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Greatest Discoveries in Mathematics

October 16, 2021

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Mathematics is a fundamental part of human thought and logic, and integral to attempts at understanding the world and ourselves. Mathematics provides an effective way of building mental discipline and encourages logical reasoning and mental rigor. In addition, mathematical knowledge plays a crucial role in understanding the contents of other school subjects such as science, social studies, and even music and art.

In this article, we bring you some of the greatest discoveries in Mathematics.

1. Trigonometry

Though Trigonometry goes back to the Greek period, the character of the subject started to resemble modern form only after the time of Aryabhata. From here it went to Europe through the Arabs and went into several modifications to reach its present form.  In ancient times Trigonometry was considered a part of astronomy.  Three functions were introduced: jya, kojya, and ukramajya.

The first one is r sin a where r is the radius of the circle and sin a is the angle subtended at the center.  The second one is r cos a and the third one is r (1 – cos a). By taking the radius of the circle to be 1, we get the modern trigonometric functions. Various relationships between the sine of an arc and its integral and fractional multiples were used to construct sine tables for different arcs lying between 0 and 90°.

Trigonometry may not have its direct applications in solving practical issues, but it is used in various things that we enjoy so much. For example, music, as you know sound travels in waves, and this pattern though not as regular as a sine or cosine function, is still useful in developing computer music. A computer cannot obviously listen to and comprehend music as we do, so computers represent it mathematically by its constituent sound waves. And this means sound engineers need to know at least the basics of trigonometry. And the good music that these sound engineers produce is used to calm us from our hectic, stress full life – All thanks to trigonometry.

• Trigonometry is used to measure the height of buildings or mountains: If you know the distance from where you observe the building and the angle of elevation you can easily find the height of the building. Similarly, if you have the value of one side and the angle of depression from the top of the building you can find and another side in the triangle, all you need to know is one side and angle of the triangle.
• Trigonometry in Video Games: Have you ever played the game, Mario? When you see him so smoothly glide over the roadblocks. He doesn’t really jump straight along the Y-axis, it is a slightly curved path or a parabolic path that he takes to tackle the obstacles on his way. Trigonometry helps Mario jump over these obstacles. As you know Gaming industry is all about IT and computers and hence Trigonometry is of equal importance for these engineers.
• Trigonometry in Construction: In construction, we need trigonometry to calculate the following:
• Measuring fields, lots, and areas;
• Making walls parallel and perpendicular;
• Installing ceramic tiles;
• Roof inclination;
• The height of the building, the width length, etc., and the many other such things where it becomes necessary to use trigonometry.
• Architects use trigonometry to calculate structural load, roof slopes, ground surfaces, and many other aspects, including sun shading and light angles.
• Trigonometry in Flight Engineering: Flight engineers have to take into account their speed, distance, and direction along with the speed and direction of the wind. The wind plays an important role in how and when a plane will arrive where ever needed this is solved using vectors to create a triangle using trigonometry to solve. For example, if a plane is traveling at 234 mph, 45 degrees N of E, and there is a wind blowing due south at 20 mph. Trigonometry will help to solve for that third side of your triangle which will lead the plane in the right direction, the plane will actually travel with the force of wind added on to its course.
• Trigonometry in Archaeology: Trigonometry is used to divide up the excavation sites properly into equal areas of work. Archaeologists identify different tools used by the civilization, using trigonometry can help them in these excavates. They can also use it to measure the distance from underground water systems.
• Trigonometry in Criminology: In criminology, trigonometry can help to calculate a projectile’s trajectory, to estimate what might have caused a collision in a car accident or how did an object fall down from somewhere, or in which angle was a bullet shot, etc.
• Trigonometry in Marine Biology: Marine biologists often use trigonometry to establish measurements. For example, to find out how light levels at different depths affect the ability of algae to photosynthesize. Trigonometry is used in finding the distance between celestial bodies. Also, marine biologists utilize mathematical models to measure and understand sea animals and their behaviour. Marine biologists may use trigonometry to determine the size of wild animals from a distance.
• Trigonometry in Navigation: Trigonometry is used to set directions such as the north, south, east, or west, it tells you what direction to take with the compass to get in a straight direction. It is used in navigation in order to pinpoint a location. It is also used to find the distance of the shore from a point in the sea. It is also used to see the horizon.

The history of the quadratic formula can be traced all the way back to the ancient Egyptians. The theory is that the Egyptians knew how to calculate the area of different shapes, but not how to calculate the length of the sides of a given shape, e.g. the wall size needed to create a given floor plan.

