MATHEMATICIAN - PIERRE DE FERMAT

Frenchman of the 17th Century, Pierre de Fermat, effectively invented modern number theory virtually single-handedly, despite being a small-town amateur mathematician. Stimulated and inspired by the “Arithmetica” of the Hellenistic mathematician Diophantus, he went on to discover several new patterns in numbers which had defeated mathematicians for centuries, and throughout his life he devised a wide range of conjectures and theorems. He is also given credit for early developments that led to modern calculus, and for early progress in probability theory.

Although he showed an early interest in mathematics, he went on study law at Orléans and received the title of councillor at the High Court of Judicature in Toulouse in 1631, which he held for the rest of his life. He was fluent in Latin, Greek, Italian and Spanish, and was praised for his written verse in several languages, and eagerly sought for advice on the emendation of Greek texts.

Fermat's mathematical work was communicated mainly in letters to friends, often with little or no proof of his theorems. Although he himself claimed to have proved all his arithmetic theorems, few records of his proofs have survived, and many mathematicians have doubted some of his claims, especially given the difficulty of some of the problems and the limited mathematical tools available to Fermat.

One example of his many theorems is the Two Square Theorem, which shows that any prime number which, when divided by 4, leaves a remainder of 1 (i.e. can be written in the form 4n + 1), can always be re-written as the sum of two square numbers (see image at right for examples).

His so-called Little Theorem is often used in the testing of large prime numbers, and is the basis of the codes which protect our credit cards in Internet transactions today. In simple (sic) terms, it says that if we have two numbers a and p, where p is a prime number and not a factor of a, then a multiplied by itself p-1 times and then divided by p, will always leave a remainder of 1. In mathematical terms, this is written: a^p-1 = 1(mod p). For example, if a = 7 and p = 3, then 7² ÷ 3 should leave a remainder of 1, and 49 ÷ 3 does in fact leave a remainder of 1.

Fermat identified a subset of numbers, now known as Fermat numbers, which are of the form of one less than 2 to the power of a power of 2, or, written mathematically, 2²ⁿ + 1. The first five such numbers are: 2¹ + 1 = 3; 2² + 1 = 5; 2⁴ + 1 = 17; 2⁸ + 1 = 257; and 2¹⁶ + 1 = 65,537. Interestingly, these are all prime numbers (and are known as Fermat primes), but all the higher Fermat numbers which have been painstakingly identified over the years are NOT prime numbers, which just goes to to show the value of inductive proof in mathematics.

Fermat's pièce de résistance, though, was his famous Last Theorem, a conjecture left unproven at his death, and which puzzled mathematicians for over 350 years. The theorem, originally described in a scribbled note in the margin of his copy of Diophantus' “Arithmetica”, states that no three positive integers a, b and c can satisfy the equation aⁿ + bⁿ = cⁿ for any integer value of n greater than two (i.e. squared). This seemingly simple conjecture has proved to be one of the world’s hardest mathematical problems to prove.

There are clearly many solutions - indeed, an infinite number - when n = 2 (namely, all the Pythagorean triples), but no solution could be found for cubes or higher powers. Tantalizingly, Fermat himself claimed to have a proof, but wrote that “this margin is too small to contain it”. As far as we know from the papers which have come down to us, however, Fermat only managed to partially prove the theorem for the special case of n = 4, as did several other mathematicians who applied themselves to it (and indeed as had earlier mathematicians dating back to Fibonacci, albeit not with the same intent).

Over the centuries, several mathematical and scientific academies offered substantial prizes for a proof of the theorem, and to some extent it single-handedly stimulated the development of algebraic number theory in the 19th and 20th Centuries. It was finally proved for ALL numbers only in 1995 (a proof usually attributed to British mathematician Andrew Wiles, although in reality it was a joint effort of several steps involving many mathematicians over several years). The final proof made use of complex modern mathematics, such as the modularity theorem for semi-stable elliptic curves, Galois representations and Ribet’s epsilon theorem, all of which were unavailable in Fermat’s time, so it seems clear that Fermat's claim to have solved his last theorem was almost certainly an exaggeration (or at least a misunderstanding).

