Tag Archives: Isaac Newton

Isaac Newton and his Contributions to Mathematics

Sir Isaac Newton. Image: Arthur Shuster & Arthur E. Shipley, via Wikimedia Commons..

In class we discussed the Fundamental Theorem of Calculus and how Isaac Newton contributed to it, but what other discoveries did he make?

Sir Isacc Newton was born on January 4, 1643, but in England they used the Julian Calender at that time and his birthday was on Christmas Day 1642. He was born in Woolsthorpe-by-Colsterworth, a hamlet in the county of Lincolnshire. His father had already passed prior to his birth and his mother remarried after his birth and left Isaac to live with her mother. He went to The King’s School, Grantham from the time he was twelve until he was seventeen. His mother removed him from school after the death of her second husband, but later allowed him to return by the encouragement of the school’s headmaster. He rose to be at the top in rankings in his school, mainly motivated to get revenge towards a bully. He began attending Trinity College in Cambridge in 1661. After receiving his degree he developed his theories on calculus over the span of two years during the plague [1].

Newton’s work in calculus intitially started as a way to find the slope at any point on a curve whose slope was constantly varying (the slope of a tangent line to the curve at any point). He calculated the derivative in order to find the slope. He called this the “method of fluxions” rather than differentiation. That is because he termed “fluxion” as the instantaneous rate of change at a point on the curve and “fluents” as the changing values of x and y. He then established that the opposite of differentiation is integration, which he called the “method of fluents”. This allowed him to create the First Fundamental Theorem of Calculus, which states that if a function is integrated and then differentiated the original function can be obtained because differentiation and integration are inverse functions [2].

Controversy later arose over who developed calculus. Newton didn’t publish anything about calculus until 1693, but German mathematician Leibniz published his own version of the theory in 1684. The Royal Society accused Leibniz of plagiarism in 1699 and the dispute caused a scandal to occur in 1711 when the Royal Society claimed Newton was the real discoverer of calculus. The scandal got worse when it was discovered that the accusations against Leibniz were actually written by Newton. The dispute between Newton and Leibniz went on until the death of Leibniz. It is now believed that both developed the theories of Calculus independently, both with very different notations. It should also be noted that Newton actually developed his Fundamental Theory of Calculus between 1665 and 1667, but waited to publish his works due to fear of being criticized and causing controversy [1].

Newton not only discovered calculus but he is also credited for the discovery of the generalised binomial theorem. This theorem describes the algebraic expansion of powers of a binomial. He also contributed to the theory of finite differences, he used fractional exponents and coordinate geometry to get solutions to Diophantine equations, he developed a method for finding better approximation to the zeroes or roots of a function, and he was the first to use infinite power series.

His work and discoveries were not limited to mathematics; he also developed theories in optics and gravitation. He observed that prisms refract different colors at different angles, which led him to conclude that color is a property intrinsic to light. He developed his theory of color by noting that regardless if colored light was reflected, scattered, or transmitted it remained the same color. Therefore color is the result of objects interacting with colored light and objects do not generate their own colors themselves [1].

Sir Isaac Newton was a truly amazing mathematician and scientist. He achieved so much in his lifetime and the amount of discoveries he made can seem almost impossible. He helped make huge advancements in mathematics and created theorems that we still use heavily to this day.

[1] http://en.wikipedia.org/wiki/Isaac_Newton

[2] http://www.storyofmathematics.com/17th_newton.html

Advertisements

The History behind Differential Calculus

Calculus is one of the most important fields of Mathematics.  Calculus is a study of rates of change and motion, which we can see by the slope of a line or a curve. There are two major branches of calculus, Differential and Integral calculus, and they are inverses of each other. Integral calculus is used to find the areas under a curve, surface area or volume, and linear distance travel. Differential calculus (which concerns the derivative) mostly goes over the problem of finding the rate of change that is instantaneous, for example, the speed , velocity or an acceleration of an object. Differentiation is especially important in natural sciences, engineering and technology.

Image: Brandon Lim.

Image: Brandon Lim.

An example of differential calculus is if you wanted to find the velocity or the acceleration of an object, for example, a car. To find the velocity of a car, you would take the first derivative of a function (position at time t : dx/dt) and to find the acceleration you would take the second derivative of a function (dv/dt : change in velocity/change in time . This leads us to Newton’s law of motion, which is Force = Mass x Acceleration, where in this context, acceleration is the second derivative of a function.

Who was the person behind the development of calculus? Well, it wasn’t actually just one person. Sir Isaac Newton and Gottfried Wilhelm Leibniz were both credited with the development of calculus. Throughout their lives, they both argued on who came up with the idea first, both have accused each other of plagiarism. Those two weren’t the only ones who contributed to the discovery of Calculus. There have been many other known mathematician of that time that also helped with the development of calculus. For example, Rene Descartes indirectly helped create differential calculus by introducing variable magnitude.

