Got Feedback?
Got Feedback?

Found a bug? Have a suggestion? Fill the form below and we'll take a look!


    Airplanes & Mobile Phones – Real World Examples Of Quadratic Equation

    There are various equations we learn and understand in the elementary of high school. Some of those equations are easy, some are complex, and some are plain boring. Whether math is easy for you or difficult, one thing is true – there is no other equation to rival the majestic quadratic equation. I am not just talking about the equation itself but also about the practical implications of the quadratic equation. Let’s be truthful, how many times you have thought that why are you learning this stuff and where it will be useful in the practical world? The answer is almost everywhere. From mobile phone to airplanes, engineers and mathematician are still using the quadratic equation to solve the complex problem.

    The story of quadratic equation was started long time ago when the Babylonians introduced the tax system to the farmer. Understandably, the farmers were not happy about the tax so they had to calculate the land needed to be cultivated to make up for the loss in taxes. They came up with the quadratic equation to make up for their loss and estimated the land on which they had to grow crop in order to make the ends meet. As ‘necessity is the mother of invention’, the solution to a simple farming problem is still being used to solve greater problems.

    There are hundreds of used of quadratic equation which we observe in the daily life. What we lack is the understanding that this equation can help us estimate the results or simply understand the phenomena. Today, I have compiled a list of practical uses of quadratic equation so the next time you are solving it – you know where it can be used.

    Quadratic Chaos – When Math Meets Biology

    The concept of quadratic chaos is defined by the quadratic equation with the focus on biology. Thhe population of a specific species can be estimated accurately with the help of quadratic equation. There is no patter to the growth of species hence it is considered as chaos. In this case x is the population while ‘n’ is the years so xn+1 is considered to be the sub function of xn. If axn was bred successfully while bxn2 dies due to overcrowding – the equation will be,

    \( xn + 1 = rxn(1 – xn) \)

    The focus of the equation is not just estimating the growth – it is more about understanding and evaluating if the prediction is accurate or hopeless.

    Mobile Phone and Complex Quadratic Equation

    The imaginary number were first discovered trying to find an answer to the cubic equation actually. When the imaginary numbers were used to solve the cubic equation, mathematicians were actually able to find the actual results. It was important to justify this mathematical fix otherwise we would have been keep on introducing new letter ultimately running out of letters. Thus the complex umbers were introduced which are the combination of number and alphabets. The incorporation of cubic equation with complex number ultimately led to solve quadratic equation that helped in predicting electron motion. It also assisted in designing the integrated circuit that can perform complex tasks. These circuits are still being used in almost every technological invention including DVDs, computer, cars, and mobile phones. Phone signals convert the speech into high frequency radio waves and this behavior was understood by the quadratic equation. You can say that without the simple quadratic equation ‘x2 = -1’, we would not have invented the mobile phones!

    Whether we think about simple machines or the more advanced one – one thing is very clear, it wouldn’t have been possible without calculation. We owe almost all of our invention to mathematics. From growing food as per the need of the population to trying to predict the natural phenomena from observing the movement of galaxies to stepping foot on moon, from calling our loved ones thousand mile away to predict the spread of infection – everything became possible for us when we dived into the realm of mathematics and its branches. One thing is very clear here – if we want to progress further, we need to reintroduce this subject to our student. Making the student understand the concept and logic behind the mathematical equation is the only way forward and we need to focus on it immediately. May be we will get lucky and find next Einstein or Newton!

    Mathematical Equations That Remarkably Impacted The World

    Calculation, equation, and, math is continuously revolutionizing our world. From the time mankind wanted to calculate the field area for growing crops – there was a thirst to know and understanding the secrets of the world. Why apple always fall down rather than flying, is there a pattern to the movement of star, what can assist in navigation, and why birds fly while we cannot – these questions of the curious minds lead to the thirst of known and the answer provided the mean to modernize the world one invention at a time!

    1- Calculus

    Due to the applicability of calculus, it is not only used in mathematics but in engineering biology, physics, chemistry, and many more branches of science. Calculus can help you in the determination of weather pattern movement of sound, movement of light, and motion of astronomical objects.
    Euler’s Polyhedra Formula
    Fourier Transform

    2- Law of Gravity

    Gravity is an undeniable force responsible for the existent of our planet. The law of gravity helps in the evaluation of weight and speed leading to significant modernizations including race car amd airplanes.

    3- Logarithms

    There are many example of the use of logarithms in the real world starting from interest rate to Google page rankings. Logarithms are also used to detect changes in multiplication and help count them.

    4- Maxwell’s Equations

    It is the set of 4 differential equations that describe the relation between electricity and magnetism. These equations are the basis for understanding the behavior of electromagnetism. From MRI scanners in the hospital to computer – the credit goes to the basic understanding of Maxwell’s equations.

    5- Navier-Stokes Equations

    These differential equation helped us understand the behaviors of flowing liquids such as smoke rising from cigarette, water moving through pipes, and air flow over plane wings. Navier-Stokes Equations are also used to model the weather and observe ocean currents.

    6- Normal Distribution

    Normal probability also known as normal distribution forms a bell curve and it is signifint in statistics. It is used in social sciences, physics, and biology to define the behavior of large groups of independent processes. Normal distribution is followed in the measurement of errors, heinght, IQ score, and blood pressure.

    7- Quadratic Equation

    There are various functions that are modeled by quadratic equation including shooting a cannon, hitting golf ball, and diving. You can calculate the expected profit you are going to get if you are using the quadratic equation. It can prevent unwanted surprises and provide you with the accurate numbers and what to expect in the future. Even in the business where you are simply selling bottled water, it can help you estimate how many bottles you have to sell to generate the profit you want.