In addition to his work in number theory, Fermat anticipated the development of calculus to some extent, and his work in this field was invaluable later to Newton and Leibniz. While investigating a technique for finding the centres of gravity of various plane and solid figures, he developed a method for determining maxima, minima and tangents to various curves that was essentially equivalent to differentiation. Also, using an ingenious trick, he was able to reduce the integral of general power functions to the sums of geometric series.

Fermat’s correspondence with his friend Pascal also helped mathematicians grasp a very important concept in basic probability which, although perhaps intuitive to us now, was revolutionary in 1654, namely the idea of equally probable outcomes and expected values.

PYTHAGORAS'S THEOREM USED IN REAL LIFE EXPERIENCES

Pythagoras was a Greek philosopher and mathematician. Whether you know it or not, Pythagoras's theorem, named after him, is used by almost everybody in real life experiences.

• Uses of Pythagoras' Theorem

• • You may have heard about Pythagoras's theorem (or the Pythagorean Theorem) in your math class, but what you may fail to realize is that Pythagoras's theorem is used often in real life situations. Gain a better understanding of the concept with these real-world examples.

    According to Pythagoras's theorem the sum of the squares of two sides of a right triangle is equal to the square of the hypotenuse. Let one side of the right triangle be a, the other side be b and hypotenuse is given by c. According to Pythagoras's theorem

    a² + b²= c²

    This is taught in every classroom throughout the world, but what isn't taught is how it can be applied outside of the classroom.

  Real Life Applications

  • Some real life applications to introduce the concept of Pythagoras's theorem to your middle school students are given below:

    1) Road Trip: Let’s say two friends are meeting at a playground. Mary is already at the park but her friend Bob needs to get there taking the shortest path possible. Bob has two way he can go - he can follow the roads getting to the park - first heading south 3 miles, then heading west four miles. The total distance covered following the roads will be 7 miles. The other way he can get there is by cutting through some open fields and walk directly to the park. If we apply Pythagoras's theorem to calculate the distance you will get:

    (3)2 + (4)2 =

    9 + 16 = C2

    √25 = C

    5 Miles. = C

    Walking through the field will be 2 miles shorter than walking along the roads. .

    2) Painting on a Wall: Painters use ladders to paint on high buildings and often use the help of Pythagoras' theorem to complete their work. The painter needs to determine how tall a ladder needs to be in order to safely place the base away from the wall so it won't tip over. In this case the ladder itself will be the hypotenuse. Take for example a painter who has to paint a wall which is about 3 m high. The painter has to put the base of the ladder 2 m away from the wall to ensure it won't tip. What will be the length of the ladder required by the painter to complete his work? You can calculate it using Pythagoras' theorem:

    (5)2 + (2)2 =

    25 + 4 = C2

    √100 = C

    5.3 m. = C

    Thus, the painter will need a ladder about 5 meters high.

    3) Buying a Suitcase: Mr. Harry wants to purchase a suitcase. The shopkeeper tells Mr. Harry that he has a 30 inch of suitcase available at present and the height of the suitcase is 18 inches. Calculate the actual length of the suitcase for Mr. Harry using Pythagoras' theorem. It is calculated this way:

    (18)2 + (b)2 = (30)2

    324 + b2 = 900

    B2 = 900 – 324

    b= √576

    = 24 inches

    4) What Size TV Should You Buy? Mr. James saw an advertisement of a T.V.in the newspaper where it is mentioned that the T.V. is 16 inches high and 14 inches wide. Calculate the diagonal length of its screen for Mr. James. By using Pythagoras' theorem it can be calculated as:

    (16)2 + (14)2 =

    256 + 196 = C2

    √452 = C

    21 inches approx. = C

    5) Finding the Right Sized Computer: Mary wants to get a computer monitor for her desk which can hold a 22 inch monitor. She has found a monitor 16 inches wide and 10 inches high. Will the computer fit into Mary’s cabin? Use Pythagoras' theorem to find out:

    (16)2 + (10)2 =

    256 + 100 = C2

    √356 = C

    18 inches approx. = C
  • REFERENCE:
  • www.brighteducation.com

ABACUS

Abacus is the most ancient tool used for the purpose of calculations. It is one of the most important calculating devices invented for the calculation purposes. Abacus is a simple device but throughout the timeline it has proved to be an important instrument of economy.

