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Posts Tagged ‘rule’

Algebraic Product Rule

Tuesday, December 29th, 2009

How to Use the Product Rule in Algebra

Description

A detailed tutorial on the algebraic product rule. Step by step tutorial including several examples of the algebraic product rule for reference.

Overview

There are many product rules in the world of math. This tutorial focuses on a product rule that is used in algebra and statistics. The product rule states that if two independent tasks T1 and T2 are to be performed, then T1 can be performed m ways and T2 can be performed n ways. Therefore, the number of ways the tasks can be performed together is m * n ways. Remember that this is only the number of possible ways to do something, not how much time it takes to do something. Also, the same method is used no matter how many different tasks you are given.

Tags: algebra, combination, multiplication, multiply, number, permutation, product, rule, statistics, task
Posted in Algebra | No Comments »

Rule of Sarrus

Tuesday, November 3rd, 2009

Rule of Sarrus Explained

Description

A detailed tutorial on the Rule of Sarrus. Step by step tutorial including several examples of the Rule of Sarrus and determinants for reference.

Overview

The Rule of Sarrus is a method used to compute the determinant of a 3×3 matrix. Mathematically stated, if you are given a 3×3 matrix, you can compute the determinant by repeating the first two columns of the matrix behind the third column, so that you have 5 columns in a row. This forms a 3×5 matrix. Then you add the products of the diagonals going from top to bottom (left to right), and subtract the products going from bottom to top (left to right). This can also be used for 2×2 matrices, but the rule used is a little different.

Tags: 2x2, 3x3, 3x5, add, algebra, bottom, column, determinant, diagonal, left, matrices, matrix, product, right, row, rule, sarrus, scheme, subtract, top
Posted in Algebra | No Comments »

Higher Order Derivatives

Friday, October 30th, 2009

How to Find Higher Order Derivatives

Description

A detailed tutorial on higher order derivatives. Step by step tutorial including several examples of higher order derivatives for reference.

Overview

A higher order derivative is a derivative with a power other than one – that is, a derivative is referred to as a first derivative, and the higher order derivatives are a second derivative, third derivative, etc. The second derivative is the derivative of the first derivative, and the third derivative is the derivative of the second derivative. When you know all the rules of taking derivatives, taking second and third derivatives are simple. Simply take the derivative and pretend it is another equation. When you go up beyond the third derivative this can get more challenging, as there will be many more parts to the equation.

Tags: antiderivative, Calculus, chain, derivative, First, higher, integral, order, power, product, quotient, rule, second, third
Posted in Calculus | No Comments »

Cross Product

Tuesday, October 27th, 2009

The Cross Product of Vectors

Description

A detailed tutorial on the cross product of two vectors. Step by step tutorial including several examples of how to find the cross product for reference.

Overview

A cross product is very similar to a dot product. However, the result of a cross product is a vector, and the result of a dot product is a scalar. In mathematical terms, the cross product can be defined as $\mathbf{a}\times\mathbf{b}=\left\|\mathbf{a}\right\|\left\|\mathbf{b}\right\|\sin(\theta)\,\mathbf{n}$ . Theta represents the meausre of the angle between a and b, and n is a unit vector perpendicular to both a and b. The vector this forms is a right-handed system.

Tags: a, algebra, b, cross, dot, n!, outer, perpendicular, product, right-handed, rule, scalar, system, unit, vector
Posted in Algebra | No Comments »

Vector Addition

Friday, October 23rd, 2009

How to Solve Vectors Using Vector Addition

Description

A detailed tutorial on how to solve vectors using vector addition. Step by step tutorial including several examples of vector addition for reference.

Overview

Vector addition involves two vectors that do not have to be equal, and could have different magnitudes and directions. The vectors are referred to as a and b. The formula for vector addition is:

$\mathbf{a}+\mathbf{b}=(a_1+b_1)\mathbf{e_1}+(a_2+b_2)\mathbf{e_2}+(a_3+b_3)\mathbf{e_3}.$

Vector addition is also occassionally referred to as the parallelogram rule, because on a picture diagram of vector addition the shape of a parallelogram is formed.

Tags: addition, algebra, direction, equal, formula, graph, magnitude, parallelogram, picture, rule, vector
Posted in Algebra | No Comments »

Parallelogram Law

Thursday, October 1st, 2009

Introduction to the Parallelogram Law

Description

A detailed tutorial of the parallelogram law. Step by step tutorial including several examples of the parallelogram law for reference.

Overview

The parallelogram law shows up in many forms, but the simplest form states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. Assuming that a rectangle has four corners A, B, C, and D, this can be expressed as:

$(AB)^2+(BC)^2+(CD)^2+(DA)^2=(AC)^2+(BD)^2.\,$

Typically, the two diagonals of a parallelogram are not equal in length. If they are, then the equation simplifies to the Pythagorean theorem. A more complicated version of the parallelogram law is often found when calculating vectors.

Tags: diagonals, Geometry, law, length, Math, parallelogram, pythagorean theorem, rule, side, square, sum
Posted in Geometry | No Comments »

Change-of-Base Rule

Thursday, October 1st, 2009

How to Solve Logarithms Using the Change-of-Base Rule

Description

A detailed tutorial on solving logarithms with the change-of-base rule. Step by step tutorial including several examples of how to solve logarithms using the change-of-base rule for reference.

Overview

The change-of-base rule is typically only used when solving logarithms with a calculator. It allows you to use a number besides the calculator presets. Tha change-of-base rule states that:

$\log_b{x} = \frac{\log_k{x}}{\log_k{b}}.$

In this formula, b must not be equal to one, as the logarithm of one is simply zero. This formula also implies that all logarithms are similar to each other.

Tags: algebra, base, calculator, change, change-of-base, log, logarithm, Math, rule, similar, theorem
Posted in Algebra | No Comments »

Modus Tollens

Thursday, September 24th, 2009

The Modus Tollens Rule Explained

Description

A detailed tutorial on the modus tollens rule. Step by step tutorial including several example problems of the modus tollens rule for reference.

Overview

Modus tollendo tollens, often simply referred to as modus tollens, is an argument in logic that states if P, then Q. Negation of Q, therefore negation of P. This is sometimes called denying the consequent, and is often confused with the indirect proof of proving by contraposition. There are several forms that the modus tollens rule can take, depending on when and how you are using it.

Logical Operator Notation: $P\to Q, \neg Q \vdash \neg P$

Basic Form: $\frac{P\to Q ~,~~ \neg Q}{\neg P}$

With Assumptions: $\frac{\Gamma \vdash P\to Q ~~~ \Gamma \vdash\neg Q}{\Gamma \vdash \neg P}$

Set Theory:

$P\subseteq Q$
$x\notin Q$
$\therefore x\notin P$

Predicate Logic:

$\forall x.~P(x) \to Q(x)$
$\exists x.~\neg Q(x)$
$\therefore \exists x.~\neg P(x)$

Tags: assumptions, discrete math, logic, logical operator, Math, modus tollendo tollens, modus tollens, negation, P, predicate, proofs, Q, rule, sequent, set theory, then, therefore, truth tables
Posted in Discrete Math | No Comments »

Modus Ponens

Thursday, September 24th, 2009

The Modus Ponens Rule Explained

Description

A detailed tutorial on the modus ponens rule. Step by step tutorial including several examples of the modus ponens rule for reference.

Overview

Modus ponendo ponens, typically shortened to just modus ponens, is an argument in logic. It is closely related to the argument modus tollens. Modus ponens states that if P, then Q. P, therefore Q. This can be expressed in either sequent form or rule form for formal notation.

Sequent Form: $P \to Q, P \vdash Q$