    8- Relativity

    Relativity opened the door to understanding – be it our understanding of the outer space or the speed of light. It provided us with the idea that light speed is universal but the time factor is different for the speed of people or objects. Relativity helped us understand the fate, structure, and the origin of the universe.

    9- Schrodinger’s Equation

    The behavior of atomic and subatomic particle is defined by Schrodinger’s Equation. It enhance the understanding of quantum physics hence played a huge role in the development of computing devices. Computational chemistry is the direct application of Schrödinger equation and it is currently being used in medication and engineered food.

    10- Second Law of Thermodynamics

    According to the second law of thermodynamics, heat flow from hot to cold environment due to the change in temperature. This is the concept used in the working of internal combustion engines used in airplane, ship, car, and motorcycles. The law is applicable to all engine cycles and led to the progress of modern vehicles.

    11- The Pythagorean Theorem

    Whenever you need to find out if a triangle in acute, right-angled, or obtuse – you can use Pythagoras theorem for that. It made the life of mathematicians easier as it help them to find the missing length of any side of a triangle.

    12- The square root of -1

    The square root of -1 = I, this process gave rise to complex numbers that are supremely elegant. In case an equation have complex number solution, it will represented by ‘I’. With the help of this equation, mathematicians were able to find symmetries and the properties of the number which are implemented in signal processing and electronics.

    13- Wave Equation

    Wave equation as the name indicates describe the behavior of waves along with ripples, guitar strings, and incandescent bulb light. It is one of the first differential equations that helped us understand other differential equations as well.

    The world of mathematics is abundant with equations that helped us revolutionize the world as we know it today. We were not only able to understand the concept behind natural phenomenon ut also manipulate them for the modern advancement. These were just the few examples, stay tuned to know more!

    Mathematicians Who Changed The World

    9 Mathematicians Who Changed The World

    Mathematicians are the true hero but you don’t really know about them. I can recall two or three mathematician and that’s too because they were famous physicist. Yes, I am talking about Newton and Einstein! If you ask me to list 5 mathematician, I will fail and not the normal fail where you get one name wrong – I will get all the names wrong apart from these famous too. If mathematicians are heroes then why don’t we hear about them? Well the answer to this question is pretty simple – because most of us hate math or just not bother with it! It is a sad reality especially when you look around yourself and get to know that almost everything in your room include some kind of math or any calculation. You are bored and browsing on the internet – math. You want to get the accurate dimension of the room for a new carpet – math. The circuit, the bulb, the plug, and even the spider web on you wall have some kind of mathematical pattern or logic behind it.

    We appreciate the glam of movie star but forgets to appreciate the inventors of camera. We love animated movies but forget to acknowledge the efforts of those who made is possible. This articles is an effort to appreciate our mathematician so if someone ask you to name a few with their invention – you know the answer!

    1- Albert Einstein

    He doesn’t need any introduction – he is probably the most known mathematician. Even though his contributions in the field of physics are notable and famous, his mathematical achievement are often not discussed. He contributed multiple equations to geometry and calculus, 10 of which are known as the ‘Einstein Field Equations’ published in 1915. He provide that how stress-energy can impact the curvature of space-time.

    2- Carl Gauss

    Carl Gauss was born in 1777 in Germany. Even though he was born in a poor family, he soon started to show the signs of a brilliant mind. He started elementary school at the age of 7 and his teacher was amazed when he summed the integers from 1 to 100 just by spotting that there were 50 pairs each summing 101. He was the discovered of Bode’s law, arithmetic – geometric mean, prime number theorem, binomial theorem, and law of quadratic reciprocity – all that while he was in college! He published his work title as ‘Disquisitiones Arithmeticae’ which provided the construction of a regular 17-gon nu ruler and compasses.

    His second book was ‘Theoria motus corporum coelestium in sectionibus conicis Solem ambientium’ which was published in 1809. It provided the accurate description of the motion and movement of the celestial bodies. He also discussed conic sections, differential equations, and elliptic orbitals in the first part of his book. The second part was focused on the refinement of estimating planet’s orbitals.

    3- John von Neumann

    John von Neumann was born in Budapest in early years of 20th century and you can call it our luck as her designed the architecture served as the base for the development of computer. His work was so accurate that even today the same architecture has been used with minor alterations or improvements. Even the most advanced computers of this days are cycling through a series of basic steps over and over again. This is the processing behind playing video or doing literally anything on the computer and it was first proposed by John von Neumann.

    He hold two Ph.D. one in mathematics and second in chemical engineering just because his father wanted it. He worked with Albert Einstein at Princeton University and made discoveries in quantum mechanics, computer science, set theory, game theory, and geometry.

    4- Alan Turing

    Alan Turing is known as the father of computer science and he was a British mathematician. He is well known for breaking Nazi crypto code during World War II. The code was protected by Enigma machine and breaking it gave the Allies advantage over Nazis. Some even say that it is one the main reason the Allies were able to defeat the Nazis.

    Besides doing this huge favor to the world, Turing set the basis for the development of modern computer. Till this day computer operate similarly to the ‘Turing machine’ designed by Alan Turing. Turing test is used to evaluate how well an Artificial Intelligence (AI) programs works. Any program that can pass the Turing test can have a conversation (chat) with a human fooling him to think that he is chatting with a human.

    His name is still a prestigious one in the field of computer science as Tuning Award is named after him – which is equivalent to Nobel Prize in Chemistry.

