6 to get 60°. Many combinations and complex shapes of snowflakes may occur, which lead some people to think that “no two are alike”. If you look closely, however, many snowflakes are not perfectly symmetric due to the effects of humidity and temperature ‘on the ice crystal as it forms.CHAPTER 1 | Nature of Mathematics Another marvel of nature’s design is the structure and shape of a honeycomb. People have long wondered how bees, despite their very small size, are able to produce such arrangement while humans would generally need the use of a ruler and compass to accomplish the same feat. It is observed that such formation enables the bee colony to ‘maximize their storage of honey using the smallest amount of wax. _ You can try it out for yourself. Using several coins of the same size, try to cover as much / >. area of a piece of paper with coins. If you arrange the coins in a square formation, there \\ are still plenty of spots that are exposed. Following the hexagonal formation, however, ‘with the second row of coins snugly fitted between the first row of coins, you will notice that more area will be covered. ‘Square Packi Hexagonal Packing (iSidhe/iStock/ThinkstockSECTION 1.1 | Patterns and Numbers in Nature and the World iC Aaa ‘Translating this idea to three-dimensional space, we can conclude that hexagonal formations are more optimal in making use of the available space. These are referred to as packing problems. Packing problems involve finding the optimum method of filling up a given space such as a cubic or spherical container. The bees have instinctively found the best solution, evident in the hexagonal construction of their hives. These geometric patterns are not only simple and beautiful, but also optimally functional. Let us illustrate this mathematically. Suppose you have circles of radius 1 cm, each of which will then have an area of x cm”, We are then going to fill a plane with these circles using square packing and hexagonal packing. \_ For square packing, each square will have an area of 4 cm?, Note from the figure that for each square, it can fit only one circle (4 quarters). The percentage of the square’s area covered by circles will be area of the circles area of the square X 100% ~ 78.54% For hexagonal packing, we can think of each hexagon as composed of six equilateral triangles with side equal to 2. em. The area of each triangle is given by side'+V3 _ Qem)'wV3 _dem'V3 _ yofleet) CHAPTER 1 | Nature of Mathematics alexmatamata/iStock/Thinkstock This gives the area of the hexagon as 6V3 cm*. Looking at the figure, there are 3 circles that could fit inside one hexagon (the whole circle in the middle, and 6 one- thirds ofa circle), which gives the total area as 37 cm’, The percentage of the hexagon’s area covered by circles will be area of thecincles ggg, _ 37 em area of the hexagon 6V3 em? 100% ~ 90.69% Comparing the two percentages, we can clearly see that using hexagons will cover a larger area than when using squares. Tigers’ Stripes and Hyenas' Spots Patterns are also exhibited in the external appearances of animals. We are familiar with how a tiger looks—distinctive reddish-orange fur and dark stripes. Hyenas, another predator from Africa, are also covered in patterns of spots. These seemingly random designs are believed to be governed by mathematical equations. According to a theory by Alan Turing, the man famous for breaking the Enigma code during World War Il, chemical reactions and diffusion processes in cells determine these growth patterns. More recent studies addressed the question of why some species grow vertical stripes while others have horizontal ones. A new model by Harvard University researchers predicts that there are three variables that could affect the orientation of these stripes— the substance that amplifies the density of stripe patterns; the substance that changes one of the parameters involved in stripe formation; and the physical change in the direction of the origin of the stripe. ‘©Goddard_Photography/iStock/ThinkstockSECTION 1.1 | Patterns and Numbers in Nature and the World [iii Jal The Sunflower Looking at a sunflower up close, you will notice that there is a definite pattern of clockwise and counterclockwise arcs or spirals extending outward from the center of the flower. This is another demonstration of how nature works to optimize the available space. This arrangement allows the sunflower seeds to occupy the flower head in a way that maximizes their access to light and necessary nutrients. (©ThinkstockImages/iStock/Thinkstock Sunflower The Snail's Shell @AndreaAstes/iStock/Thinkstock Snail’s shellRamialege CHAPTER 1 | Nature of Mathematics We are also very familiar with spiral patterns. The most common spiral patterns can be seen in whirlpools and in the shells of snails and other similar mollusks. Snails are born ith their shells, called protoconch, which start out as fragile and colorless. Eventually, these original shells harden as the snails consume calcium. As the snails grow, their shells also expand proportionately so that they can continue to live inside their shells. This process results in a refined spiral structure that is even more visible when the shell is