Imaginary Complex Number

Introduction:

We will discuss about imaginary number. An imaginary number is a number in the form bi where b is a non-zero, real number and i, defined by i2 = − 1, is known as imaginary unit.An imaginary number bi can be added to real number a to form a complex number of the form a + bi,a and b are called , "real part" and "imaginary part" .Imaginary numbers can thought of complex numbers where real part is zero.


Applications of Imaginary Numbers


Real numbers (or even rational numbers) propose a satisfactory portrayal of data.
Fraction such as ⅔ and ⅛ are worthless to a personality including stones, but fundamental to a person compare the size of poles apart collection of gravel.
Pessimistic numbers such as −3 and −5 are insincere when measure the mass of an object, but essential when keeping track of pecuniary debit and credit.
Complex Imaginary numbers have indispensable physical application in a variety of sciences and allied areas such as signal processing, control theory, fluid dynamics, quantum mechanics, cartography, and trembling analysis.
In the complex number demonstration the same apparatus are referred to as real and invented. When the complex number is with the sole purpose make-believe, such as a real part of 0 and an imaginary part of 120. Understanding Number Sense is always challenging for me but thanks to all math help websites to help me out.

Control theory


In control theory, systems are often misshapen from the time domain to the regularity domain using the Laplace transform.
The system pole and zeros are then analyzed in the intricate seaplane.
In the root locus method, it is more than increasingly important whether the pole and zeros are in the omitted or right half hydroplane, i.e. have real part superior than or a reduced amount of than zero.
In the right half plane, it will be unbalanced.
All in the left half plane, it will be stable,
On the pretend axis, it will have unimportant immovability..
If a system has zeros in the right half plane, it is a minimum phase system.

Solve Basic Probability Problems

Introduction:

Probability is one of the important topic in intermediate and college mathematics. Probability plays an important role in the day to day life. It is way of expressing that an event will occur or has occurred. It is also defined as the possible outcomes to the total number of events.

A probability of an event A occurred is represented by a real number in the range from 0 to 1 and written as P(A), p(A) or Pr(A). The probability of impossible event is given as 0, and a certain event has a probability of 1. However, the conversations are not always true: probability 0 events are not always impossible, nor probability 1 events certain. Basic proabbility problems are easy to solve. When the concept of basic probability is understood very well, then it is easy to solve those problems.

Examples problems for basic probability:

Basic probability Problem 1:

What is the probability of rolling a 3 on a dice?

Solution:

Here the number of possible out comes on the event is 6
The number of outcomes in rolling a dice is 3
Solve for the probability,

number of outcomes in Event
Prob(E) = --------------------------------------------------------------------
Total number of possible outcomes of the event

Prob(E) = 3/6

Prob(E) = 1/2

Basic probability problem 2:

What is the probability of selecting card of hearts from a deck of 52 cards?

Solution:

The total number of cards in a deck is 52
The number of cards of hearts fro a deck of cards is 13
Solve for the probability,

number of outcomes in Event
Prob(E) = ----------------------------------------------------------------------
Total number of possible outcomes of the event

Prob(E) = 13/52

Prob(E) = 1/4

Basic probability problem 3:

What is the probability of selecting number 9 from a deck of 52 cards?

Solution:

The total number of cards in a deck is 52
The number 9 from a deck of cards is 4
Solve for the probability,

number of outcomes in Event
Prob(E) = ----------------------------------------------------------------------
Total number of possible outcomes of the event

Prob(E) = 4/52

Prob(E) = 1/13

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Basic probability problems practice:

What is the probability of rolling a 3 on a dice?

Answer:

The answer is 6/6 = 1

2) What is the probability of selecting two balls from a basket containing 20 balls?

Answer:

The answer is 2/20 = 1/10.

Basic Terms and Definitions

Introduction to basic terms and definitions:

Mathematics is the group of sciences (arithmetic, geometry, algebra, calculus etc) dealing with quantities, magnitudes, and forms and their relationships, attributes etc by the use of numbers and symbols. There are many branches in mathematics. There are several terms and definitions involved in mathematics. Some of the basic terms and definitions involved in mathematics are defined below: Is this topic Translations Math hard for you? Watch out for my coming posts.


Basic terms and definitions:


Arithmetic:

It is a branch of mathematics that deals with the study of number specially with regard to basic operations such as addition, subtraction, multiplication and division etc and their applications to solution of problems.

Algebra:

Algebra, like arithmetic is a branch of the science of calculation. In arithmetic, however, we calculate by means of figures only, whereas in algebra, we use symbols also to denote quantities.

Algebraic equation:     It refers to any equation that contains only algebraic expressions and signs of operations.

Algebraic expression: It consists of various algebraic terms connecting with the help of signs of operations.

Algebraic function: Any function that involves only algebraic expressions and  signs of operations.