    5- Benoit Mandelbrot

    Benoit Mandelbrot discovered fractal geometry based on the complex shape built on simple formula. They are the basis of computer animation and graphics so you can think why Benoit’s name is included in this list. Without fractal much wouldn’t have been possible as they are also used to design computer chips and cellphone antenna.

    These are just the few name who made it big in the field of mathematics. There are hundreds of other mathematicians who are neglected and their work is still being used in different sector. It is important that we know our ancestors and the effort they put into making this word an advanced place!

    From Father of Math to Father of Algebra

    Mathematics is a bit dry for me. I start to solve an equation and suddenly the lizard on the wall looks interesting as compared to the equation I am supposed to solve. My lack of focus is the only thing that made me research the importance of the subject more. The more I got bored, the more I tried to understand the reason behind it. The question I asked myself was why I am solving this equation followed by who came up with this subject, and are there any mathematicians who got famous doing math? Well, as you have guessed – one question lead to another and soon I was searching for famous mathematicians. Turned out there is not one but hundreds of famous people who are mathematician. This article is a tribute to those who (unlike me) were brilliant at math. They not found the value of their ‘x’ but help millions to find it too. So let’s dive into the realms of mathematics and recognize the efforts of notable mathematicians who shaped the world as we know it today!

    1- The Pythagoreans

    Perhaps the first known mathematician and his students known as the Pythagoreans and let me make this clear – they were obsessed with number. Pythagoras presented a theorem about right angled triangles ad still considered to be the Father of Geometry for this feat. Number was not only a subject for the Pythagoreans – it was the spiritual qualities of numbers that made it the focus of effort.

    2- Euclid

    Euclid was a Greek mathematician and wrote a book titled ‘Elements’. It was the primary textbook till 19th century. He was the brain behind Euclidean Geometry which provide the basis for modern geometry. He was also the first person to formalize mathematical proofs as the method of exposition.

    3- Archimedes

    Even though the achievements of Archimedes are not limited to the field of mathematics he thought that it is the only field worth pursuing. He is known as one of the greatest mathematicians as he perfected the integration methods helping him find the surface area, volume, and center of mass of the geometrical figures. He also gave us the accurate approximation to pi and help us find the mass of object immersed in liquid.

    4- Muhammad ibn Musa al-Khwarizmi

    He is often known as the grad father of computer as he proposed the concept of algorithm which was later used in the development of first computer. We owe our understanding of algebra to Musa al- Khwarizmi. He also worked on solving quadratic equations, surveying plot of land, and dividing up inheritance.

    5- John Napier

    John Napier is known for the invention of logarithms which are still being used in different mathematical calculations. He was the first mathematician who worked towards bringing the decimal point into common usage. His logarithms helped with trigonometric calculations in astronomy and navigation.

    6- Johannes Kepler

    With German roots, Johannes Kepler was the first mathematician and astronomer who provided the idea that earth and other planets revolve around the sun in elliptical orbitals. He was also the first mathematician to provide the mathematical treatment of the close packing of equal spheres that explained the shape of honeycomb cells. He also provided a method to find the volume of solids and contributed to calculus.

    7- Rene Descartes

    The credit of using alphabets in math goes to Rene Descartes. He is known as the ‘Father of Analytical Geometry’ because of the discovery of plotting two- dimensional points on a mathematical plane. Descartes also developed four rules for deductive reasoning which are not only applied to math but to other fields of study as well.

    8- Blaise Pascal

    He is famous for Pascal’s triangle and the number used in this triangle are binomial, each number is the sum of number above. After coming out with Pascal’s triangle, he presented the probability theory. Pascal was also the first mathematical to experiment with the atmospheric pressure and came up with the statement that vacuums exit in the real world too.

    9- Isaac Newton

    Apart from his outstanding research in the field of physics, he is known as the ‘Founder of Calculus’. Newton not only gave us three laws of motions but also discovered colored spectrum along with the discovery and calculation of gravity ultimately helping us to calculate the mass of an object.

    10- Gottfried Wilhelm Leibniz

    The credit of discovering binary numeral system or the base-2 system goes to Gottfried Wilhelm Leibniz. The binary system is still used in computers and related devices ascertaining that Gottfried’s work was value in the development of compute – the most widely used device nowadays. Gottfried Leibniz discovered infinitesimal calculus which is focused on differentiation, integration, and limits of function.

    11- Thomas Bayes

    He was the first mathematician to use probability inductively along with the establishment of mathematical basis for probability inference (calculating the frequency considering the past trials). Bayes’s theorem is also a significant achievement of Bayes as it can find probability of an event in past or future based on the pattern of available data.

    12- Leonhard Euler

    Leonhard Euler worked in different fields including mathematics, physics, astronomy trigonometry, geometry, algebra, and calculus. He introduced the notation for a function f(x). He also popularized the use of the Greek letter π to denote the ratio of a circle’s circumference to its diameter.

    13- William Playfair

    Playfair was one of the most important developer of statistical graphics focusing on the idea that chart and graphs can represent data more effectively. He devised and published pie charts, statistical line graph, and the bar chart. He provided a common language to data representation.

    These were some of the notable mathematicians who changed the society as we know it. Do you know any other – tell us in the comment section so we can learn about them too!

    Asians Are Good At Math – Reality, Myth, Or Plain Racism?

