Fundamental Identities

Let us study about fundamental identities,
If an equation contains one or more variables and is valid for all replacement values of the variables for which both sides of the equation are defined, then the equation is known as an identity. The equation x2 + 2 x = x( x + 2), for example, is an identity because it is valid for all replacement values of x.

If an equation is valid only for certain replacement values of the variable, then it is called a conditional equation. The equation 3 x + 4 = 25, for example, is a conditional equation because it is not valid for all replacement values of x. An equation that is said to be an identity without stating any restrictions is, in reality, an identity only for those replacement values for which both sides of the identity are defined. For example, the identity
is valid only for those values of α for which both sides of the equation are defined.

The fundamental (basic) trigonometric identities can be divided into several groups. First are the reciprocal identities.
Hope the above explanation helped you.

Introduction of Complex numbers

Let us study about complex numbers,
The complex numbers are the field of numbers of the form , where and are real numbers and i is the imaginary unit equal to the square root of , . When a single letter is used to denote a complex number, it is sometimes called an "affix." In component notation, can be written . The field of complex numbers includes the field of real numbers as a subfield.

The set of complex numbers is implemented in Mathematica as Complexes. A number can then be tested to see if it is complex using the command Element[x, Complexes], and expressions that are complex numbers have the Head of Complex.

Complex numbers are useful abstract quantities that can be used in calculations and result in physically meaningful solutions. However, recognition of this fact is one that took a long time for mathematicians to accept.

The Triangle Inequality Theorem

Let us learn Triangle Inequality Theorem :

In Δ TAB (Figure 1 ), if T, A, and B represent three points on a map and you want to go from T to B, going from T to A to B would obviously be longer than going directly from T to B. The following theorem expresses this idea.

Figure 1

Two paths from T to B.

Theorem (Triangle Inequality Theorem): The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Example 1: In Figure 2 , the measures of two sides of a triangle are 7 and 12. Find the range of possibilities for the third side.

Figure 2

What values of x will make a triangle possible?

Using the Triangle Inequality Theorem, you can write the following:

7 + x > 12, so x > 5

7 + 12 > x, so 19 > x (or x < 19)

Therefore, the third side must be more than 5 and less than 19.

Hope the above explanation helped you.

Difference between rational and irrational numbers

Let us learn the difference between rational and irrational numbers,

A rational number is a number that can be written as a fraction.

The numerator and denominator of the fraction must be whole numbers.

A number that cannot be written as fraction is said to be an irrational number. Most irrational numbers contain surds (roots) and constants like Pi or e.

Example 1

Is 0.75 a rational or irrational number?

Well you can write 0.75 as ¾. So 0.75 is a rational number as it can be written as a fraction.

Example 2

Is 0.463 a rational or irrational number?

Again 0.463 can be written as a fraction as 463/1000. So 0.463 is a rational number.

Example 3

Is 0.3˙ a rational or irrational number?

Yes it is a rational number as 0.3 recurring is 1/3.

Example 4

Is 13 a rational or irrational number?

All whole numbers can be written as fraction, as you can make the denominator equal to 1 (13/1). So 13 is a rational number.

Example 5

Is √9 a rational or irrational a number?

√9 is 3, and 3 can be written as 3/1. So √9 is a rational number.

All the examples so far have been rational numbers (in fact most numbers are rational).

Hope the above explanation helped you.

Factorization

Factorization:

If a polynomial can be written as the product of two or more expressions, then each expression is called the factor of the given polynomial.If a polynomial can be written as the product of two or more expressions, then each expression is called the factor of the given polynomial.We can understand this topic even better by learning about the different methods of factorization.

Methods of Factorization:

(i) Common factors

(ii) By expressing as difference of squares

(iii) By grouping

(iv) Trinomials

(v) Sum or difference of cubes

Let us learn more about factoring methods,in pre algebra, Factoring is done to represent the polynomial in terms of factors. On multiplying the factors written in factored form it gives the polynomial again. The various methods employed to find the factored form of a polynomial are,

• Grouping method
• Using formula
• Greatest common factor

Hope you like the above example of Factorization.Please leave your comments, if you have any doubts.

