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Browse High School Transcendental Numbers

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Infinity, Pi, and Stranger Things [7/15/1996]
Imagine there's an infinite string attached to the top of your head, and you're dangling from it... somewhere along your spine, you have an itch you can't scratch...

The Number e [6/3/1996]
Is "e" a number like Pi? How does "e" relate to continued fractions?

Numbers [12/10/1997]
How do integers and whole numbers, rational numbers, transcendental numbers, and counting numbers relate to each other?

Why is "e" so important? [1/8/1995]
Why is "e" so important? How significant is "e" compared with "pi?" How did it come about? How is it defined? Why is it taught only at higher level mathematics? Are there other numbers like "e"?

"Accidental Order" in Pi, e [04/12/1999]
Using a result of Dirichlet's to prove that a given sequence will appear in infinitely many prime numbers.

Alabama legislature and pi [04/15/1998]
Did Alabama really vote that pi should have its biblical value of 3?

Are Transcendentals Irrational? [9/2/1995]
Are transcendental numbers a subset of the irrationals?

Complex Powers [09/28/2001]
How do I show that abs(z^i) is less than exp^pi where z is a complex number not equal to 0?

Compounding Interest and e [11/11/1998]
How is the number e derived from compounding interest? Does it come from the series expansion of e?

Derivative of e^x by Limit Definition [05/21/2002]
Show that the derivative of e^x is e^x, using the limit definition but not the Taylor series expansion.

The Difference Between Log and Natural Log [2/8/1995]
What is the difference between log and natural log?

Doesn't Pi Ever Repeat? [7/8/1996]
How do people know that pi goes on forever without repeating?

e^(i*pi) = -1: pi = 0 ? [10/17/1997]
I have been playing with the equation e^(i*pi) = -1 and have found some disturbing things...

e^(pi*i) = -1: A Contradiction? [8/17/1996]
I know that e^(i*Pi) = -1. But squaring and taking a natural log of both sides, you get 2*i*Pi = 0. Please explain.

e^pi vs. pi^e [03/20/2002]
Which is greater, e^pi or pi^e? I would like to have a simple proof.

e - The Transcendental Number [5/16/1996]
Can you help us find a history of "e", the transcendental number?

Euler Equation [01/21/1997]
What is the meaning behind e^(pi*i) = -1?

Euler Formula: e^(pi*i) = -1 [6/5/1996]
Why does e^(pi*i) = -1?

e^x Appreciation [7/9/1996]
What is the history of e^x? What do you think of it?

Facts about e [11/09/1998]
Where does the number "e" come from and what uses does it have in the real world?

Find the Flaw [08/02/2001]
I don't understand where the following proof goes wrong...

Gregory's Series [06/14/1997]
Where does this series 4[1 - 1/3 + 1/5 - 1/7 + ....] for pi come from?

History and Applications of the Natural Logarithm [03/02/1998]
I'm so surprised at how often the number e comes up. Where did it come from? Who first derived it? Why is it so common in the field of biology?

How Can Pi Not Repeat? [06/01/1999]
Won't pi repeat when it runs out of number combinations?

How Pi is Calculated [08/07/1997]
If pi is calculated by dividing the circumference of a circle by its radius, how can computers generate pi to billions of decimal places?

Intriguing Limit [04/30/1997]
Why is the following true: lim x --> oo (1-1/x)^x = 1/e?

Introducution to Algebraic Numbers and Integers [04/15/2008]
While reading about the classifications of complex numbers, such as real, irrational, and so on, I came upon a reference to 'algebraic numbers' and 'algebraic integers'. Can you tell me about those?

The Irrationality of e [09/04/1998]
Are there any simple proofs to show that a number like e is transcendental? What about irrational?

Irrational Pi [7/15/1996]
Why is pi considered an irrational number when it's defined as the ratio of the circumference of a circle to its diameter?

The Limit of (1+1/x)^x As x Approaches Infinity [02/17/1998]
How Euler calculated e, and what it has to do with the equation (1+1/x)^x.

My Calculator Gives Wrong Answers for Large Values of x ... [01/20/2010]
Doctor Vogler reveals how a calculator's imprecision leads to increasingly large rounding errors -- and misleads a student trying to make sense of the formula for the base of natural logarithms.

Natural Logarithms [11/23/1998]
Why are logarithms to the base e called "natural"?

The Nature of e [02/23/1997]
Why is e considered natural?

Numbers that represent pi [08/08/1997]
If pi is an inexact and transcendental number, how in the world did we obtain a certain series of numbers to represent pi?

Pi and Polygons [03/14/1999]
Derive a formula to find the angle of an n-sided polygon with x sides.

Proof of the Infinite Series That Calculates 'e' [02/04/2004]
Is there a proof about this infinite series that gives the value of e: 1 + 1/1! + 1/2! + 1/3! + 1/4! + . . . + 1/n! where n goes to infinity?

Proofs of e [03/21/2002]
I found out that the definition of e is: e = lim, as n approaches infinity (1+1/n)^n. Is there a proof for this or for 1+2(e-1/n)?

Proof that e^i(pi) = -1 [06/02/1999]
How can it be proven that e^[i(pi)] = -1? And why does it matter?

Proving a Number is Transcendental [4/12/1996]
How does one prove a number is transcendental?

Proving Pi is Irrational [01/06/1999]
How do you know that pi never repeats? How would you show that it is irrational?

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