Geometry:

A branch of mathematics concerned with the properties of lines, curves, and surfaces usually divided into pure, algebraic, and differential geometry in accordance with the mathematical techniques utilized.

Calculus:

It is a branch of mathematics that deals with differentiation (differential calculus) and integration (integral calculus) of functions. It finds applications in different areas of natural, physical and social sciences.

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Additional basic terms and definitions:


Trigonometry:

Trigonometry includes the measurement of triangles and includes the relation between the angles, sides, and triangles.

Trigonometric equation: Equations whose variables are the argument of trigonometric functions are the trigonometric equations

Trigonometric identities: Identities involving trigonometric function are called as trigonometric functions.

Probability:

The Numerical measure of the likelihood of an event to occur is referred as probability.

Statistics:

Statistics is the science of making effective use of numerical data relating to groups of individuals or experiments. In other words, Statistics is the numerical data collected systematically, summarized, and tabulated.

Definition of Algebra

Definition of algebra:

Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures. -Source from Wikipedia.

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Simple problems On Algebra:


Example 1:

Solve the algebraic expression X-3= 12

Solution:

We need to find the value of x

Step 1: X-3+3=12+3 (we need to add 3 on both the sides)

Step 2: X=15.

Example 2:

Solve the algebraic expression 3a-9=63

Solution:

We need to find the a value

Step 1: 3a-9+9=63+9 (first we need to add 9 on both the sides)

Step 2: 3a=72 (Now we divide using 3 on both the sides)

Step 3: a=24

Example 3:

Solve the algebraic expression 10+B+1=2(B+1)

Solution:

Step1 :( 11+B)=2B+2

Step 2: 11+B-B=2B-B+2 (subtract B on both the sides)

Step 3: 11=B+2

Step 4: 11-2=B+2-2 (Subtract 2 on both the sides )

Step 5: 9=B.

Example word problems:

Word Problem 1:

Marissa spent `$` 42 for shoes.  This was `$` 14 less than twice what she spent for a dress.  How much was the dress?

Solution:

Step 1: Let assume x be how much she spent for the dress.

Step 2: So, 2x-14=42

Step 3: 2x+14-14=42+14 (Here we need to add 14 ob both the sides)

Step 4: 2x=56(Now we divide using 2 on both the sides)

Step 5: x=56/2

Step 6: x= 28

So the dress cost is $28.

Word Problem 2:

The sum of two numbers is 84, and one of them is 12 extra than the other.  What are the two numbers?

Solution:

In this problem, we need to find two numbers. So, let x be one of them.  Let x be the first number.

We know that the other number is 12 more, x + 12.

Here the sum is 84:

So it can be written as x+x+12=84.

Now we need to solve this:

So, x+x+12=84.

Step 1:  2x+12 =84

Step 2: 2x-12+12=84-12 (subtract 12 on both the sides)

Step 3: 2x=72 (now we need to divide using 2 on both the sides)

Step 4: x= 36 .So the first number is 36

Now we need to find another number:

x+12 =36

So, 36+12 = 48.The another number is 48

(The sum of 36+48 is 84)

Basic Geometry Test

Introduction:

Geometry is used to learn all kinds of shapes and their properties. There are two main classifications in geometry. Basic geometry is the straight edge in every side of a figure completely lies in the same plane. Basic geometry learns about all shapes such as lines, circles, etc. The term geometry means learn of properties of figures and shapes and the relationship between them. Please express your views of this topic What is a Straight Angle by commenting on blog.


Basic geometrical concepts:


Point
Line
Ray
Line segment
Plane
Point:

In basic geometry test, place a dot with the sharp tip of your pencil or with a pin on a sheet of paper. Learn about dot gives you an idea of a point. The point represents a particular location. Thus we say that a point has an exact position. Point does not contain length, breadth or thickness.

Line:

In basic geometry test, two points are joined in any direction. It has no height and thickness. It is a straight one dimensional figure. It can extend in both directions infinitely.

Ray:

In basic geometry test, we learn that the sun is emitting rays of light. We can also see the rays of light in the torch light. A ray of light originates from a point in the sun or from the torch light and extends endlessly in one direction.

Line segment:

In basic geometry test, a line segment is a part of the line consisting of two end points on it.

Plane:

In basic geometry test, a plane is a flat surface which extends endlessly in all directions. No boundaries are present in the plane. A plane cannot be drawn of our own. But a branch of a plane can be drawn on any flat surface. Is this topic 2 Parallel Lines hard for you? Watch out for my coming posts.


Basic Geometry example


Example 1:

Let XY be a ray.

(1) What is the starting point of the ray?

(2) May Y be the starting point of the ray?

(3) Where does the point Y lying on the ray?

(4) Are the ray XY and YX different?

Solution:

1) Starting point is X.

2) No.

3) Y can be marked anywhere on the ray in order to name the ray.

(4) XY and YX are different because they are in different directions.

