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CSM Spring 2017 Honors Project: "The Enigma e"
- 1. The Enigma: e by Andrew Emad Behbahani
- 2. What is e? e is a mathematical constant that represents a base of a natural logarithm. It is approximately 2.718281828... It is used frequently in Calculus as well as other sciences.
- 3. History of e Before e there were many walls in the study of mathematics that prevented discovery of it. The concept of infinity Complex exponents were avoided and many did not want to work often with irrational numbers.
- 4. e and Logarithms The discovery of e was a result of the invention of logarithms. A logarithm is a mathematical operation used to find the exponent for a number given its original base.
- 5. John Napier - The Creator of the Logarithm John Napier was a Scottish mathematician that would create the logarithm. He would create a comprehensive table of logarithms. He did this in order to find a way to multiply or divide ridiculous numbers.
- 6. Logarithms and the Path to Discovering e Napier wanted to express a number as ab where a is the base. He chose to use a base of a equals 0.9999999. The term b would be known as a “logarithm” which roughly translates from Latin as “Root Number”.
- 7. Napier’s Logarithmorum Canonis Descriptio In 1614, Napier published “Description of the wonderful canon of logarithms”, which defined logarithms and how they work, as well as include his log and “anti-log” tables for solving complicated calculations.
- 8. The Bernoulli Brothers The Bernoulli Brothers were brothers who were both mathematicians. Jacob took the first step in getting closer to e than any other mathematician during his time.
- 9. The Compound Interest Equation
- 10. Compound Interest Jacob had found that the limit of the compound interest equation as n approaches infinity would eventually reach e.
- 11. Leonhard Euler (1707-1783) Leonhard Euler, a student of Johann Bernoulli was a Swiss mathematician, who was sometimes known as the “Mozart of Mathematics”. For decades he wrote many volumes on mathematics, even when he had became blind.
- 12. Euler’s Findings in Mathematics Euler is known for creating the notation for modern-day functions, summation, the imaginary number i, and notation for trigonometry. Although he has been credited for revolutionizing math, he is noted for his new ideas on e, which still resonate with mathematics.
- 13. Euler and e Euler used the calculated the limit of the compound interest equation to 18 digits. Euler is widely credited for using the letter “e” for the notation of this number, and making it the base of the natural logarithm. But he did not name e after himself, contrary to popular belief.
- 14. Euler’s Identity and Euler’s Formula
- 15. Curve Analysis and e [mechanics] The area under 1/x from 1 to e is 1 (the blue area).
- 16. The Derivative of the Exponential Function
- 17. Applied Uses of e Rate of Decay for Radioactive Substances Population Growth (follows an exponential pattern) Memory Retention/Forgetting Curve (pictured right)
Editor's Notes
- -e is a mathematical constant that represents the base of a natural logarithm. -It is present in many studies involving mathematics. It appears very frequently in Calculus. Not only it has uses in math, but is also used in studies like Biology and Physics. -e = 2.7182818284590452353602874713527… (there is at least 115,000 known digits — Steve Wozniak is documented the latest person to have calculated it accurately using an Apple II computer in 1978)
- Before e was discovered, before calculus was invented, and before calculators were produced, there were more arduous challenges to calculate numbers. For example: “Infinity did not exist!” - During ancient Greek times when Geometry was pioneering a new era of mathematics, Greek mathematicians were very reluctant in trying to use infinity in math at all, because it did not represent a physical value. This would further slow down the discovery of calculus (I will explain why this was important in discovering e later on) “Exponents are pretty scary” - For thousands of years, exponents have existed, but what about using fractions as exponents? Many mathematicians could not answer a question like this.
- Now before we can talk about what a natural logarithm is, we need to figure out what a logarithm is by definition.
- John Napier (1550-1617) was a Scottish mathematician who would devise the main component that would lead to the discovery of e. This main mathematical component is commonly known as a logarithm. A logarithm is a function used to find the exponent of a power and a number. However, this sole discovery of logarithms took him over twenty years to formulate, because it required tediously extraneous calculations.
- During his time as a mathematician, Napier became heavily interested with Geometric Series and Exponents. Napier was curious on how exponents worked, and was figuring out a shortcut to fill out the gaps in power functions, perhaps ways to find complex roots, or fractional exponents more easily, as well as find a way to multiply and divide bigger by adding and subtracting them. Napier would use the expression (1-10^-7)^x to formulate an expression where x would represent any real number. This would be the closest value to 1 without compromising the value of the actual base. He would name this expression the “logarithm” which roughly translates from Latin as “Root Number”.
