Scientific Notation Rules

 Often we come across abnormally large numbers or abnormally small numbers like 400,000,000,000 or very, very small number like0.00000000002. Especially in the field of science such abnormal size of numbers is quite common.

In calculations involving such numbers, the work is cumbersome and possible errors may occur.

A special type of notation is used to simplify the dealings with these type of numbers and it is named as scientific notation as such numbers are very predominant in the field of science.

However a scientific notation must be used following a set of rules, universally accepted.

Let us take a closer look.
Scientific Notation Rules – Description

Since very large or very small numbers can be rounded to the appropriate place value, the power of 10 is used as base in scientific notation. The established general form is,

                                       a x 10n, where,  1= I a I < 10

If  ‘n’ is positive, it represents the number of zeroes from ‘a’ to the right of unit place and hence  it denotes large numbers.

For very, very small decimal numbers ‘n’ is negative and represents the number of zeroes before ‘a’ up to the decimal point.
Scientific Notation Rules – Examples

The velocity of light is about 310,000,000 kilometers per second.

We find there are seven 0s after the digits 1. That is the number is same as 31 times 10 to the power 7. But as per scientific notation rules, 31x 107 is not allowed and hence the given is modified as 3.1 x 108

Thus in scientific notation, 310,000,000 = 3.1 x 108

Similarly it is equally incorrect to mention (as a scientific notation) 310,000,000 as 0..31 x 109

Let us consider another example with a very  small number.

The wave lengths of a certain color is experimentally found to be 0.0000015 millimeter.

There are 6 decimal places to cross the digit 1from right. That the given number is (1/1000000)th of 1.5.

Hence in scientific notation it is written as 1.5 x 10-6

Importance of Derivatives of Trigonometric Functions

Trigonometric Functions 
In Mathematics, trigonometric functions or circular functions are ratios of the sides of a right angled triangle, having the given angle.  Trigonometry deals with functions of an angle present in a triangle. It relates the angles of the given triangle with the length of its three sides. It also helps to find the length and angle of a triangle. Trigonometric functions are widely used in real life in various fields such as carpentry, surveys, engineering, physics, navigation, astronomy etc.

Trigonometric functions are sin p, cos p, tan p, cot p, sec p, csc p.  Among them sine, cosine and tangent are the most familiar and basic trigonometric functions. Before going into derivatives of trigonometric functions, let us first see what these trigonometric functions refer to.

Let us consider a circle of radius 1. If a ray originates at the origin which makes an angle with x axis, what will the sine, cosine and tangent of this angle provide?
• The sine of this angle gives the length to which the triangle has risen on the other side i.e. the y component

• The cosine of this angle gives the x component length

• The tangent of this angle gives the slope of the triangle, which is obtained by dividing Y component by the X component.

Thus, trigonometric functions can also be defined as the lengths of all line segments from a given unit circle.

Derivative of Trigonometric Function
With this basic understanding on trigonometric functions, let us explore on the derivatives of trigonometric functions namely sin p, cos p, tan p, cot p, sec p, csc p.

The derivatives for each of these trigonometric functions can be represented as follows:

Derivative of Sinx
The symbolic expression of Sinx derivative is given by

d/dx (sin x) = cos x   or sin’ x = cos x


It is easy to find the derivatives of sin x. This is because, the derivative of sin x can be obtained by just calculating the cos x.

Derivative of Cosx
The Cosx derivative is given by

d/dx (cos x) = - sin x or cos’ x = - sin x


Derivative of Tan p
The derivative of tan p is given by “tan-1 p = sec^2   p”

Derivative of Cot p
The derivative of cot p is given by “cot-1 p = - coesc^2 p”

Derivative of Sec p
The derivative of sec p is given by “sec’ p = sec p tan p”

Derivative of Csc p
The derivative of csc p is given by “csc’ p = - csc p cot p”

Different types of Matrices

Matrix is a rectangular array of numbers or functions which are called the elements or the entries of the matrix. The elements or entries are arranged in rows and columns in either () or [] brackets. The elements of a matrix are denoted by aij where ‘i’ is the row number and ‘j’ is the column number. We can only add or subtract two matrices if they both have same number of rows and columns.

Order of matrices: An order of the matrices is written as m X n where m is the number of rows and n is the number of columns in it. That means a matrix of order 2 X 2 can only be added to a matrix of order 2 X 2 and a matrix of order 3 X 3 can only be subtracted from a matrix of order 3 x 3.

