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------------------ 1952 - Transcendental Numbers : “e” and “pi”
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- Transcendental means “beyond human experience“; but, not beyond human knowledge. Supernatural, but still natural numbers. Numbers are just inventions man created for counting. 0, 1,2,3,4,5, ….. How could numbers be supernatural? We started by simply counting our fingers. We started with positive numbers and it took a while for man to accept negative numbers. For several centuries man could not believe that there could be something less than nothing. Now we have the government to thank for that.
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- Finally, a number line was invented that extended positive numbers to the right all the way to infinity, and to the left all the way to negative infinity. Later, man ran into the problem of taking the square root of a negative number. The only way that was possible was to invent “i” the imaginary number that was the square root of negative one, ( i = -1^½), and i^2 = -1.
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- Now, we have real numbers and imaginary numbers together. We make the number line the real axis , or x axis, and the imaginary line the vertical, or y axis, and that plane can define all real and imaginary numbers that exist. 4 + 3i is an imaginary number, a point 4 counts to the right and 3 counts up on the imaginary axis is a unique point for this number on the plane.
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- If we plot all the points of radius one on the graph where x^2 + y^2 = 1 then we define the equation of a unit circle, of radius one. This is simply the Pythagorean Theorem where any right triangle inside the circle has three sides r^2 = x^2 + y^2. But, in this case the hypotenuse is the radius, “r”. Remember, the sum of the squares of two sides of a right triangle are equal to the hypotenuse squared. All points on a unit circle can be defined by the equation x^2 + y^2 = 1.
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-Now, put 2 radii at right angles and the hypotenuse is a cord across the circle. The cord according to the Pythagorean Theorem is equal to the square root of 2. If we do this 4 times around the unit circle we create a square. The area of the square is simply 2 square units. The sides are each the square root of 2 which is an “irrational number“. A decimal number 1.414.….. that goes to infinity never repeating itself. Two infinite series decimal numbers multiplied together equal the simple number 2.
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- Now, we can define the number “pi” which is the ratio of the perimeter, or circumference, of a circle divided by the diameter. “pi” = c / d. or, c = 2*pi*r. “pi” is one of those Transcendental numbers. It is “supernatural“. It appears to not be an invention of man but an invention of nature. The area of the unit circle is “pi” = 3.14.….. to infinity. The circumference of the unit circle is 2*pi = 6.28.……. to infinity. A circle with a simple radius 1 has an area and a circumference that are irrational numbers whose decimal numbers go on to infinity.
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- “pi” in addition to being a Transcendental number is an Irrational number. Irrational numbers are real numbers that can not be written as a fraction of two integers. A Rational number can be written as a fraction, like 1/3 = 0.33333333 …….. Its decimal expansion goes on to infinity. But, for Rational numbers the pattern always repeats itself. For Irrational numbers the pattern goes to infinity and NEVER repeats itself. The square root of 2 is an Irrational number just like “pi”.
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------- 2^½ = 1.414213562 …….. goes to infinity but the pattern never repeats itself.
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------- “pi” = 3.141592654 …….. goes to infinity but the pattern never repeats itself, it is Irrational. Is it not amazing that “pi” is defined as a fraction of circumference/diameter of a circle but it can not be defined as a fraction of any two integer numbers?
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- Another way to say this is that Irrational numbers can not be written as a termination or a recurring decimal. But, “pi” is also a Transcendental number because it also can not be written as a polynomial equation with rational coefficients of which “pi” is a root. This statement is abstract and Transcendental in math takes a little getting used to. Both “pi” and “e” are Transcendental numbers that fit this definition. A little later we will create “e” using polynomials but never with the root “e”.
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- “pi” defines circles. “e” defines everything that grows. Let’s use $1,000 to illustrate. Suppose you put a thousand dollars in the bank and the bank gave you a 100% annual interest rate. At the end of a year you would have $2,000 with interest.
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- That is $1,000 * ( 1+1.00) = $2,000 (If it were 6% interest it would be $1,000*1.06 = $1,060.) If we kept our money in the bank for another year the total would be $1,000*2*2 = $4,000. The third year we would have $1,000*2*2*2 = $8,000. The equation is $1,000 * (1+1.00)^n, where “n” is the number of years. But, what if the bank compounded the interest monthly, instead of annually. Then the equation becomes $1,000 *(1 + 1.00/12)^12 = $1,000 *(1.0833)^12 = $2,613. You made $613 more by compounding interest monthly instead of yearly.
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- Compounding grows things faster.
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- How about if the bank compounded interest daily?$1,000 *(1+1.00/365)^365 = $1,000*(1.1.0027397)^365 = $2,715. You made $715 more by compounding daily. Not that much more. What would happen if the bank simply compounded continuously? Notice we are doing two things at once. We are making the base in the equation a smaller and smaller number by compounding more often. And, at the same time, we are making the exponent larger and larger. Whenever you grow anything like this in the limit you approach the magical, transcendental number “e” . The limit of (1+1/n)^n always equals 2.718281828459045 ……. We define this number as “e”. Therefore , the maximum we can earn compounding 100% interest continuously for a year is $2,718.