To solve the practical problem, by around 1500 BC, Egyptian mathematicians had created a table for the area and side length of different shapes. This table could be used, for example, to determine the size of a hayloft needed to store a certain amount of hay.

While this method worked fine, it was not a general solution. The next approach may have come from the Babylonians, who had an advantage over the Egyptians in that their number system was more like the one we use today (although it was sexagesimal, or base-60). This made addition and multiplication easier. It is thought that by around 400 BC, the Babylonians had developed the method of completing the square to solve generic problems involving areas. A similar method also appears in Chinese documents at around the same time.

It was the Indian mathematician, Brahmagupta, who came up with the solution to the quadratic equation, in his 628 AD treatise Brāhmasphuṭasiddhānta (‘Correctly Established Doctrine of Brahma‘).

Quadratic equation came into existence because of the simple need to conveniently find the area of squared and rectangular bodies, but from the days of its origin, this popular maths equation has now come a long way to prove its significance in the real world.

• Sports analysts and team selectors use different quadratic equations to analyze the performance of athletes over a period of time. Moreover, sporting events such as javelin and basketball use quadratic formulas to find the accurate distance, speed, or time required to score more.
• Military and law enforcement units use quadratic formulas to calculate the speed of missiles, moving vehicles and aircraft. The landing coordinates of planes, tanks, and jets are also determined using the formulas from quadratic equations.
• Auto parts such as brakes and curved elements are designed on the basis of the quadratic formula. Pension plans, insurance models, employee work performance; all these parameters are calculated using quadratic equations. Apart from these, the boundaries in agricultural lands and the area of fields with the highest yield are also measured by the means of the quadratic formula.
• The construction of monuments, offices, flats, roads, bridges, and more involves complex calculations and area measurements, so all these mathematical complications are dealt with using different quadratic formulas.
• The angles at which a satellite dish is set to catch the signals are also determined using the quadratic equations. Also, to figure out the way a dish receives signals from multiple satellites at the same time, a quadratic equation is taken into account.

3. Euler’s Identity

As a famous mathematical equation, Euler’s identity is often referred to as a mathematical jewel. Euler’s identity is the famous mathematical equation e^(i*pi) + 1 = 0 where e is Euler’s number, approximately equal to 2.71828, i is the imaginary number where i^2 = -1, and pi is the ratio of a circle’s circumference to the circle’s diameter approximately equal to 3.14. It is named after Leonhard Euler, a Swiss mathematician who discovered this formula in the 1700s.

Why is this worth remembering? It is worth remembering because it is the only equation that so simply links together the mathematical constants of pi, i, and e along with 0 and 1.

Mathematicians love Euler’s identity because it is considered a mathematical beauty since it combines five constants of math and three math operations, each occurring only one time. The three operations that it contains are exponentiation, multiplication, and addition. The five constants that this equation combines are the number 0, the number 1, the number pi, the number e, and the number i.

We know the numbers 0 and 1. We recall that the number pi is approximately 3.14 and it goes on forever. The number e, like the number pi, continues forever and is approximately 2.71828. The number i is our imaginary number where i^2 is equal to -1.

Why is this so beautiful for mathematicians? It is a beauty because it is such a simple equation that shows the relationship of so many constants of math. Can you think of other equations that are just as simple and that relate just as many constants together?

Euler’s identity is actually a special case of Euler’s formula, e^(i*x) = cos x + i sin x, when x is equal to pi. When x is equal to pi, cosine of pi equals -1 and sine of pi equals 0, and we get e^(i*pi) = -1 + 0i. The 0 imaginary part goes away, and we get e^(i*pi) = -1. Moving the -1 over to the other side by adding gives us Euler’s identity. Looking at Euler’s formula, e^(i*x) = cos x + i sin x, we see that e taken to an imaginary power equals a complex number consisting of a real part (the cosine part) and an imaginary part (the sine part).

4. Fibonacci Numbers

The Fibonacci sequence is one of the most famous formulas in mathematics. Each number in the sequence is the sum of the two numbers that precede it. So, the sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, and so on. The mathematical equation describing it is Xn+2= Xn+1 + Xn

A mainstay of high-school and undergraduate classes, it’s been called “nature’s secret code,” and “nature’s universal rule.” It is said to govern the dimensions of everything from the Great Pyramid at Giza, to the iconic seashell that likely graced the cover of your school math textbook.