sliced. This figure, called an equiangular spiral, follows the rule that as the distance from the spiral center increases (radius), the amplitudes of the angles formed by the radii to the point and the tangent to the point remain constant, This is another example of how nature seems to follow a certain set of rules governed by mathematics ‘©vectortatw/iStock/Thinkstock A Logarithmic Spiral Flower Petals * Flowers are easily considered as things of beauty. Their vibrant colors and fragrant ‘odors make them very appealing as gifts or decorations. If you look more closely, you will note that different flowers have different number of petals. Take the iris and trillium, for example. Both flowers have only 3 petals. Sudnitskaya/iStock/Thinkstock § 2 5 4 3 oul i ‘Trilliumz People celebrating the Songkran New Year Festival in Bangkok, Thailand SECTION 1.1 | Patterns and Numbers in Nature and the World [IEE Flowers with five petals are said to be the most common, These include buttercup, columbine, and hibiscus. Among those flowers with eight petals are clematis and delphinium, while ragwort and marigold have thirteen. These numbers are all Fibonacci numbers, which we will discuss in detail in the next section and in the next chapter, lupiterimages/PHOTOS.com/Thinkstock Buttercup flower with five petals World Population ‘As of 2017, it is estimated that the world population is about 7.6 billion, World leaders, sociologists, and anthropologists are interested in studying population, including its growth, Mathematics can be used to model population growth, Recall that the formula for exponential growth is A= Pe”, where A is the size of the population after it grows, is the initial number of people, ris the rate of growth, and tis time. Recall further that ¢ is Euler's constant with an approximate value of 2.718. Plugging in values to this formula would result in the population size after time ¢ with a growth rate of r. [SZUEBE Population Growth The exponential growth model A = 30e™ describes the population of a city in the Philippines in thousands, r years after 1995. ‘a, What was the population of the city in 1995? b. What will be the population in 2017?BREERD cu arte 1 | Nature of Mathematics the rate “r” being a negative value, The time "” should coincide with the given rate, ie. ifthe time is in years, then the growth rate should also be a yearly growth rate. Solution ‘a, Since our exponential growth model describes the population t years after 1995, we consider 1995 as t= 0 and then solve for A, our population size. A= 30 A= 30e%200 Replace ¢ with r= 0. A= 30° A=30(1) a =30 ‘Therefore, the city population in 1995 was 30,000. b. We need to find A for the year 2017. To find t, we subtract 2017 and 1995 to get = 22, which we then plug in to our exponential growth model. A=30e A= 3060 Replace t with t= 22. = 30e# A = 30(1.55271) te 155271 A= 46.5813 Therefore, the city population would be about 46,581 in 2017. (Tas ‘The exponential growth model A = 50e™” describes the population of a city in the Philippiites in thousands, r years after 1997. a. What is the population after 20 years? 1b, What is the population in 2037?SECTION 1.2 | The Fibonacci Sequence EE SAAN ™ Determine what comes next in the given patterns. 7. A= 1,240,000; r= 8% per year; t= 30 years 1. ACE,G,L,— 8. A= 786,000; P = 247,000; 2. 15 10 14 10 13 10 ___ = 525,600; 3. 3612244896 __ = Answer completely. 4, 2730333639 10. Suppose the population of a certain bacteria in a laboratory sample is 100. If it doubles in population 5. 41 39 37 35 33 __ every 6 hours, what is the growth rate? How many = Substitute the given values in the formula A = Pe" to bacteria will there be in two days? find the missing quantity. 6. P = 680,000; r= 12% per year; 1= 8 years RECTION 1.2 The Fibonacci Sequence As we have seen in the previous section, the human mind is hardwired to recognize patterns. In mathematics, we can generate patterns by performing one or several ‘mathematical operations repeatedly. Suppose we choose the number 3 as the first number in our pattern. We then choose to add 5 to our first number, resulting in 8, which is our second number. Repeating this process, we obtain 13, 18, 23, 28, ... as the succeeding numbers that form our pattern. In mathematics, we call these ordered lists of numbers a sequence. Sequence A sequence is an ordered list of numbers, called terms, that may have repeated values. The arrangement of these terms is set by a definite rule. EEA cenerating a Sequence Analyze the given sequence for its rule and identify the next three terms. a. 1, 10, 100, 1000 b, 2,5,9, 14,20PS) CHAPTER 1 | Nature of Mathematics Solution a, Looking at the set of numbers, it can be observed that each term is a power of 10: 1 = 10°, 10 = 10', 100 = 10°, and 1,000 = 10°, Following this rule, the next three terms are: 10* = 10,000, 10° = 100,000, and 10° = 1,000,000. b. The difference between the first and second terms (2 and 5) is 3. The difference between the second and third terms (5 and 9) is 4. The difference between the third and fourth terms (9 and 14) is 5. The difference between the fourth and the fifth terms is 6. Following this rule, it can be deduced that to obtain the next