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- We can generalize this equation mathematically to become: e^x = (1+x/n)^n
We can generalize the financial equation starting with a principle “p”, with an interest rate, “r” and “t” years with interest compounding continuously to:
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------------Principle with Interest = p*e^r * t = $1,000*e^1.00 * 1 = $1,000 * 2.1718 = $2,718 at the end of the first year. After 10 years you would have: $1,000*e^1.00*10 = $1,000 * 22,026 = $22,026,466. You would be a millionaire 22 times over in just 10 years if you could get a 100% compounded interest rate.
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- Another way to create “e” is with polynomials. A polynomial is a math expression of 2 numbers added together and raised to some power. For example: ( x+2)^2 = (x+2)(x+2). If we continuously make the base smaller and the exponent larger we eventually get to “e” using this polynomial series: (1+1/10)^10 = 2.59374246 ….. (1+1/100)^100 = 2.704813829 …… (1+1/1000)^1000 = 2.716923932 …. As the base gets closer to 1 and the exponent gets larger and larger we approach “e” = 2.718281828 ……..
“e” surfaces as a tug of war between 1 and infinity.
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- “e” and “pi” are Transcendental numbers that are found everywhere in nature. “e” is linked to calculus both integration and differentiation. Integration is the summation of tiny rectangles to calculate the area under a curve. If you have a curve where x*y = 1, or, y = 1/x then the area under that curve from 1 to “e” is 1.
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- In differentiation, calculus is the calculation of a rate of change of one variable versus another. Velocity is miles / hour, the rate of change of distance with time, or the differential of distance versus time. It is the slope of the curve. A slope is a rate of change. Velocity is the slope of the curve of distance versus time. The slope of another curve y = e^x is e^x = dy/dx, at every point along the curve. e^x is its own differential, e^x. A curve y = e^x always has a slope of e^x.
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- “e” = an infinite series of factorials. A factorial is the multiplication of a series of numbers 1 less than the last number. 4 factorial = 4! = 4*3*2*1 = 24.
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-------------- “e” = 1 + 1/1! + 1/2! + 1/3! + 1/ 4! + 1/5! + ……
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-------------- “pi” is also related to an infinite series:
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“pi“^2/6” = 1/1^2 + 1/ 2^2 + 1/ 3^2 + 1/ 4^2 + ………………………….
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“pi”/ 4, which is 45 degrees, = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - 1/15 + ………………………………
- But, the most intriguing equation of all relates “e” and “pi” together:
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--------- e^i * pi +1 = 0 …… TAKE SOME TIME TO STUDY THIS EQUATION.
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“e” raised to an imaginary “pi” is equal to -1.
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“e” raised to (-1)½ * pi = -1.
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This the most amazing equation in all mathematics, bar none. A transcendental number raised to the power of an imaginary number +1 becomes nothing. REPEAT FOUR TIMES….. Just so it sinks in a little.
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- It is easy to memorize “pi” out to 15 places:
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------------- “How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.”
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------------ “How(3) I (1) want(4) a (1) drink(5) , alcoholic(9) of(2) course(6) , after(5) the(3) heavy(5) lectures(8) involving(9) quantum(7) mechanics(9) .”
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------------- 3.14159265358979.……. The sequence goes on forever and never repeats itself. Pi shows up in many equations in physics.
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Heisenberg’s Uncertainty Principle in Quantum Mechanics states that you can not determine with certainty the position and the momentum ( mass*velocity) of an atomic particle. The better you know one the less you can know the other. Mathematically, (delta x)(delta p) = h / 4*”pi”. Where, the delta is either a certainty of position ( x) or a certainty in momentum (p) equal or greater than Planck’s constant (h) / 4*”pi”. There is always a trade-off knowing one or the other, and “pi” is in there.
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- Einstein’s general Theory of General Relativity is:
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---------------- R ik - ½ gik*R + lamda gik = 8*pi*G / c^4*Tik
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---------------- I will not try to explain this equation, but there is “pi” again.
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- Everybody that remembers school remembers the “Bell Curve”. This is the curve of a random distribution. And, it was used to grade the class for A’s, B’s, C’s, D’s, and F’s. The equation for the Bell Curve is f(x) = 1/pi^½* e^-x^2. A rather imposing equation that represents a Normal distribution, or a Random distribution of unrelated events. Note that both “e” and “pi” are in the picture.
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- The mean of the Bell Curve is “mu” and the variance is “sigma” , one standard deviation from the mean. If you calculate one sigma on either side of the mean you include 68% of the distribution. Those were the C students. If you go 2 sigma you encompass 95% of the distribution. Those were the B or D students depending on which side of the “mu” you were on. And, +or- 3 sigma includes 99 % of the Bell Curve distribution. Those were the A and F students. Under a normal distribution just as many students should flunk as get A’s. Many teachers did not like to grade on the curve. They prefer to give A’s.
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- “e” and “pi” are Transcendental numbers and they exist in their own right as part of the natural world. Other illustrations are in the math that describes population growth. Or, in the math for radioactive decay. These two numbers are even finding their way into Quantum Mechanics. The more we study “e” the more important it reveals itself to be.
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- Google in 2004 announced that its revenue target was $2,718,281,828. When Google went recruiting it used as its hiring scheme, the first 10-digit prime number found in the consecutive digits of e.com. Google wanted only math savvy applicants to apply for the jobs.
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- We have introduced only “e” and “pi” as two Transcendental numbers, but, would you believe there are actually infinitely more Transcendental numbers then there are integers and fractions. That happens to be a conjecture that I take on faith and will not attempt to prove. Math is just too amazing for words, that is the reason we use numbers.
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