Many sources claim it was first discovered or “invented” by Leonardo Fibonacci. The Italian mathematician, who was born around A.D. 1170, was originally known as Leonardo of Pisa, said Keith Devlin, a mathematician at Stanford University. Only in the 19th century did historians come up with the nickname Fibonacci (roughly meaning, “son of the Bonacci clan”), to distinguish the mathematician from another famous Leonardo of Pisa, Devlin said. [Large Numbers that Define the Universe]

But Leonardo of Pisa did not actually discover the sequence, said Devlin, who is also the author of “Finding Fibonacci: The Quest to Rediscover the Forgotten Mathematical Genius Who Changed the World,” (Princeton University Press, 2017). Ancient Sanskrit texts that used the Hindu-Arabic numeral system first mention it, and those predate Leonardo of Pisa by centuries.

But what exactly is the significance of the Fibonacci sequence? Other than being a neat teaching tool, it shows up in a few places in nature. However, it’s not some secret code that governs the architecture of the universe.

It’s true that the Fibonacci sequence is tightly connected to what’s now known as the golden ratio (which is not even a true ratio because it’s an irrational number). Simply put, the ratio of the numbers in the sequence, as the sequence goes to infinity, approaches the golden ratio, which is 1.6180339887498948482… From there, mathematicians can calculate what’s called the golden spiral, or a logarithmic spiral whose growth factor equals the golden ratio.

• Fibonacci Betting System: Gamblers have always wanted to find their way to money, or rather, find successful ways of betting which will win them more than losing. So, the Fibonacci betting system came to be. The system is most often used with roulette and craps, for the don’t pass and pass bets and when betting on the outside bets in roulette. Some use it when playing baccarat while others use it when playing blackjack. Firstly, you need to identify a unit, a starting unit. In the sequence, the first unit is 1, because zero doesn’t really count. Then, you need to select how much you are going to bet per unit. Let’s say, five dollars. Your first bet will be five dollars. If you lose, you move up in the sequence, betting another five. If you lose again, you would bet ten dollars. Losing moves you up the sequence, winning takes you down. If you ended up winning with your ten-dollar bet, you would go down the sequence and bet five dollars once more. Take note that betting systems most often do not work and that betting by itself is gambling, meaning random occurrences which you have almost no effect on with your system or otherwise.
• Converting Kilometres to Miles: The ratio of the Fibonacci numbers is very close to the Golden Ratio, 1.618034. Since 1 mile is roughly 1.609 kilometers, this is very close to the Golden Ratio. You can calculate miles from kilometers by shifting the register down in the Fibonacci sequence.
• Fibonacci Series in Nature: The Fibonacci Sequence is found all throughout nature, too. It is a naturally occurring pattern.
• Tree branches: Although we all usually see trees everywhere in our day-to-day life, how often have you looked for the patterns in them? In trees, the Fibonacci begins in the growth of the trunk and then spirals outward as the tree gets larger and taller. We also see the golden ratio in their branches as they start off with one trunk which splits into 2, then one of the new branches stems into 2, and this pattern continues.
• Storms: Your eye of the storm is like the 0 or 1 in the Fibonacci sequence, as you go on in the counter-clockwise spiral you find it increasing at a consistent pattern. This pattern is much like the Golden Ratio.
• Seashells: When cut open, nautilus shells form a logarithmic spiral, composed of chambered sections called camerae. Each new chamber is equal to the size of the two camerae before it, which creates the logarithmic spiral. This proportional growth occurs because the nautilus grows at a constant rate throughout its life until reaching its full size.
• Flower Petals: The petals of a flower grow in a manner consistent with the Fibonacci. Of the most visible Fibonacci sequence in plants, lilies, which have three petals, and buttercups, with their five petals, are some of the most easily recognized.
• Galaxies: The golden spiral can be found in the shape of the “arms” of galaxies if you look closely. It can’t be told if galaxies follow a perfect spiral, because we can’t measure a galaxy accurately, but on paper, we can measure it and see the size.
• Flower Heads: Most of the time, seeds come from the centre of the flower head and migrate out. A perfect example of this is sunflowers with their spiraling patterns. At points, their seed heads get so packed that their number can get extremely high, sometimes as much as 144 and more. When analyzing these spirals, the number is almost always Fibonacci.
• Parts of the Human Body: You are an example of the beauty of the Fibonacci Sequence. The human body has various representations of the Fibonacci Sequence proportions, from your face to your ear to your hands. You have now been proven to be mathematically gorgeous.