three terms, we should add 7, 8, 9, respectively, to the current term. Hence, the next three terms are 20+7 = 27,27 +8 = 35,35 +9=44. | | 1 | | | < Analyze the given sequence for its rule and identify the next three terms. a, 16, 32, 64, 128, a 1,1,2,3,5,8 ‘The sequence in Check Your Progress 2 Item B is a special sequence called the Fibonacci sequence. It is named after the Italian mathematician Leonardo of Pisa, who was better known by his nickname Fibonacci. He is said to have discovered this sequence as he looked at how a hypothesized group of rabbits bred and reproduced. The problem involved having a single pair of rabbits and then finding out how many pairs of rabbits will be born in a year, with the assumption that a new pair of rabbits is born each month and this new pair, in turn, gives birth to additional pairs of rabbits beginning at two months after they were born. He noted that the set of numbers generated from this problem could be extended by getting the sum of the two previous terms. Starting with 0 and 1, the succeeding terms in the sequence can be generated by adding the two numbers that came before the term: O+1=1 01,1 l41=2 0,1,1,2 142=3 0,1,1,2,3 24325 0,1,1,2,3,.5 34528 0,1,1,2,3,5,8 S+8=13 0,1,1,2,3,5,8, 13, 0,1,1,2,3,5,8, 13, While the sequence is widely known as Fibonacci sequence, this pattern is said to have been discovered much earlier in India. According to some scholarly articles, Fibona sequence is evident in the number of variations of a particular category of Sanskrit and Prakrit poetry meters. In poetry, meter refers to the rhythmic pattern of syllables. Fibonacci sequence has many interesting properties. Among these is that this pattern is very visible in nature. Some of nature’s most beautiful patterns, like the spiral arrangement of sunflower seeds, the number of petals in a flower, and the shape of a snail’s shell—things that we looked at earlier in this chapter—all contain Fibonacci numbers. It is also interesting to note that the ratios of successive Fibonacci numbers approach the number @ (Phi), also known as the Golden Ratio. This is approximately equal to 1.618.SECTION 1.2 | The Fibonacci Sequence [iE saiaaaaaa 2 2.0000 31.5000 34 21.6190 2 21 31.6667 3 81.600 8-1 6182 55 The Golden Ratio can also be expressed as the ratio between two numbers, ifthe latter is also the ratio between the sum and the larger of the two numbers. Geometrically, it can also:be visualized as a rectangle perfectly formed by a square and another rectangle, which can be repeated infinitely inside each section. 3 § & 5 2 & Golden rectangle with the golden spiral ‘Shapes and figures that bear this proportion are generally considered to be aesthetically pleasing. As such, this ratio is visible in many works of art and architecture such as in the Mona Lisa, the Notre Dame Cathedral, and the Parthenon, In fact, the human DNA molecule also contains Fibonacci numbers, being 34 angstroms long by 21 Angstroms wide for each full cycle of the double helix spiral. As shown in the list above, this approximates the Golden Ratio at a value of about 1.619 (1 Angstrom = 10° meter or 0.1 nanometer),PL CHAPTER 1 | Nature of Mathematics ©wendylfiviStock/Thinkstock ‘The Parthenon is said to exhibit the application of the Golden Ratio. ® Let Fib(n) be the nth term of the Fibonacci sequence, with Fib(1) = 1, Fib2) Find Fib (8). Find Fib (19). If Fib (22) = 17,711 and Fib(24) = 46,368, what is Fib(23)? Evaluate the following sums: a, Fib (1) + Fib (2)=__ b, Fib (1) + Fib (2) + Fib 3) =__ ¢. Fib (1) + Fib (2) + Fib (3) + Fib @) 5. Determine the pattern in the successive sums from the previous question, What will be the sum of Fib (1) + Fib(2) + ... + Fib(10)? = Answer completely. 6. If you have a wooden board that is 0.75 meters wide, how long should you cut it such that the Golden Ratio is observed? Use 1.618 as the value of the Golden Ratio.SECTION 1.3 | Mathematis for our Wold Mathematics for our World We have seen in the preceding sections how evident mathematics is in the natural world, specifically in how the patterns that we observe in nature follow logical and mathematical structures. Its, therefore, important that we learn mathematical concepts and apply them in solving societal problems. Mathematics for Organization A lot of events happen around us. In the blink of an eye, several children have already been born, liters of water have been consumed, or thousands of tweets have been posted. For us to make sense of all available information, we need mathematical tools to help us ‘maké sound analysis and better decisions. For instance, a particular store can gather data on the shopping habits of its customers and make necessary adjustments to help drive sales. Scientists can plot bird migration routes to help conserve endangered animal populations. Social media analysts can crunch all online postings using software to gauge the netizens’ sentiments on particular issues or personalities. an ee coe coin NEF TS m sult ADIGA ViStock/Thinkstock ORawpii Software can generate a map of words that are most talked about in social media. The bigger the font, the more netizens are talking about the concept or topic. Mathematics for Prediction