Abacus Importance

If we go deep into the history books we come to know that abaci was used from the ancient times to perform basic arithmetical problems by the people related to any field of life. Businessmen have used it for centuries to perform basic calculations for the purpose of their routine business.

It is a true assistant for anyone who wants to perform his calculations very quickly and with full authority. This is the reason behind the fact that even in this modern era there is thought to be a need to teach children to learn perform calculations using abacus.

The reason being the simplicity of the device and its look that makes children feel it just like their toy and never let them feel bored while performing calculations on it. It is also very easy to learn and makes calculations easier for the juvenile to learn.

When the calculations are performed on abacus, a little stress is made on the mind. People who regularly perform calculations on abacus can perform them more quickly than their other counterparts who perform those calculations on electronic calculators.

Need of Abacus

There may arise a question in one’s mind that why is there a need to teach children to use abacus or to teach them solving arithmetical problems using abacus in this time of advanced technology. But the fact is that even if the advancements are made in the technology making it possible for human beings to perform complex calculations and to explore new dimensions in the field of knowledge, the mind of the child is completely empty and it is to be written on. Child’s mind could grab some things with difficulty and has the tendency of forgetting learnt topics after some time.

With the help of abacus simple arithmetical functions are easy to learn especially multiplication which may seem easier to an adult to perform but are very difficult for the juvenile to understand.

Learn While Playing

Nowadays abaci are also used as toys by children. These are among the things that they see quite often and are familiar with them. When children are asked to learn arithmetical calculation on their abaci they feel like playing and in this way they learn a lot without being bored at all. Children feel happy with something like abacus because of its shape. Anything, which has rods and beads attached to it, makes them feel good and relaxing while learning.

This process is easy to adapt for the beginners and puts less stress on the mind of children. Every bead of the abacus is assigned a particular value and it is relatively easy to learn those values and keep performing calculations on the device, as the beads are assigned certain values therefore there is no need to keep anything in mind rather keep fingers on the beads and place beads in their calculated positions. The result can be easily inferred from the positions of the beads on the rod.

Abacus Necessity

Sometimes it is needed to teach children how to use abacus. For instance blind children are taught to use abacus because they cannot perform calculations on the paper. Abacus has proved out to be an ideal tool for those people as it allows them to calculate easily and quickly while being reliable. The use of abacus is easy to understand for the new beginner and it also helps in developing the mind of the children.

At the initial learning stage of a child it is almost impossible to learn all of the things in a flash. Abacus has this innate tendency to design the model for the good development of a child’s brain. The study has shown that the children which have used abacus in their past could perform arithmetical calculations more quickly and comfortably when they are grown ups.

This is due to the fact that they have the golden rule of calculating long figures with the help of broken packets of that figure in their minds and it allows them to perform more conveniently in the later stage of their life.

WHY IS GEOMETRY IMPORTANT IN DAILY LIFE?

 

Mathematical thinking and reasoning begins for students long before it is taught through any sort of schooling.  Beginning as infants, humans are attracted to patterns, designs and shapes.  Parents reinforce this by often purchasing toys or mobiles with brightly colored shapes, pictures or designs.  Babies are attracted to these items before they are able to reach, grasp or manipulate them in anyway.  Later, toys are manipulated in such a way as to provide further hands on learning to develop these types of skills.  These shapes and designs are the very foundational level of the mathematical field of geometry.

Geometry is everywhere.  Angles, shapes, lines, line segments, curves, and other aspects of geometry are every single place you look, even on this page.  Letters themselves are constructed of lines, line segments, and curves!  Take a minute and look around the room you are in, take note of the curves, angles, lines and other aspects which create your environment.  Notice that some are two-dimensional shapes while others are three-dimensional shapes.  These man-made geometrical aspects please us in an aesthetic way.