    From mathematic to biology, from chemistry to physics, from theoretical to practical interventions – Asians are making their mark and it’s high time you start thinking how? Ever noticed a pattern in the results of any mathematic contest? Ever wondered why there are so many research papers being submitted by Asian countries? No, well you are not the first one to ignore this. We have been so used to celebrate the inventions of western scientist that we got used to the idea that all inventions are western.

    Most of the time we ignore the fact that we are not making as much progress in the scientific world as our Asian counterparts and this is base of our problem. What could have been a healthy competition has turned into blatant ignorance. In case the achievement of the Asian professionals are highlight – they are mostly tainted with racism. Yes – I final said the word. With the recent stark evidences, we cannot ignore the fact that we still don’t have a cure for racism in our society.

    Let me give you a recent example. The cases of Covid – 19 were first reported in Wuhan, China. It shake the basis of world economy. China was the first country to report the virus and the first to control it. Now as we are dealing with the consequences associated with the spread of Covid – 19 such as loosing job, poverty, deaths and much more – how many of us are thinking about how china was able to control it. Let’s just think about it, with the hub of virus, the largest population of the world was able to defeat a virus they don’t know much about. The answer to this question in strict control as well as intensive research. With the threat came the approach to solve it, from total shut down to multiple therapeutical approaches – China defeated the virus while we are still living under the protection of masks and calling Covid – 19 the Chinese virus.

    When we go through the scores of different countries in math contests – Asians are performing better. They are academically better and have a stronger grasp of the concept. The reason of this outstanding performance if not limited to a single factor rather it is a combination of various factors that work together to make Asians better. The first factor is old and simple hard work – Asian are known to be hard worker. With centuries of oppression and being undervalued, they had to work extra hard to make their make and thus they embrace the quote that ‘hard work pays off’. In most of the international tests, a student have to fill a detailed questionnaire and by detailed I mean around 100 questions, with persistence and hard work Asians attempt those questions while western student gets bored and leave the questions unanswered. This is a clear example of the importance of persistence, concentration, and hard work in action.

    The second factor is the parental involvement in the choice of academic career. Even though there is an ongoing debate about different parenting styles, it seems that Asians are doing something right! They have a significant role in the academic choices of their children. With the efforts of first few years when parent are teaching their children the basis of academics like alphabet and counting, they shape the mind of the child to put extra effort in the academics.

    Another reason of Asians better in math is based on simple logic – they don’t get as much vacations as we do. A Japanese school year us 243 days long, the South Korean is 220, while the average American school years is just 180 days long. With longer school years, there is increased chances of learning something new. Even if the children are not learning anything new, they are revising what they have learned previously and hence they are getting better and better every day & there are more than 50 days of getting better each year!

    Another reason for Asians being good at math is their numbers are easier. Take the example of this sequence, 4, 8, 5, 3, 9, 7, 6. Human have a 2 second span of memorizing the digits. If you speak English you are going to recall this number sequence right 50% of the time – if you are Chinese, you are going to get it right every time.
    yī = one
    èr. = two
    sān. = three
    sì = four
    wǔ = five
    liù = six
    qī = seven
    bā = eight

    As we most easily memorize whatever we can say or read within that two-second span, you can learn the digits easily. Chinese digits are brief so they fit into 2 second memory loop and help you get them right every time! These are the facts that make Asian better in math and other subject. So the next time you want hear someone throwing a racist statement away at Asian – tell them what they are doing wrong and how they can get better too. After all tolerance and information is the only way out of this dire times.

    There Is More Than One Ways To Solve The Quadratic Equation

    When it comes to stepping up your math game from linear equations to the more complex ones – most people get scared. One of the best examples of an equation that can frighten beginners is the quadratic equation. Even though the equation itself is pretty innocent but the combination of those alphabets can definitely give you a scare if you don’t know what to do.

    What is Quadratic Equation?

    Simply put a quadratic equation is an equation that looks like this “ax² + bx + c = 0”. As you can guess ‘x’ is the variable that need to be solved. A, b, and c are numbers where ‘a’ is not 0. What makes this equation quadratic is the squared term which is (ax2).

    Why Quadratic Equation is Important?

    In daily life quadratic equation can be used to calculate the area of a room. You can measure the room to lay a new carpet or to install new furniture that will actually fit in your room. Quadratic equation is also commonly used to figure out the profit and being used in almost all of the business nowadays. It can also help you with the profit you need to make to overcome tax loss. Quadratic equation is also being used in the field of sports especially athletics. For the games based on throwing objects such as shot put, it can help you estimate the time it will reach to a specific target at a certain distance. Manipulating the force of throw can give you the results you want!

    The use of quadratic equation in the practical arena cannot be denied. It is often considered to be one of the most sophisticated mathematical equation. It can calculate the values for you if you are trying to grow more crops or planning to fly a plane. This diverse implementation of the quadratic equation leads to a question – how to solve the equation.

    Solving Quadratic Equation

    There are more than one methods to solve a quadratic equation and it totally depends on you how you want to solve it. These methods are going to provide you with an introduction to four methods so you can pick the method which is easiest for you. Today, we are going to provide you with not one but four simple methods you can use to solve a quadratic equation which are stated below.

    1. Square Root Method
    2. Completing the Square
    3. Factoring
    4. Quadratic Formula

    Let’s discuss each of these methods in detail so you can know which one is easier for you. By the end of this article – you are going to find your favorite method to solve quadratic equations.

    Square Root Method

    The Square Root method is used to solve any quadratic equation when ‘bx’ is 0. Move the constant to the right side of the equation and then divide both sides by ‘a’, now you will take the square root of the both side that will give you two values of ‘x’ – one will be positive and the other one will be negative.

