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Gottfried Wilhelm Leibniz

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Kathryn Brink

on 27 May 2010

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Transcript of Gottfried Wilhelm Leibniz

Gottfried
Wilhelm Leibniz Leibniz expanded the binary system, which eventually led to the creation of binary-based technology, such as computers and calculators. The binary system is still used today as the foundation of computer operating systems and other similar devices. Leibniz also contributed significantly to calculus, particularly in methods of solving differential equations. His ideas have various applications, including:
determination of rates of decay
projection of population growth
calculation of interest Exponential population growth, used primarily for microorganism populations, can be modeled by the equation below, which is also derived from a differential equation. Estimating the population of a community at a given time is useful for various reasons. Projecting population growth can help to approximate when a population's resources will be exhausted. It can also help scientists model the growth of bacteria or other organisms in a lab. A(t)=total money at a given time P=principal r=rate of compounding t=time Leibniz's work with differential equations also applies to the calculation of compound interest. By using the equation below (derived using a differential equation similar to that for population growth), it is possible to determine how much money will have accumulated in an account after a certain amount of time at a defined interest rate. In additon to differential equations, Leibniz made important strides in developing new techniques for finding the "quadrature of a curve," or integrating. His work with integration, another part of calculus, can be applied in determining:
displacement
force
work Leibniz, a German mathematician and philosopher, made many important contributions to mathematics in the late seventeenth century. By Kathryn R. Brink These include advancements in: Binary System Differentiation Integration k=constant t=time A=initial population P(t)=population at a given time 0 Leibniz's work with differential equations applies to determining rates of decay, such as radioactive decay. By using the equation below (derived from a differential equation that involves the rate of decay of a sample), it is possible to determine how much of a sample will remain after a given period of time. This information is useful in the storage and transportation of radioactive materials. N=amount of substance remaining N =initial amount of substance k=constant t=time Using principles based on Leibniz's ideas, integration can be used to determine displacement, or change in position, based on an equation for velocity. This is useful in physics for describing the motion of particles in terms of distances based on velocity data. Similarly, an equation for velocity can be calculated based on acceleration data using integration. Integration also applies to problems using a variable force to determine work. Variable forces occur in many situations, such as those involving: In addition, integration can be used to determine the force exerted by a liquid, given certain information on the liquid and its container. This information is useful in the development of hydraulic systems, for example: Springs Electric Charges Gravity Filling or Emptying
a Tank Car Lifts Log Splitters Large Hydraulic Machines To begin, here is some background on His Life Leibniz was born in 1646 to a semi-noble family in Leipzig, Germany. During his youth, Leibniz studied language and philosophy both at home and at school, developing an interest in deductive reasoning and systemization. Later, in 1661, Leibniz began to study at the University of Leipzig. Though formally receiving a degree in law, Leibniz focused his studies on philosophy. In 1666, Leibniz published his Dissertation De Arte Combinatoria, which lacked substantial mathematical significance. Instead, this publication demonstrated Leibniz's interest in mathematics as a method of explaining his philosophical ideas. Leibniz became earnestly interested in the study of mathematics after arriving in Paris in 1672. It was during this time that he developed most of his best-known mathematical concepts, including calculus. He also invented a calculating machine during that time, but it was received poorly by his fellow mathematicians. Furthermore, he developed much of the notation used in calculus today. Not long after Leibniz published his findings in calculus, Newton published his version. This led to a debate about the originality of Leibniz's work, which continued even after his death in 1716. Now, to some applications of Leibniz's work. All in all, Leibniz's advancements in mathematics have many applications today, even though he developed them over three hundred years ago. e=mathematical constant e=mathematical constant e=mathematical constant Works Cited
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