Types of matrices

1. Column matrix: A matrix is said to be a column matrix if has only one column. For example: - a matrix of order 4 X 1 is said to be a column matrix as it has four rows but only one column. In general, a matrix of order m X 1 is considered as column matrix where m can be any real number.
2. Row matrix: A matrix is said to be row matrix if it has only one row. For example: - a matrix of order 1 X 4 is said to be a row matrix as it has four columns but only one row. In general, a matrix of order 1 X n is considered as row matrix where n can be any real number. A example of row matrix is [2 , 6, 11, 7].
3. Zero matrix: A matrix is said to be zero matrix if all its elements or entries are zero. It is also called a null matrix. For example: - [0], [0 0] are zero matrices. We denote zero matrix by O.
4. Square matrix: A matrix is said to be square matrix if its number of rows are equal to number of columns. Thus m X n is a square matrix if m = n and is known as a square matrix of order ‘n’.
5. Unit matrix: A matrix is said to be unit matrix if all its elements or entries are one. For example: -
[1], [1 1], [1 1 1] are all unit matrices.

Polynomial Factoring

Factoring polynomial is similar to factoring numbers, but you will be dealing with expressions instead of numbers. In order to factor a polynomial, you have to determine the polynomial that divides the original polynomial evenly.

Polynomial Factoring

How to factor a polynomial?
Different approaches are used to factor the polynomial; most common approaches are simple factoring, and factoring in pairs. In this article, you will find examples for these two approaches of polynomial factoring.

Simple Factoring
In simple factoring, you identify a common factor for the expression and place it before the parenthesis. What is a common factor? To know that, look at the example below:
5x + 60 = 5 (x) + 5(6) = 5 (x+12)

Now the common factor of the above expression is 5. To cross verify, multiply the factor with each term in the expression and ensure if you are getting the original polynomial back. For example,

5(x+6) = 5 multiplied with x + 5 multiplied with 12 = 5x + 60 (the original polynomial)

Factoring in Pairs
“Factoring in Pairs” is yet another factoring polynomials solver. This approach is used when you cannot find any common factor for the terms in the expression. In this approach, the expression is split into pairs of terms and then each pair will be factored separately. Here is an example:

Find the factors of ab-5b-2a+10.

This polynomial algebraic expression includes 2 variables namely a and b. There is no common factor for these 4 terms. In this case, take the first two terms and find a common factor. Similarly, take the last 2 terms and find a common factor, as shown below:

ab-5b-2a+10 = b (a-5) -2 (a-5) = (b-2) (a-5)

In the above step, the first two terms ab-5b are taken into consideration. Take the common factor out, it is b. Now it becomes b (a-5). Similarly take the next two terms -2a+10. The common factor is 2 here. When you put it as -2, the expression becomes -2 (a-5). Now the expression can be further simplified as (b-2) (a-5).

If the consecutive terms do not have any common factors, then you can rearrange the terms based on the commonality and then do the factoring.

Rounding Decimals

Let's learn about rounding decimals in today's post.

Rounding decimals is nothing but turning a number to the nearest tens, hundreds, thousands or so on. Suppose we are adding a decimals and the answer we get is 4.6, here we can round the decimal to 5. If the number after decimal is 5 or more than 5, it can be rounded by adding 1 and if it is less than 5, it can be rounded by removing the decimal value. This method is the method of rounding decimals.

Next time i will help you with the concept of decimals number line.

statistics independent events

Independents Events is a key word often referred to in probability calculations. Knowingly or unknowingly , they use Independents Events often in the theory and application of probability. events are independent events, if the occurrence of event does not affect the probability of other variables. i.e.., positive or negative.

Algebra Solver

algebra solver step by step:

Algebra is one of the a lot of basal aspect of mathematics in which, we about-face from basal accession to variables. Algebra has assorted subdivisions like polynomials, graphing, arrangement of equations, logarithms, etc. Polynomial is aswell a allotment of algebra. Expressions of an exact ambit end to end with variables (x, y) and constants are alleged as polynomials. Polynomials absorb with operations like accession and subtraction. Algebra footfall by footfall solver can be acclimated for ths affectionate of problems. The algbra footfall by footfall solver solves the expressions accustomed in it with footfall by footfall explanation. The afterward sections shows the footfall by footfall solver free.