    Square Root Method Example:

    4x² – 9 = 0
    4x² = 9
    x² = 9 / 4
    x² = 3 / 2 or – (3 / 2)

    (take square root of both sides, and remember, every square root has a negative counterpart)

    2. Completing the Square

    Another pretty simple method to solve quadratic equations is completing the square. You can create a perfect square on the left side of the equation by adjusting your constant (c). After squaring, this perfect squared can now be factored into two binomials which will be identical. You can now use these identical binomials to obtain the value for ‘x’.

    Completing the square:

    x² – 4x + 9 = 0 (original equation)
    x² – 4x = -9 (subtract 9 from both sides)
    x² – 4x + 4 = 5 (take half of b, then square it and add that value to both sides of the equation)
    (x – 2)² = 5 (break your perfect square into its binomials, made up of x, the sign in front of b and the square root of your new c)
    x – 2 = √5 or – √ 5 (take the square root of both sides)
    x = 2 + √5 or 2 – √ 5 (add 2 to both sides)

    3. Factoring

    The second approach to solve a quadratic equation is factoring. The concept behind factoring is to find the number pairs that can be multiplies together to produce ‘c’ and then summed to produce ‘b’. If you are able to find the right combination, you will get two binomials. These binomials can now be solved individually to get the value of ‘x’.

    3. Quadratic Formula

    The quadratic formula is one of the most widely used approach when it comes to solving quadratic equation. The formula looks like this

    You put the values and get the answer which will be the value of ‘x’. There are various online calculators available to solve quadratic equation and trust us – they are going to live your life much easier.

    With the help of these methods, you can now finally stop hating the quadratic equation. Understand the logic behind the equation and then select the method you want to invest in. It will take practice but after patient and time, you will be able to solve it correctly every single time!

    Exponential Growth Trajectory – The Math Behind Corona Spread

    Let me ask you a question – how is the social distancing going? Our lives were going smooth, we were partying, enjoying our lives, bitching about the governments, and going to work (even if we hate it), and living our lives normally. Suddenly there was a murmur about a new deadly virus affecting Chinese citizens living in China. We didn’t spare any thought as the virus was not impacting us – it was not our problem. Few days and it started impacting us as the reported cases skyrocketed. We were concerned, a little anxious, but it was just a news of faraway land and BOOM, the stats hit us with a force which is unmatchable to anything we have seen before. 

    Suddenly your neighbor tested positive and Covid is now your problem. It is few step away from you and the governments shut down entire countries to prevent the spread of infection. As it is now our problem, we are hoarding toilet papers and food to survive the pandemic. We are ordering hundreds of masks off amazon to protect ourselves and our families.

    How did we come to this when it is expected that the virus initially infected a single person. We can call this infectious person patient 1. Patient one was the host to the virus, he was mingling, drinking, clubbing, and socializing without knowing that he is infecting other. The dilemma of the spread of Corona virus is severe and the answer to the questions are related to mathematics. Exponential growth trajectory is your answer to all the questions related to the spread of Corona Virus.  

    Exponential growth is one of the most powerful tool of nature. Let me prove this statement with an example. In 1859, there was an English Farmer Thomas Austin. He migrated to Australia and brought 24 rabbits with him. The rabbits liked the environment and started to reproduce. By 1865, in less than 6 years, the number of rabbit population raised to 2 billion. The exponential growth is one of the most effective tool to reproduce and Corona is no exception.

    Patient 1 was responsible for spreading the virus to 2.5 individuals, each of them infected 2.5 person and the virus transmission rate exploded. Each infected person is expected to infect a certain number of people which is estimated to be around 2.5 right now, who each in turn go on to infect 2.5 more, and on and on, unless drastic measures are taken to reduce social contact and isolate the infected from others. To evaluate the scope of the infection, think about counting doubles – 1, 2, 4, 8, 16, 32, 64 and so on. How many times would you have to double to get to more than 1 million? 20. How many doubles to get to more than 16 million? 24. Human population also follow the exponential growth pattern hence we are short of natural resources as the population of the world is growing exponentially. Since past few decades, we are listening to the debates discussion alarming rate of population growth and now the debate switched to covid spread.

    Why are the governments focused on social distancing, let me tell you all about this dilemma. There was patient of covid, patient 31, he is responsible of infecting almost 1,160 healthy individuals and introduced a new cluster of infection. As he was initially asymptomatic and he was social (maybe too social), he lived his normal life and introduced a whole cluster of infection, an achievement which is not remarkable at all. Moral of the when hospitals are overcrowded, medical supplies are limited, population is stressed, and hospital staff story – don’t be like patient 31!

    In this dire situation is overworked – prevention of infection through social distancing is our only hope. As we don’t have the treatment options, prevention is our only weapon. When new hosts are not exposed to the virus, the curve of infection will drop indicating that the condition is under control. 

    There is a hope that the infection cannot always grow exponentially, as people become resistant and new intervention are introduced, the rate of the infection is bound to drop. Till then we have to utilize the only option we have – prevention via distancing ourselves. Right now what we all need is play our part and stick to the quarantine regime so we can buy valuable time for healthcare providers to deal with the situation and come up with an effective treatment plan. 

    Currently the number of Corona cases are doubling after every 6 days and this is a clear indication of how fast the virus is infecting. Till we come up with the strategy to treat the symptoms, it’s all about limit your social activity and stay at home – not only for your own sake, but for the sake of your loved ones too!

    Geometrical views you would love to see.

    Geometry is a branch of mathematics dealing with questions of shape, size, the relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer.