An angle is formed when two rays come together at the same point (end point).  The distance between the two rays is measured in degrees using a tool known as a protractor.  Angles can be found on the human body as well as in the many structures we have created for living and working.  On your body, each joint as it is moved creates different sized angles based on how far apart the body parts are located.  An example of angles with in a home might include the brackets holding a shelf to the wall.  Angles are created as shapes come together. 

Shapes are unique representations with specific properties to define them.  Shapes can be two- or three- dimensional.  There are numerous defined shapes.  Shapes include things such as polygons, which include squares, circles, rectangles, triangles, etc.., quadrangles, which include parallelograms, rhombus, trapezoids, etc…solids, which include cylinders, pyramids, prisms, etc…  Each item in our tangible world is created by combining shapes of some sort together.  Thinking of a soda can as a cylinder or a refrigerator as a combination of squares, cubes and rectangles provides a deeper understanding of how shapes can be combined together to create the world around us.

A line is the path, which is always straight, and extends out infinitely (forever).  A line will not necessarily extend forever, but in order for it to be considered a line, it has the potential to, if continued on, to never end.  Lines are represented by a straight line with arrows on both ends, indicating that it could extend forever.  Line segments are similar to lines, in that they are always straight, but they do not extend out forever, instead they end at specific points, known as endpoints.   Line segments are typically represented by a straight line with two dots at each end, representing the end points.  These end points are generally given a label such as line segment AB.   A curve is similar to a line segment in that it has two specific end points, however it is never straight.  A curve would be represented in the same manner; however, instead of being straight the portion between the two end points would be curved.

Nature also has an abundance of geometry.  Patterns can be found on leaves, in flowers, in seashells and many other places.  Even our own bodies consist of patterns, curves and line segments.  It is through the observation of nature that scientists have begun to explore and explain the more basic principles now accepted as scientific truths.  These observations and realizations have lead to the progression of new learning in both science and geometry.  This began with the simple repetitive patterns such as the orbiting of the planets or the back and forth motion of a pendulum.  It continues today as new theorems and natural events are explained and represented through geometric representations, thinking and principles. 

At the most basic level, geometric principles occur all around us.   Mankind craves the geometrical principles and to explain events occurring within the natural world.   Home builders, interior designers, landscape designers all rely on geometric principles to attract the eye of prospective customers.  In nature, animals use the patterns and other geometric ideas as part of the reproduction process, defense mechanisms, and as a method to attract others.  In some cases, the geometry found in nature has provided inspiration for man-made items.  While in other cases, it is the natural events which have provided the inspiration for further developments and understanding of geometric principles and ideas.

Homes maximize their geometric aspects to draw the eye of potential buyers.  Curves are added to break up traditional rectangular patterns.  Spiral stairways might be added to replace the traditional straight staircases.  Patterns are found in every single part of the home including:  painting designs, window placements, carpeting, and numerous other examples.  In addition to the aesthetic principles, geometric thinking is needed to ensure homes and buildings are structurally sound.  Understanding which angles provide make for stronger and safer buildings has helped to change trends in construction.  Additionally, laws, in this case building codes, have been changed to include the implementation of the known laws of geometry into various trades.   These changes have significantly increased the safety of many things in our world.

Therefore, many professions require at least a foundational understanding of geometry.  Sports, construction, weaving, sewing, decorating, as well as many others require the use of the concepts learned through the study of geometry.  In many of these professions, the knowledge learned through a complete understanding of geometric principles has provided not only an increase in safety, but also an increase in the creation of tools, skill level enhancement, and aesthetically pleasing arrangements.  In sports, an understanding of angles might allow a baseball player to better catch or hit a ball farther.  In weaving and sewing, aesthetics can of course be increased by using geometric designs, but other designs can be introduced, perhaps increasing sales for a new clothing designer.  

In fact, almost all professions require some basic understanding of the more basic principles of geometry.  Whether it includes the idea of shapes or on the more complex end, the understanding of the process involved in proving a supposition, geometry is a direct or indirect influence.  Part of geometric thinking is the understanding of thinking in a critical manner and deducing answers given specific facts.  In this way, geometry is a life-long skill.  It is this type of thinking which is used to create new inventions or discover solutions to a variety of life's problems.   In the end, geometry surrounds all of us, making it a safer and more productive place.