5. Binary Numbers

The formulation of the binary number system essentially laid the groundwork for digital circuitry, computers, and the field of computer science, as we know it in today’s technologically advanced world. As our world has traversed technologically from simple mechanics all the way to quantum modeling, the need to count hasn’t diminished over time, by humans and machines alike. The primary system used by humans for calculation is the Decimal Number System, however, the need for a more sophisticated, straightforward number system in digital computers and computer-based devices caused the adoption of the binary number system.

The binary number system is very literal in its nomenclature. Simply put, it is literally a numbering system that represents numbers using only two unique digits – typically, 0 and 1. The numbering system is also known as the base-2 number system. Computers utilize this numbering system to store and manipulate their data which includes numbers, words, music, graphics, and more. In fact, the term ‘bit’, which is the smallest possible unit of digital technology, actually stems from the words ‘BInary digiT’. Today, programmers use the hexadecimal or the base-16 number system as a more compact way to represent these binary numbers. Why? Because it is simpler for computers to convert from binary and hexadecimal and vice versa, and it is significantly harder to do this with the commonly-used decimal number system.

The modern binary number system was first researched in Europe in the 16th and 17th centuries by Gottfried Leibniz, and a few other mathematicians. However, systems similar to the binary system appeared in the antiquity of various cultures and civilizations. The I Ching also called the Classic of Changes or the Book of Changes, is among the oldest Chinese texts, which dates all the way back to the 9th century BCE. In this text, the concept of Yin-Yang describes the interconnection between forces in the world.  In I Ching, yin-yang is represented by using trigrams and later renditions of the text utilize hexagrams. This is one of the very first versions of binary notations which, at the time, was used to interpret the quaternary divination technique, based on the duality of yin and yang. Later on, the Song Dynasty scholar, Shao Young, repositioned the hexagrams in such a format that strongly resembled modern-day binary numbers.

Even before these developments in China, the ancient scribes found in Egypt used something known as the Horus-Eye fractions, which was one of the two methods the Egyptians used to represent fractions. The Horus-Eye fractions are actually a binary numbering system that was used for representing fractional quantities of grains, liquids, and other measures at the time. This system can be found in documents from the Fifth Dynasty of Egypt in 2400 BCE, while more-developed hieroglyphic forms date back to the Nineteenth Dynasty of Egypt in 1200 BCE.

The Indian scholar Pingala, the author of Chhandahshastra, was also known to be one of the earliest inventors of the binary system in the 2nd century BCE. According to researchers, his work described the binary numeral system using fixed patterns of short and long syllables when describing prosody (the basic rhythmic structure of a verse in poetry). This is also similar to Morse code. The short syllables were termed laghu (0) while the long ones were called guru (1). Pingala’s system was similar to the modern-day binary system since it started at one which had four short laghus which represented 1 and so on. The numerical value simply adds one to the sum of all the place values. The difference between the modern-day binary system and Pingala’s invention is that the latter’s system starts with one instead of zero, and the binary representations increase towards the right and not the left like in the modern rendition.

6. Euclid’s Elements

No list of mathematical achievements would be complete without the inclusion of the most seminal and influential mathematical work to come out of Greek antiquity. Written around 300BC, Euclid’s work built the foundation for modern mathematics by introducing a set of axioms and proceeding to demonstrate by mathematical rigor a collection of theorems that naturally followed. Covering subjects ranging from algebra to plane geometry (also now known as Euclidean Geometry), Elements remained a cornerstone of mathematical teaching for over 2,000 years following its creation. Elements influenced the thinking of great minds ranging from Dostoevsky to Einstein, and Abraham Lincoln’s inclusion of the phrase “dedicated to the proposition” in his Gettysburg address is often attributed to his readings of Euclid.

The thirteen volumes of Euclid’s “Elements” contain 465 formulas and proofs, described in a clear, logical style using only a compass and a straight edge, it contains formulas for calculating the volumes of solids such as cones, pyramids, and cylinders. The books discuss perfect numbers and primes; proof and generalization of Pythagoras’ Theorem, Euclid proved that the diagonals of the regular pentagon cut each other in “extreme and mean ratio”, which is now more commonly known as the golden ratio or golden section.

Euclidean Geometry is still as valid today as it was 2,300 years ago, it is widely used in many disciplines, including art, architecture, science, and engineering, to name but a few.