It is sometimes said that history repeats itself, As much as we can use mathematical models using existing data to generate analysis and interpretations, we can also use them to make predictions, Applying the concept of probability, experts can calculate the chance + of an event occurring. The weather is a prime example, Based on historical patterns, meteorologists can make forecasts to help us prepare for our day-to-day activities. Theypiace CHAPTER 1 | Nature of Mathematics can also warn us of weather disturbances that can affect our activities for weeks or months. Astronomers also use patterns to predict the occurrence of meteor showers or eclipses. In 2017, announcements were made about heavenly phenomena such as the Draconid Meteor Shower and “The Great American Eclipse.” They were able to tell when these phenomena would occur and where would be the best places to view them. A total solar eclipse Mathematics for Control We have demonstrated by means of examples around us that patterns are definitely present in the universe. There seems to be an underlying mathematical structure in the way that natural objects and phenomenon behave. While photographers could capture a single moment through a snapshot, videographers could record events as they unfold. Painters and sculptors could create masterpieces in interpreting their surroundings, poets could use beautiful words to describe an object, and musicians could capture and reproduce sounds that they hear. These observations of nature, as well as their interactions, and relationships, could be more elegantly described by means of mathematical equations. ‘As stated by astrophysicist Brian Greene, “With a few symbols on a page, you can describe a wealth of physical phenomena.” Itis interesting then to ponder on how mathematics, an invention of the human mind, seems to permeate the natural laws that hold the universe together. There have been instances when a natural phenomenon has been speculated to exist because mathematicsSECTION 1.3 | Mathematics for our World (NLA says so but no hard evidence have been found to support its existence. Such phenomena were proven to exist only when advancements in technology have allowed us to expand our horizons. For example, in 1916, Albert Einstein hypothesized the existence of gravitational waves based on his theory of general relativity. This is when “ripples” are formed in the fabric of space-time due to large and violent cosmic events, very much like when a pebble is thrown on a stagnant pond. About a hundred years later, the Laser Interferometer Gravitational Wave Observatory (LIGO) announced that it found evidence of this phenomenon, i ‘ & i 6 A large cosmic disturbance could cause ripples in space-time, like a pebble thrown in a pond. Though the use of mathematics, man is also able to exert control over himself and the effects of nature. The threat of climate change and global warming has been the subject of much debate over the years. It is believed that unless man changes his behavior, Patterns are said to indicate that sea levels could rise to catastrophic levels as the polar caps’melt due to the increase in global temperatures. To ensure that greenhouse gas concentrations in the atmosphere are kept at levels that would not interfere with the climate system, the United Nations Framework Convention on Climate Change (UNFCCC) was signed in 1992 and has 197 parties as of December 2015.ES carter 1 | Nature of Mathematics Mathematics is Indispensable How is it possible that mathematics, a product of human thought that is independent of experience, fits so excellently the objects of reality? Albert Einstein In this chapter, it was highlighted how mathematics plays a huge role in the underpinnings of, our world. We have seen it in living creatures and natural phenomena. We have also looked : at examples of how mathematical concepts could be applied. Whether you are on your way to becoming a doctor, an engineer, an entrepreneur, or a chef, a knowledge of mathematics will be helpful. At the most basic level, logical reasoning and critical thinking are crucial skills that are needed in any endeavor. As such, the study of mathematics should be embraced a8 it paves the way for more educated decisions and in a way, brings us closer to understanding the natural world. [EXERCISE SET 1.3 = Answer the following questions 41. Vlad had a summer job packing sweets, Each pack should weigh 200 grams. Vlad had to make 15 2. A certain study found that the relationship between the students’ exam scores (y) and the number of hours they spent studying (x) is given by the packs of sweets. He checked the weights, in grams, equation y = 10x + 45. Using this information, what correct to the nearest gram. Following are his Till be the ectiniied core of «staat akan spent aac 4 hours studying? 212, 206-203, 206199 3. The distance traveled by an object given its initial 196 197 197 209-206 velocity and acceleration over a period of time ¢ is given by the equation d = vot + Yeat’. Find the 198 191-196-206. 207 distance traveled by an airplane before it takes off : if it starts from rest and accelerates down a runway What is the most frequent data? ai aso ae erat S s