Learning Basic Algebra Free

Introduction to learning basic algebra free:

Algebra, a branch of mathematics deals with the expressions of various order, their operations, relations and construction and concepts of expressions.Algebra is the combination of numerical numbers, alpha-numeric variables, expressions and etc., the basic aim of the algebra is balancing the algebraic equations on both sides.In learning basic algebra free, you can learn about the various expressions and order of expressions, with their standard forms.In this article for learning basic algebra free we shall discuss about the basics of algebra. Is this topic Improper Fractions to Mixed Numbers hard for you? Watch out for my coming posts.


Basics of Algebra – learning basic algebra free:


Adding and Subtracting ,
Multiplication of algebra ,
Division of algebraic and
Solving Equations.


Adding and Subtracting Algebraic Expressions:

This shows how to solve the addition and subtraction in algebraic expression.

eg:

Addition:

2a^2 + 2b + 5a^2= 7a^2 +2b.

Subtraction:

15a^2 – 31b -10a^2 = 5a^2 – 31b.

Multiplication of algebra expressions:

This shows how to solve the Multiplication in algebraic expression.

eg:

multiplication:

2(a + 3b) = 2a +6b.

Division of algebraic expressions:

This shows how to solve the Division on in algebraic expression.

Eg:

Division:

12ab2 / 31ab2

Solving Equations:

This shows how to solve the system of algebraic expression.

Eg:

Solve:

5 - (x + 2) = 0.

-5x + 10 = 0

-5x = -10

x = 2

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Example for solving algebra - learning basic algebra free:


Solve the following linear equations: x + 2y + 3z = 14, 3x + y + 2z = 11, 2x + 3y + z = 11.

Solution:

x + 2y + 3z = 14 (1)

3x + y + 2z = 11 (2)

2x + 3y + z = 11 (3)

Consider the equations (1) and (3)

(1) ? x + 2y + 3z = 14

(3) × 3 ? 6x + 9y + 3z = 33 (subtracting)

–5x – 7y = –19

5x + 7y = 19 (4)

Consider the equations (2) and (3)

(2) ? 3x + y + 2z = 11

(3) × 2 ? 4x + 6y + 2z = 22 (subtracting)

–x – 5y = –11

x + 5y = 11 (5)

Consider the equations (4) and (5)

(4) ? 5x + 7y = 19

(5) × 5 ? 5x + 25y = 55 (subtracting)

–18y = –36

? y = 2

Substitute y = 2 in (5) we get

x + 5(2) = 11

x + 10 = 11

x + 10 - 10 = 11 - 10

? x = 1

Substitute x = 1, y = 2 in (3) we get

2(1) + 3(2) + z = 11

2 + 6 + z = 11

8 - 8 + z = 11 - 8

? z = 3

The solution is x = 1, y = 2, z = 3.

Inference and Models

Introduction:

Inference is the process of drawing a conclusion by applying clues (of logic, statistics etc.) to observations or hypotheses; or by interpolating the next logical step in an intuited pattern.( Source : Wikipedia).The last pattern of drawing is called inference. In the field of logic, the valid inference models are described. Human inference can be studied with the field of cognitive psychology; and then artificial intelligence researchers developed inference systems for the human inference. Understanding Margin of Error Examples is always challenging for me but thanks to all math help websites to help me out.


Inference and Models

Accuracy of Inductive Inference:

A conclusion is made infered from the multiple observations, this process is known as inductive reasoning. The conclusion we made is correct or incorrect, degree of accuracy is incorrect, or it should be correct in some situations. Conclusions for multiple observations can be tested by additional observations. Let us see about inference models below.

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Example Problems for Inference Models

Examples of Deductive Inference:

The inference models are below:

1.All women are indian.

2.Sarojini Naidu is a woman.

3.Therefore, Sarojini Naidu is Indian.

Consider the following symbological track models are below:

1.All dolphin are elephant.

2.  Cat is dolphin.

3.Therefore, an elephant is dolphin.

All the three parts of an infernce models are false but it is valid.

1.All cucumber are black.

2.A apple is an cucumber.

3.Therefore, a apple is black.

The above inference models are true and the premises need to be true.

Now we move to the invalid form.

1.All Sparrow are parrot

2.Pigeon is a parrot.

3.Therefore, pigeon is sparrow.

To show how the inference is invalid and by below premises to prove as a false conclusion.

1.All pizzas are beverages.

2.Burgars are beverages.

3.Therefore, burgars are pizzas.

A valid argument with false premises, and may provide false conclusion.

1.All slim people are American.

2.Indira Gandhi was slim.

3.Therefore, Indira Gandhi was American.

A valid argument can be used to derive true conclusion from the false premises.

1.All fat people are singers.

2.Himesh Reshammiya was fat.

3.Therefore, Himesh Reshammiya was a singer.

In this above statement, and we have two false premises that implies a true conclusion.