Algebra and Linear Algebra: Complex numbers and logarithms

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来自 Georgia Institute of Technology 的课程

工程考试复习基础

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Georgia Institute of Technology

工程考试复习基础

207 个评分

The purpose of this course is to review the material covered in the Fundamentals of Engineering (FE) exam to enable the student to pass it. It will be presented in modules corresponding to the FE topics, particularly those in Civil and Mechanical Engineering. Each module will review main concepts, illustrate them with examples, and provide extensive practice problems.

从本节课中

Mathematics

This module reviews the basic principles of mathematics covered in the FE Exam. We first review the equations and characteristics of straight lines, then classify polynomial equations, define quadric surfaces and conics, and trigonometric identities and areas. In algebra we define complex numbers and logarithms, and show how to manipulate matrices and determinants. Basic properties of vectors with their manipulations and identities are presented. The discussion of series includes arithmetic and geometric progressions and Taylor and Maclaurin series. Calculus begins with definitions of derivatives and gives some standard forms and computation of critical points of curves, then presents grad, del and curl operators on scalar and vector functions. Differential equations are calcified and to methods to solve linear, homogenous equations are presented. Fourier series and transforms are defined along with standard forms, and finally Laplace transforms and their inverse are discussed. In all cases, basic ideas and equations are presented along with sample problems that illustrate the major ideas and provide practice on expected exam questions.Time: Approximately 4.5 hours | Difficulty Level: Medium

Analytic Geometry and Trigonometry: Straight Lines 8:13

Analytic Geometry and Trigonometry: Polynomials and Conics9:23

Analytic and Geometry and Trigonometry: Trigonometry7:41

Algebra and Linear Algebra: Complex numbers and logarithms5:39

Algebra and Linear Algebra: Matrices and determinants7:29

Vectors: Basic Definitions and operations13:05

Vectors: Examples8:18

Series: Arithmetic and geometric progressions 10:13

Calculus: Derivatives and curvature10:32

Calculus: Integration5:38

Calculus: Gradient, divergence and curl7:37

DifferentialEq: Classification6:11

DifferentialEq: Solutions7:58

DifferentialEq: Fourier series7:09

DifferentialEq: Laplace7:51

与讲师见面

Dr. Philip Roberts
Professor
School of Civil and Environmental Engineering

>> The next topic in mathematics that I want to cover is Algebra and

Linear Algebra.

First of all, we'll look at complex numbers and logarithms and

then matrices and determinants.

First of all,

in complex numbers, the definitions of complex numbers are given here.

And the definition is Z, a complex number,

is equal to a+jb where in that equation A is the real component and

B is the imaginary component and J is the square root of -1

although many disciplines and more commonly I is used instead of j, but

here we'll use j just to follow the notation of the handbook.

So a complex number can be displayed in terms of the diagram which is shown here,

which is sometimes called an Argand diagram,

where the horizontal axis is the real axis or the value of the real component or

the real part of the imaginary number and the vertical axis

is the imaginary axis which shows the magnitude of the imaginary part.

So, the point here Z, is A plus JB and

this can also be expressed in polar form where the radius here

is C and the magnitude of the radius is just equal to the square

root of a squared plus b squared and the angle theta.

And manipulations on complex numbers can be performed fairly easily.

For example, the summation of two complex numbers, Z1 is A + JB1.

Z2 is A2 + JB2, then to sum those equations we simply add up

the real components and add up the imaginary components as shown here.

In the Polar Coordinates System, we can express it like this

where r is the radius, or the magnitude of the vector, or the imaginary number.

And, in this case, we have r is equal to cosign theta and

Y, X is equal to r cosign theta and

Y component is equal to sign theta, etcetera.

And the rest of the equations as given here.

Another convenient form to express this in terms of Euler's identity,

exponential E to the J theta is equal to cosign theta

plus J sign theta, is another way to express the complex number.

So let's do an example on that.

The polar coordinates of a complex number are magnitude four and angle 56 degrees.

It's rectangular components or coordinates are most nearly which of these.

So here we simply use the definitions that x, the real component,

is equal to r cosign theta is equal to four cosign 56 degrees is 2.24.

Y, the imaginary component, is r sign theta which is equal

to 4 sign 56 degrees which is equal to 3.32.

So the answer is C and you can also always confirm for

yourself you want that indeed the magnitude or the radius

is equal to the square root of the sum of the squares of those two components.

Next we look at logarithms and the basic definition of the logarithm of a number

say x to the base b is written like this.

Log of b of x is equal to c where b to the raised

to the power c is equal to x, is the definition of a logarithm of base b.

And in particular most commonly we use so called natural logarithms written as

LN of X which is a logarithm raised to the base E.

And also, logarithms to the base 10 usually written as LNG

without the subscript 10 is written like that, are the most common ones.

To convert the basis of two logarithms, we have this equation here.

Log B of X is equal to log A of X divided by log A of B.

So, for an example, we given that log of six is equal .778.

And because it doesn't show the base there,

implicitly this is log to the base 10 and similarly the Logarithm 8 is .9 over 3.

The question is what is the value

of log to the base six of eight of these alternatives?

So we use our general equation for converting between bases.

Log b of x is log a of x divided by log a of b.

So in this case, we have logarithm to the base b of x is equal to

log 10 of x divided by log 10 of b where x and

b are the two values which were given so therefore the logarithm to the base 6

of 8 is log to the base 10 of 6 divided by log to the base 10 of 8.

And these values we were given as .778 and point-nine-oh-three.

And the ration when we compute that is equal to point-eight-six-two, so

the answer is B.

And this concludes our discussion of logarithms.

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