Matrices

Matrices:

Definition of a Matrix:

A rectangular array of entries is called a Matrix. The entries may be real, complex or functions.
The entries are also called as the elements of the matrix.
The rectangular array of entries are enclosed in an ordinary bracket or in square bracket. Matrices are denoted by capital letters.There are several types of matrices. They are:

* Row matrix.
* Column matrix.
* Zero matrix or null matrix.
* Square matrix.
* Diagonal matrix.
* Unit matrix.
Let us now learn the uses of matrices,

* Matrices are used in computer animation for plotting the diagrams and For compressing and transforming computer graphics it is used.
* It is very useful for scientists for recording the data from their experiments and for engineers, they are recorded Math reports. In architecture also, it is used.
* Matrices are used in solving partial differential equations like designing airplanes or cars.
* It is also used to analyze the structure of the universe.
* In architecture, Matrices are used for computing the measurements.
* For example, Like our problems of homework, It is very useful to figure out things like price and quantity.
* We can use matrices to manipulate colors in computer graphics and animation. By varying the red, green, and blue colors we can manipulate. Transformation of matrix is also used in computer graphics that means it is very helpful for maintaining the objects that is rotating, transform a movie clip or bitmap.

Hope you like the above example of Matrices.Please leave your comments, if you have any doubts.

Solved Word Problems in Algebra

Solved Word Problems in Algebra:

In math, problems can be given word format (verbal) which is called as word problems. In these type of problems the word format of the problem has to be converted into mathematical form and then solve. To convert the word problems to mathematical form requires good verbal knowledge, since it is difficult to understand word problems without good verbal knowledge. Almost all kinds of problems are available in word format.We can understand the word problem only by looking at a hard core example of a word problem in algebra.

Solve algebra word problem 1:

Two consecutive numbers have a sum of 81. What are the numbers?

Solution :

To begin solving this problem, define the variable in algebraic form. You do not know what the first consecutive number is, so you can call it x.

Let x = The First Consecutive Number

Since the numbers are consecutive, meaning one number comes right after the other, the second number must be one more than the first. So, x + 1 equals the second number.

Let x + 1 = The Second Consecutive Number

The problem says that the sum of the two numbers is 81. This can be shown in the equation like the following:

x + (x + 1) = 81

The equation which you just wrote can be solved as follows:

Initial Equation

x + (x + 1) = 81

After combining like terms

2x + 1 = 81

After subtracting 1 from each side

2x = 80

After dividing each side by 2

x = 40

Hope you like the above example of Word Problems in Algebra .Please leave your comments, if you have any doubts.

Congruent Triangles:

Image 1



Congruent Triangles:

Congruent Triangles are a type of similar triangles. Similar triangles have the same type of shape; however their size may be different. On the other hand congruent triangles have the same shape as well as same size also.Let us learn about congruency. Congruence is fundamental; it is the counterpart of equality for numbers. In analytic geometry,congruence may be defined intuitively thus: two mappings of figures onto one Cartesian coordinate system are congruent if and only if, for any two points in the first mapping, the Eucledian distance between them is equal to the Euclidean distance between the corresponding points in the second mapping.

As we can see above that the shape of the congruent triangles is the same but the size differs.If we learn to identify the congruent triangles we can solve the problems related to congruent triangles too.Let us now look at an example problem related to Congruent Triangle:As given in the Figure above Image 1

Given SP=SR AND

QP=QR

Is SQP SQR are congruent

Solution:

From the SQP and SQR

We have,

SQ = SQ

SP = SR

QP = QR

As the three sides of TRI SQP and the three sides of TRI SQR are equal, we write

SQP SQR

Hope you like the above example of Congrunt Triangles.Please leave your comments, if you have any doubts.

Division Algorithm for Polynomials

Let us study what is meant by Division Algorithm for Polynomials,
Let us consider the cubic polynomial
x3 – 3x2 – x + 3. If we tell you that one of its zeroes is 1, then you know that x – 1 is
a factor of x3 – 3x2 – x + 3. So, you can divide x3 – 3x2 – x + 3 by x – 1, to get the quotient x2 – 2x – 3.
Next, you could get the factors of x2 – 2x – 3, by splitting the middle term, as
(x + 1)(x – 3). This would give you
x3 – 3x2 – x + 3 = (x – 1)(x2 – 2x – 3)
= (x – 1)(x + 1)(x – 3)
So, all the three zeroes of the cubic polynomial are now known to you as
1, – 1, 3.
Let us discuss the method of dividing one polynomial by another in some detail.
Before noting the steps formally, consider an example.
Example : Divide 2x2 + 3x + 1 by x + 2.
Solution : Note that we stop the division process when
either the remainder is zero or its degree is less than the
degree of the divisor. So, here the quotient is 2x – 1 and
the remainder is 3. Also,
(2x – 1)(x + 2) + 3 = 2x2 + 3x – 2 + 3 = 2x2 + 3x + 1
i.e., 2x2 + 3x + 1 = (x + 2)(2x – 1) + 3
Therefore, Dividend = Divisor × Quotient + Remainder

Let us now extend this process to divide a polynomial by a quadratic polynomial.