- In 1614, Napier would publish a Latin treatise known as Mirifici logarithmorum canonis descriptio, which is Latin for “Description of the wonderful canon of logarithms”. In this publication, Napier would describe logarithms and how they work, as well as include a comprehensive college of logarithmic values and “anti-logarithmic” values that could be used to make more complicated calculations. Astronomer Johannes Kepler among the many well known scientists who used Napier’s log tables in Geometric calculations of planetary orbits. With logarithms, he proved that Earth and other planets in our solar system revolved around the sun.
- Jacob had found that the limit of the compound interest equation as n approaches infinity would eventually reach e. He had also found that the sum of 1/n^2 converges. Johann Bernoulli is also known for authoring the l’Hopital’s Rule, even though credit is given to Guillaume de l’Hopital. Jacob Bernoulli (1654-1705) and Johann Bernoulli (1667-1748) were Swiss mathematicians who specialized in many areas of sciences, specifically mathematics. Johann originally was a student of Jacob. They had strong dislike towards each other. They were known for stealing each other’s credit. Johann contributed to and published works on calculus as well as differential equations. He was the first to apply calculus to determining the lengths and areas of curves.
- In the area of finance, compound interest is used to calculate the amount of money when applying interest rates that are compounded for a period of time. Compound interest equations date back to Babylonian times and has all been calculated different ways throughout history. But one version of the equation helped in finding e.
- Leonhard Euler, a student of Johann Bernoulli was a Swiss mathematician, who was sometimes known as the “Mozart of Mathematics”. He specialized in many areas of math such as number theory, topology, analysis, and even hydrodynamics. For decades he would write volumes upon volumes of mathematical discoveries and notes, making almost as much contributions to mathematics as Newton did. Although he has been credited for notations such as function notation, sigma notation, pi, and i as an imaginary number. He is noted for conceptualizing immense ideas relating to the the number e that still resonate with mathematics today. Namely, using the letter e for the notation of the number.
- Euler is known for creating the notation for modern-day functions, summation, the imaginary number i, and notation for trigonometry. Although he has been credited for revolutionizing math, he is noted for his new ideas on e, which still resonate with mathematics. He had dealt with the konigsberg bridge problem
- After the Bernoulli brothers learned to approximate e by using the compound interest equation, Euler used the equation to approximate it even further to 18 digits. Euler is widely credited for using the letter e for the notation of the number, and making it the base of the natural logarithm. Euler is widely credited for using the letter e for the notation of the number. It has been a general misconception that he named the number after himself, but it was rather likely that Euler could not use the letter “a” because it was already used in his other works to represent sums.
- Euler was able to draw connections with e, imaginary numbers, and trigonometry, he did so in Euler’s Formula. He created Euler’s Formula by using the power series for e sinx and cosx. Using this formula, Euler created Euler’s identity by using pi as an x-value for his formula. Which is commonly known as one of the most beautiful equations in math because of its simplicity and use of multiple rational and irrational numbers.
- Meaning the exact area under the curve of this function between 1 and e is exactly 1. This was discovered by Dutch mathematician, Christiaan Huygens in 1661. The definite integral of 1/x from e to 1 is 1. Meaning the exact area under the curve of this function is exactly 1. This was discovered by Dutch mathematician, Christiaan Huygens in 1661.
- Interestingly enough, e^x is a function where the derivative and its indefinite integral is itself. The function y=ax has a tangent line through the point (0,1). The slope of that tangent line depends on a. There is an a where that slope is 1. For that a, we call the constant e. Looking deeper into this function, we find this is because at any certain point the y-value is equal to slope at that point. When analyzing it, we notice that ex grows continuously, this could be seen as a monotone increasing function.
- Rate of Decay of a radioactive substance is m=m0e-at where m0 represents the initial mass of a substance, a represents rate of decay in a substance and how much radiation a substance emits, and t represents time. Newton’s Law of Cooling: T= T1+(T0-T1) e-at is a function that represents the rate of cooling a hot object at temperature T0 when placed in an environment of a lower temperature T1. Newton states that an object cools at a rate that is proportional to the difference between that object’s temperature and the temperature of the surrounding environment. Lambert’s Law: Measures the intensity of a sound wave as distance increases. This is represented by the equation, I=I0e-axwhere I represents intensity and x represents distance. Population Growth: exponential patterns.