Newton and Leibniz essentially created integral and differential calculus. They were both interested in objects that are in motion. However, they both looked at different aspects of this. Newton was more involved with the speed of a falling object and Leibniz with the slopes of curves to illustrate the rate of change. Although they both looked at different things, they both came up with the same results, hence the accusations of stealing the other’s ideas. However, combining both of their ideas, fundamental theorem of calculus was created, which links the concept integration to derivation.

It is hard to see the difference between the function and its derivative without having a visual presentation. In math, graphs are usually used to show what a function and its derivative look like. Any value of the first derivative at a given point is equal to the slope of the tangent to the graph of the function at that point. As we all know that in a graph, positive means increasing, so when the derivative is positive, the function must be increasing and when the derivative is negative, the function must be decreasing. When the value is zero at a point, the tangent is horizontal, and the function changes from increasing to decreasing, or from decreasing to increasing, depending on the value of the second derivative. The second derivative basically represents the curvature of the function. Since the first derivative shows the rate of change, the second derivative shows the rate of change of the rate of change. When the second derivative is positive, the function concave upwards and when the second derivative is negative, the function concave downwards.

To find a derivative of a function we have to make sure that the two x values are as close as possible so we can receive an accurate result. Derivative is defined by the limit of slope formulas as the x values become closer to each other. For example, we take a point which is on a curve, now we take another point that is closest to x, x+delta x. All we need to do now is plug this into the slope formula, one more thing, since we want the closest value to x, delta x has to be very small, so we find the derivative as delta x goes to 0; now we have the entire formula for derivative shown in the image.

IMG_20141220_102335

Differential Calculus helped evolve Math in many ways. It is used in many different fields of science, such as, physics, biology, and engineering.

Work Cited

http://www.edinformatics.com/inventions_inventors/calculus.htm

http://www.encyclopediaofmath.org/index.php/Differential_calculus

http://www.math10.com/en/maths-history/history5/origins-differential-integral2.html

http://science.jrank.org/pages/1134/Calculus-Differential-calculus.html

http://www.wyzant.com/resources/lessons/math/calculus/differentiation

 

Invented or Discovered?

A philosophical question about math that has been asked since the times of the ancient Greeks (and possibly even before then) is whether mathematics is discovered or is it invented by man. People seem to think it has to be one or the other, but what if it is actually both?

Gottfried Wilhelm von Leibniz. Image: Christoph Bernhard Francke, via Wikimedia Commons.

Gottfried Wilhelm von Leibniz. Image: Christoph Bernhard Francke, via Wikimedia Commons.

Math is just a language, and like any other language that uses words to describe something (strings of symbols), math also uses symbols. Written language was developed both independently and simultaneously in ancient times. One person got an idea to use a written symbol to represent a tangible object. Sometimes multiple people got this same idea independently of each other, and other times a person would see this writing, it would spark the idea in their heads, and they would go on to develop their own written language. The same language was not developed by different people, rather each person used different symbols to represent different words. (Guns, Germs and Steel- Jared Diamond Chapter 12) The same can be said for math. Calculus was developed simultaneously, but independently by Isaac Newton and Gottfried Leibniz. Both developed different ways of doing calculus and each way gave the same results. Other times mathematicians have relied on the work of others to further their results.

Isaac Newton. Image: Sir Godfrey Kneller, via Wikimedia Commons.

Isaac Newton. Image: Sir Godfrey Kneller, via Wikimedia Commons.

The fact that math has been developed independently and yet yielded the same results shows that math is discovered. Math is the language used to describe the natural world and as long as the world exists someone can, at any time, develop a language to describe it. It may not be the same math that we use today (the Babylonians used an arithmetic system very different from our modern one), but it would still yield the same results. Given enough time, one would think, they would eventually be able to build the same skyscrapers and the same rocket ships that we have.

On the other hand, math was invented. We invent the symbols and decide what they represent; we invent the axioms and the particular system that we use. Newton invented infinitesimals in the use of calculus while Leibniz invented his own notation for calculus. The ancient Egyptians invented a different way to calculate the area of a circle than the one we use today. (A History of Mathematics- Uta c. Merzbach and Carl B. Boyer) Math does not exist without someone to invent the symbols we use to describe it.

Many people ask, and for good reason, if this question is even important, and it just may be. What if the concept of zero or negative numbers were never invented? Without these simple concepts would we still be able to build the same skyscrapers and rocket ships? It is possible that someone could have invented a concept similar to these but using different concepts? It is even possible that someone may have invented a way around them so we could avoid them altogether and this new invention could have even lead to a much simplified math system.