    In two dimensions there are 3 geometries: Euclidean, spherical, and hyperbolic.

    What about in three dimensions, which corresponds to space we actually live in? It has been shown that in three dimensions there are eight possible geometries.

    There is a 3-dimensional version of Euclidean geometry, a 3-dimensional version of spherical geometry and a 3-dimensional version of Hyperbolic geometry. There is also a geometry which is a combination of spherical and Euclidean, and a geometry which is a combination of hyperbolic and Euclidean.


    What is Euclidean Geometry?

    The study of plane and solid figures on the basis of axioms and theorems captured by the Greek mathematician Euclid. In its irregular outline, Euclidean geometry is the plane and solid geometry often taught in secondary schools.

     In Euclid’s great work, the Elements, the only tools employed for geometrical constructions were the ruler and the compass—a reduction retained in fundamental Euclidean geometry to this day.

    What is Non-Euclidean Geometry?

    Non-Euclidean geometry is a type of geometry. This geometry only uses some “postulates” (assumptions) that Euclidean geometry is built on. In normal geometry, parallel lines can never meet. In non-Euclidean geometry they can meet, either boundless many times (elliptic geometry), or never (hyperbolic geometry).

    The non-Euclidean geometries grew along two different historical threads. The first thread started with the search to understand the movement of stars and planets in the obviously hemispherical sky.

    An example of Non-Euclidean geometry can be seen by drawing lines on a round object, straight lines that are parallel at the equator can meet at the poles.

    What Is Spherical Geometry?

    Photo by Zane Lee on Unsplash

    From early times, people noticed that the shortest distance between two points on Earth were great circle routes. For example,  in geography

    the Greek astronomer Ptolemy wrote:

    “It has been demonstrated by mathematics that the surface of the land and water is in its totality a sphere…and that any plane which passes through the centre makes at its surface, that is, at the surface of the Earth and of the sky, great circles”

    Great circles are the “straight lines” of spherical geometry. This is an outcome of the properties of a sphere, in which the shortest distances on the surface are great circle routes.

    There are many ways of projecting a portion of a sphere, such as the surface of the Earth, onto a plane. These are known as maps or charts and they must certainly buckled distances and either area or angles.

    For example, Euclid wrote about spherical geometry in his astronomical work circumstances. In addition to looking to the heavens, the earliest attempted to know the shape of the Earth and to use this understanding to solve problems in navigation over long distances. These activities are features of spherical geometry.

    What Is Hyperbolic Geometry?

    Hyperbolic geometry, also called Lobachevskian Geometry, a non-Euclidean geometry that turned down the soundness of Euclid’s fifth, the “parallel,” postulate. Clearly declared, this Euclidean postulate is: through a point not on a given line there is exactly one line parallel to the given line. In this geometry, through a point not on a given line there are at least two lines parallel to the specified line. The doctrine of hyperbolic geometry, however, confess the other four Euclidean postulates.

    Although many of the theorems of hyperbolic geometry are uniform to those of Euclidean, others differ. In Euclidean geometry, for example, two parallel lines are taken to be everywhere equidistant. In hyperbolic geometry, two parallel lines are taken to converge in one direction and split in the other. In Euclidean, the sum of the angles in a triangle is equal to two right angles; in hyperbolic, the sum is less than two right angles. In Euclidean, polygons of different areas can be similar; and in hyperbolic, similar polygons of differing areas do not exist.


    Geometry applied to many fields, including art, architecture, as well as to other branches of mathematics.


    Mathematics and art are related in a variation. For instance, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of panorama geometry.


    Photo by Drahomír Posteby-Mach on Unsplash

    Mathematics and architecture are associated, since, as with other arts, architects use mathematics for multiple reasons. Apart from the mathematics required geometry when engineers start constructing buildings, architects  to create forms considered melodious, and thus to lay out buildings and their surroundings according to mathematical, decorative and sometimes religious principles; to decorate buildings with mathematical objects such as tessellations; and to meet environmental goals, such as to keep down wind speeds around the bases of tall buildings.

    How to solve multiple algebraic expressions?

    An algebraic expression is an expression is made up of integer constants, variables, and algebraic operations (addition, subtraction, multiplication, division, and exponent. For example, 8x2 + 4xy – z is an algebraic expression.

    We are now leading you from basic arithmetic to algebra.

    In algebra, letters are used to represent numbers.

    Types of Algebraic expressions:

    There are 3 main branches of algebraic expressions which include:

    • Monomial Expression
    • Binomial Expression
    • Polynomial Expression

    Monomial Expression:

    An algebraic expression which has only one term is known as a monomial.

    Examples of monomial expression include 2×3, 4xy, etc.

    Binomial Expression:

    A binomial expression is an algebraic expression which has two terms.

    Example 1:

    Since 8x and 2x are like terms then,

    8x+2x =10x

    Since 8 and 2 are the multiples of x, therefore 8 and 2 are the coefficients.


    Add 8x and 2

    Here both the terms are unlike so, the sum will be, 8x+2

    Polynomial Expression:

    Polynomial expression having more than one terms with non-negative integral exponents of a variable. Example 2x2+5x+9-4x2+2x -6

    Further branches of Expression:

    Including monomial, binomial and polynomial branches of expressions, an algebraic expression can also be categorized into two additional branches which are:

    • Numeric Expression
    • Variable Expression
    1 – Exponent (power), 2 – coefficient, 3 – term, 4 – operator, 5 –  constant, xy  – variables

    Numeric Expression:

    In numeric expression, never include any variable. These only consist of numbers and operations. Some of the examples of numeric expressions are 20+5, 15÷3, etc.

