How about if I start a new thread on Numerical Methods for Buying-LOW-&-selling-HIGH stock? Maybe it won't be so alien to you then.

Hi maximR & RED-HAIR-SHANKS,

Don’t let your brain rust! This is fun and predictably IRRATIONAL.

A very good suggestion. Preparing the tutorial notes in MSWord, convert to picture, resize, touch-up and upload are very inconvenient. Therefore, I second that!

As you can see, I typed the following equation using LaTeX commands.

e^{i\pi} + 1 = 0

Can anyone identify the most compact equation in all of mathematics?

Find out the answer in http://www.codecogs.com/latex/eqneditor.php

Can you name that beautiful equation?

Hint: See Post #226 on Page 12.

I'm back! Wow, what a fruitful discussion...

In logic, accordingly, the statement “There exist irrational numbers a and b such that a^b is rational” means “At least one, but not all cases, where two irrational numbers a and b will produce a rational value of a^b.”

This post has been edited by Critical_Fallacy: Dec 13 2013, 11:39 PM

Appreciate v1n0d's efforts in guiding both of you.

The whole idea is to develop your HOTS, even without the knowledge of Gelfond-Schneider Theorem.

The simplest irrational number we know is √2. So, in the simplest form, we have .

However, we do not know whether is rational or irrational. Thus, we have two cases here.

CASE 1: If is rational, then it is self-evident that the theorem is correct because both a and b are irrational from the beginning.

CASE 2: If is irrational, then we can reassign and raise it to the power of , so that , that is obviously rational, which in turn proves the theorem is correct.

CONCLUSION: Therefore, whichever the case is, we can deduce that the theorem is readily in a position to be validated, even when a ≠ b, or having two distinct irrational numbers, e.g. .

P.S.: Gelfond-Schneider Theorem only tells you that is irrational.

This post has been edited by Critical_Fallacy: Dec 14 2013, 01:16 AM

The most compact equation in all of mathematics is surely Euler’s Identity, or .

In this equation, the five fundamental constants coming from four major branches of classical mathematics – arithmetic (0, 1), algebra (i), geometry (π) , and analysis (e) , – are connected by the three most important mathematics operations – addition, multiplication, and exponentiation – into two non-vanishing terms.

Some of my Tutorial followers (in Complex numbers) are probably aware that Euler’s Identity is but one of the consequences of the miraculous Euler formula , because when , , , it follows that .

This post has been edited by Critical_Fallacy: Dec 14 2013, 01:39 AM

Hi ailing tan, RED-HAIR-SHANKS & maximR, this is my 6th Critical Tutorial on Differentiation. Learning differentiation is a prerequisite for Maclaurin Series and Taylor Series. In addition, if you master differentiation, learning integration will be a lot easier because you will be dealing with the antiderivatives of the functions. However, please take note that hyperbolic functions and parametric equations are not covered in this tutorial.

Prerequisites: SPM Differentiation, Geometry Coordinates, Trigonometry

Part 1 :: Derivative of a Function, Constant rule, Power rule , Sum Rule

Part 2 :: Difference Rule, Derivative of , Product Rule

Part 3 :: Quotient Rule, Chain Rule, Derivative of

Part 4 :: Derivatives for and

Part 5 :: Implicit Differentiation

Continue to Post #307

ailing tan, RED-HAIR-SHANKS & maximR,

Part 6 :: Applications of Implicit Differentiation

Part 7 :: Derivative of an Inverse Function, Derivatives of Inverse Trigonometric Functions

Part 8 :: Derivative of and , Logarithmic Differentiation

Part 9 :: More examples on differentiation, Newton-Raphson method

Part 10 :: Applications of Newton-Raphson method

Pre-print version:  Tutorial_6_Differentiation.pdf ( 284.54k ) Number of downloads: 11

30 bottles are randomly selected from a single batch.

The data from 10 and 20 measurements are obtained from the same batch (parent population).

Batch production occurs when many similar items are produced together in which small batches of product are made one at a time. Say there are 5 batches in daily production. Suppose that 30 bottles of bottled water are randomly selected in each of the five batches. We usually perform a single test called analysis of variance, for the hypothesis “all five population means are equal” for QA/QC purposes.

Have you asked mumeichan?

Here is the Applied Physics based ACME network:


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Because two elements of the matrix come with desired values, , only a maximum of seven (ERO) operations are required to get . In fact, ERO is nothing simpler than the basic arithmetic operations + − × ÷. By the way, ERO is NOT a formula, but an algorithm, meaning a process or set of rules to be followed in calculations.

Could you post your workings here?

This post has been edited by Critical_Fallacy: Dec 18 2013, 01:36 PM

Like I promised, you need only seven ERO steps to find the matrix inverse. In real exam, you don't have the luxury of time to write many steps and matrices. You need to learn how to use an algorithm to perform the optimum ERO steps. The idea behind the algorithm is NORMALIZE to 1 and EMPTY to 0. Keep practicing to familiarize with the algorithm. Most textbooks do not tell you how to perform ERO efficiently.

Most textbooks wouldn't tell you how to make the entries 1 or 0. I discovered the method when I understand the concept behind ERO and Gauss Elimination procedure.

We called the diagonal elements a11, a22 and a33 pivot elements where they are normalized to 1. The easiest way to normalize the diagonal elements is to divide the element by itself. This is the reason you see the entire row is divided by the diagonal element. You cannot normalize all three diagonal elements at the same time. You must normalize one diagonal element and then empty the subsequent entries in the same COLUMN of the normalized element. For example, if a11 is normalized to 1, then subsequent entries in the same Column 1, a21 and a31 must emptied to 0. Only then, you can normalize the second diagonal element, either a22 or a33.

Here is my Gauss Elimination algorithm:



This post has been edited by Critical_Fallacy: Dec 18 2013, 10:29 PM

Vector Calculus is an advanced topic of Calculus. You can refer to Paul's Online Math Notes :: Calculus III.

More advanced topic is Fractional Calculus.

Go to your library and borrow Larson & Edwards' Calculus, Stewart Calculus, and Thomas' Calculus.

Vector Calculus also covered in Engineering Mathematics, by K.A. Stroud, which is Learner-friendly.

This post has been edited by Critical_Fallacy: Dec 18 2013, 10:43 PM

Oh Shana-chan, I miss you! Although the problem contains many linguistic confusions, it can decoupled by linearizing "John" as "J" and "Me" as "M", and then write the governing algebraic equations.

This is integration by parts: .

The bad news is that has no antiderivative, when you separate them by parts. The good news is that there is an easy way, called "integration by substitution"! Always check the relationship between u and v, before resorting to integration by parts.

I know you've heard it a thousand times before. But it's true - hard work pays off. If you want to be good, you have to practice, practice, practice. If you don't love something, then don't do it. Moreover, you can't hire someone to practice for you.

Therefore, solve the following linear algebraic equations using Gauss Elimination algorithm in Post #342.

Whether Shanks simplifies Rows 1 & 2 or not, it doesn't really matter, because the ultimate objective is to normalize all diagonal elements. It does not change the fact that Shanks still can reduce the augmented matrix to row echelon form within the minimum 6 STEPS in Gauss Elimination of a 3-by-3 matrix. before performing the back-substitution.