    Variable Expression:

    A variable expression is an expression which occupies variables along with numbers and operation to define an expression. A few examples of a variable expression include 4x+y, 6ab + 25, etc.

    An algebraic expression is a mathematical phrase that depends on numbers and/or variables. Though it cannot be solved because it does not contain an equals sign (=), it can be simplified. Algebraic equations, however, solve, which contain algebraic expressions separated by an equals sign.

    Following are the fundamental steps to simplify an algebraic expression:

    You need to simplify an algebraic expression before you evaluate it. This will make all your calculations much easier. Here are the main steps to follow for simplifying an algebraic expression:

    1. Remove parentheses by multiplying factors
    2. In terms with exponents. Use exponent rules to remove parentheses 
    3. Merge like terms by adding coefficients
    4. Merge the constants.

    Let’s work through an example.

    In this expression, we can use the distributive property to get rid of the sets of parentheses. When simplifying an expression, the first thing you need to look for clear parentheses. We multiply the factors to the terms inside the parentheses.

    = 3(3+x)+4(2x+2)+(x2)2

    We have to get rid of the parentheses in the term with the exponents by using the exponent rules. When a term with an exponent is raised to a power, we multiply the exponents, so (x2)2 becomes x4.

    = 9+3x+8x+8+x2

    The next step in simplifying we have to simplify the like terms and combine them. Here 3x and 8x are like terms, because they have the same variable with common powers, the first power since the exponent is understood to be 1. We can combine these two terms to get 11x.

    = 9+11x+8+x4

    = 17+11x+x4

    Finally, our expression is simplified. Keep in mind one more thing that algebraic expression is usually written in a certain order. We start with the terms that have the largest exponents to the constants. Using the commutative property of addition, we rearrange the terms and put this expression in correct order, like this.

    = x4+11x+17

    1. Difference between an algebraic expression and an algebraic equation:

    An algebraic equation can be solved and does include a series of algebraic expressions separated by an equals sign.

    An algebraic expression is a mathematical phrase that can contain numbers and/or variables. It does not contain an equals sign and cannot be solved.


     3X + 9  or 4x-8


    4x +2=18 or 9x-2=14

    2. Solve an algebraic equation with exponents.

    If the equation has exponents, then you have to isolate the exponent on one side of the equation and then to solve by “removing” the exponent by finding the root of both the exponent and the constant on the other side. Let’s know how you do it:

    • 2x2 + 6 = 14

    First, subtract 12 from both sides.

    • 2x2 + 6 -6 = 14 -6 = 0
    • 2x2/2 = 8/2 = 0
    • x2 = 4

    Solved by taking the square root of both sides, since that will turn x2 into x.

    √x2 = √4

    State both answers:x = 2, -2

    Solve an Algebraic Expression with subtraction

    Subtract (3x+2y) from (6x+7y)

    = (6x+7y) – (3x+2y)

    Now, we multiply negative sign to (3x + 2y):

    = 6x+7y-3x-2y

    Now, we arrange the expression as:

    = 6x-3x+7y-2y

    So, we got the answer 3x+5y

    Example: 2 

    Subtract (4X2 – 2X + 6) from (2x2 + 5x + 9)

    = (2x2 + 5x + 9) – (4X2 – 2X + 6)

    = 2x2 + 5x + 9 – 4X2 + 2X – 6

    Now, arrange the expression:

    = 2x2 – 4X2 + 5x + 2X + 9 – 6

    = -2x2+7x+3

    Solve an Algebraic Expression with Multiplication:

    Example # 1

    We have the algebraic expression (2x2)(3)

    We multiply (2x2) by 3 

    So we got the answer 6×2

    Example # 2

    Multiply (x + 2) by (3x – 5)

    = x (3x – 5) + 2 (3x – 5)

    = 3x2 – 5x + 6x – 10

    = 3x2 + x – 10

    Solve an Algebraic expression with division:

    Example # 1

    = -25×3 by 5x

    = -25x3 / 5x= -5x

    Example # 2

    Divide (12x<sup>2</sup> + 3x + 9) by 3

    =12x2+3x +9 / 3

    =12x2 + 3x3 + 9 / 3

    =4x2 + x + 3

    Solve an algebraic expression with fractions. 

    If you want to solve an algebraic expression with fractions, then you have to cross multiply the fractions, combining like terms, and then isolate the variable. Here’s an example of how you would do it:

    • (x + 3)/6 = 2/3

    First, cross multiply to remove the fraction. You have to multiply the numerator of one fraction by the denominator of the other.

    • (x + 3) x 3 = 2 x 6 =
    • 3x + 9 = 12

    Now, combine like terms. Combine the constant terms, 9 and 12, by subtracting 9 from both sides.

    • 3x + 9 – 9 = 12 – 9
    • 3x = 3

    Isolate the variable, x, by dividing both sides by 3 and you’ve got your answer.

    • 3x / 3 = 3 / 3
    • x =1

    Solve an algebraic expression that contains an absolute value.

    The absolute value is always positive.

    For example, the absolute value of -7 (also known as |7|), is simply 7. To find the absolute value, you have to separate the absolute value and then solve for x twice, solving both for x with simply removed the absolute value, and for x when the terms on the other side of the equals sign have changed their signs from positive to negative and vice versa. Here’s how to do it.

    • |6x +2| – 4 = 8 =0
    • |6x +2| = 8 + 4 =0
    • |6x +2| = 12 =0
    • 6x + 2 = 12 =0
    • 6x = 12
    • x = 2

    Now, solve again by flipping the sign of the term on the other side of the equation after you’ve isolated the absolute value:

    • |6x +2| = 12 =0
    • 6x + 2 = -12
    • 6x = -12 -2
    • 6x = -14
    • 6x/2 = -14/2 =0
    • x = -7

    Now, just state both answers: x = 3, -7

    Understanding the “inside out” of number system: (Part 2)

    As we have learned conversion from other systems to the decimal system in our article “Understanding the ‘inside out’ of number system: (Part 1 )”. Now we are going to understand the “ conversion between:

    1. Binary and octal: In this conversion, we are going to focus on how can we convert binary to octal numbers and how to convert octal to binary digits.
    2. Binary and hexadecimal: In this conversion, we will understand in detail how to convert binary to hexadecimal and how to convert hexadecimal to binary digits.

    Conversion between binary and octal:

    Each digit of an octal number is equivalent to three binary digits because any octal number ( 0 to 7) can be represented by three binary digits as shown in the given table.

    Conversion from Binary to Octal number system:

    The binary digits are split into two groups of three digits starting from right to left. Zeros can be added at the left end to make a complete group of three digits if required. Each group represents an equivalent octal digit.

    Octal number system provides an appropriate way of converting large binary numbers into more dense and smaller groups.

    There are multiple ways to convert a binary number into an octal number. You can transform using direct methods or indirect methods. First, you need to convert a binary into other base systems. 

    Since there are only 8 digits (from 0 to 7) in the octal number system, so we can represent any digit of octal number system using only 3 bit as following below.

    Octal SymbolBinary equivalent

    So, if you make each group of 3-bit binary input number, then replace each group of binary number from its equivalent octal digits. That will be the octal number of a given number. Note that you can add any number of 0’s in leftmost bit (or in most significant bit) for integer part and add any number of 0’s in rightmost bit (or in the least significant bit) for fraction part for completing the group of 3 bit, this does not change the value of input binary number.

    Here we have the binary numbers which we are going to convert to octal numbers: (10111010001111)

    Binary:    010  111  010  001  111

    Octal:      2 7   2 1 7

    So, we got the octal numbers (27217).

    Conversion from Octal to Binary number system:

    An octal number can be easily converted to a binary number by replacing each octal digit with the corresponding three binary digits.

    There are various direct or indirect methods to convert an octal number into a binary number. In an indirect method, you need to convert an octal number into other number systems (e.g., decimal or hexadecimal), then you can convert into binary number by converting each digit into binary numbers from the hexadecimal system and using conversion system from decimal to binary number.

    Here we have an octal number which we are going to convert in binary digits: ( 42153 )8

    Octal: 4   2   1   5 3

    Binary: 100 010 101  011

    So, we got the answer ( 100010001101011 )2

    Conversion between Binary and Hexadecimal:

    Like the octal system, the hexadecimal system can be easily derived from the binary system. Each Hexadecimal digits (0-9 and A- F) is equivalent to four binary digits as shown in the table given under:

    Decimal Hex Binary
    0 0 0
    1 1 1
    2 2 10
    3 3 11
    4 4 100
    5 5 101
    6 6 110
    7 7 111
    8 8 1000
    9 9 1001
    10 A 1010
    11 B 1011
    12 C 1100
    13 D 1101
    14 E 1110
    15 F 1111
    16 10 10000
    17 11 10001
    18 12 10010
    19 13 10011
    20 14 10100
    25 19 11001
    26 1A 11010
    27 1B 11011
    28 1C 11100
    29 1D 11101
    30 1E 11110
    31 1F 11111
    32 20 100000
    33 21 100001
    34 22 100010

    Binary to hexadecimal Conversion:

    Conversion between binary and hexadecimal is simply accomplished by grouping the binary numbers into four digits replacing each group with a hexadecimal equivalent digit.

    Hexadecimal number system provides an appropriate way of converting large binary numbers into more compact and smaller groups. There are various ways to convert a binary number into a hexadecimal number. You can convert using direct methods or indirect methods. First, you need to convert a binary into other base systems (e.g., into a decimal, or into octal). Then you need to convert it into a hexadecimal number.

    Since there are only 16 digits (from 0 to 7 and A to F) in the hexadecimal number system, so we can represent any digit of hexadecimal number system using only 4 bit as following below.


    Here we have the binary numbers to convert in hexadecimal number system (111010000110111000)2

    In this conversion we make the pair of four digits, so we have to add two leading zeros to make perfect pairs like:

    Binary numbers :  0011  1010  0001  1011  1000

    Hexadecimal:        3 A 1 B 8

    We got the answer (3A1B8)16. So, if you make each group of 4-bit binary input number, then replace each group of binary number from its equivalent hexadecimal digits. That will be a hexadecimal number of given numbers. Note that you can add any number of 0’s in leftmost bit (or in most significant bit) for integer part and add any number of 0’s in rightmost bit (or in the least significant bit) for fraction part for completing the group of 4 bit, this does not change the value of input binary number.

    Conversion from Hexadecimal to Binary number system:

    Conversely, a hexadecimal number can be converted into binary by replacing each digit by the equivalent four binary digits.

    Here we have the hexadecimal number system to change in binary digits: (5DC7).

    Hexadecimal number: 5 D C 7 

    Binary Numbers: 0101 1101 1100 0111

    So, we got the answer 0101110111000111

    This method of using the binary equivalent of digits of numbers in Interbase conversion is called “Direct Method”.