Algebraic identities or equations

  1. (x+y)(x+y) = x²+2xy+y²
  2. (x-y)(x-y) = x²-2xy+y²
  3. (x+y)(x-y) = x²-y²
  4. (x+a)(x+b) = x²+(a+b)x+ab
  5. (x+y+z)² = x²+y²+z²+2xy+2xz+2yz
  6. (x+y)³ = x³+y³+3x²y+3y²x  or  x³+y³+3xy(x+y)
  7. (x-y)³ = x³-y³-3x²y+3y²x  or  x³-y³-3xy(x-y)
  8. x³+y³+z³-3xyz = (x+y+Z)(x²+y²+z²-xy-xz-yz)
  9. x³+y³ = (x+y)(x²-xy+y²)
  10. x³-y³ = (x-y)(x²+xy+y²)
  11. if x+y+z = 0 then x³+y³+z³ = 3xyz

Equations for surface area and volume of geometrical shapes

surface area

Area of rectangle = l x b -length x birth

Area of the circle = πr²

Surface area of cuboid = 2( lb+bh+hl) where l= length, b= breth h = height

Surface area of cube = 6a² where a= edge of the cube

Lateral Surface area of cube = 4a² where a= edge of the cube

Curved surface area of a cylinder = 2πrh where r= radius of the cynder, h= height of the cylinder. 

Total surface area of the cylinder =  2πr(r+h) where r= radius of the cynder, h= height of the cylinder.

Curved surface area of cone = πrl, where r= radius of the cone, l is slant height of cone.

Total surface area of a cone = πr(l+r), where r radius, l= slant height

if we know height and radius of the cone we can find the slant height (l) of the cone by  l= √(r²+h²)

Surface area of a sphere = 4πr², where r is the radius of the sphere.

Curved surface area of a hemisphere (half sphere) = 2πr² 

Total surface area of a  hemisphere (half sphere) = 3πr² 

Volume

volume of cuboid = l x b x h where l= length, b= breth, h= height

Volume of the cube = a³ where a is the edge of the cube

Volume of the cylinder = πr²h where  r= radius, h = height

Volume of the cone = 1/3(πr²h) where r= radius of the cone , h= height of the cone

Volume of the sphere = 4/3(πr³) where r = radius of the sphere

 

How to find out the square root by long division method

If the square root of the number N is  x .  Then x *x will the number N. So in this post we will see how to find out the square root of a number in long division method.

Step-1

Consider the Fig1-a  we are finding out the square root of 225.  First we have to group the 225 as group of 2 numbers from right to left as shown in fig. If the leftmost digit is single we can leave as it is. Now consider each group from left to right. So first we consider the 2. Then find the  perfect root number who squire is just below or same to the first group of number (2). So in this case 1*1=1 and 2*2=4.  As 4 is greater than 2 we can’t consider 2 as denominator and quotient . So we will take 1 as denominator and quotient. And the product of denominator and quotient(1*1)=1 will be write below the first group of the number (2). And  subtract the product from (2) and write down the remainder in the second step.

Step-2

In the second step 1 is the current  numerator. Bring down the second group of number (25). The new numerator is 125. In order to get  first part of the denominator, add the last step quotient with last denominator. (1+1=2). Now put a new number in the right side of the denominator and multiply it with that new number. The result should be just less than or same as the numerator. In this case 25*5 =125. So  new denominator is 25 and quotient is 5. this product will subtract from numerator and the remainder will write in the third step. In our case the remainder is zero. So the current quotient is 15 and that is the square root of the number 225.

Notes

If we are finding the root of a big number we have to repeat the steps. as shown in the fig1-c. In each step  first part of denominator can be obtain by adding the last quotient and last denominator.

If we are finding out the square root of a number which not have a perfect root. Then the remainder will not become zero, after all the group of nuber bring down. in this case we can put decimal point in the quotient. And bring down group of zeros (2 zero) to the new steps. this can be repeat until how much decimal point we require. This one we can see in the fig2-a. 

 

Square root of a number with decimal point

Now we can see how to find out the square root of a number with decimal point. We can follow the above step itself. But we have to group the number after the decimal point from left to write.  And also we have to ad one zero if the rightmost digit if it is single. Please place the decimal point in the quotient, just before bring down the first decimal number group. also we can see the example in the fig2-b.

If any one need more help on this topic please contact me through the forum.

How to find out the surface area of sphere

Surface area of sphere = 4πr²

How to find out the surface area of sphere. Before starting to explain the equation, we will see how a sphere is forming. Please consider below figures.

The FIG-1a is a circle cut it from a sheet of paper. Please fix a string on the top side of the circular paper as shown in the fig-1b. And if we rotate the string we can see our circular paper as shown in the fig-1b. Now rotate the string much more speed. And we can see our circular paper as a sphere as shown in fig-1c.

Consider the center of the circle is C in the fig1-a. So the center of the circle C will become the center of sphere in fig-1c. Please remember, the circle is a closed figure whose every point lies at constant distance (called radius). From a fixed point (called center). So we can now say, A sphere is a three dimensional figure (solid figure) .Made up with all points in the space which lie in equal distance (called radius of the sphere). from a fixed point (called center of the sphere).

Surface area of the sphere

Now we can see how to find out the surface area of sphere. Consider a rubber ball (nothing but a sphere). And fixe a nail on it surface as shown in the fig-2a. Now take a long string and start to wind it on the ball from the nail without any space and complete the all surface of the ball with string. cut the remaining string.Now we can unwind the string and keep it in some place.

Assigning the Area of the sphere in to circle

Cut four  circular paper with the same radius of the ball. Now start to fill the four circles, one by one  with the string which is used to  wind on the ball. After filling each circle cut the string and use remaining string for the  next circle as shown in fig-2b. After filling four circle, we can see that the entire string is used to cover all the surface area of the four circle. Therefore we can say that the surface area of the sphere is equal to the surface area of four circle with the same radius of the sphere.

As we know the area of the circle is πr²

so the surface area of a sphere is 4πr²

Where r is the radius of the sphere.

OBSERVATION

Now all ready we studied how to find out the surface area of sphere.  As we see a sphere have only and only one surface. And that is a curved surface area. The surface area of the sphere can be calculated with equation 4πr² . If cut the sphere through it diameter we can get two supprate figure as shown in the fig-3.

This figure is nothing but a half sphere. It can also be called as hemisphere. But in this case we can see that the hemisphere have two surfaces. One of it surface is curved surface.and the other one is flat circle. The curved surface area of the hemisphere is the half  surface area of the sphere. (as it is half of the sphere). There for the curved surface area of a hemisphere is 2πr²

And the flat circle surface area of the hemisphere is πr²

So the total surface area of the hemisphere  = 2πr²+πr² = 3πr²

If need more clarification please contact through comments or through the forum.

Thanks and regards

How to find out the surface area of the right circular cone

Surface area of the cone = πr(l+r) 

Where r is the radius of the circular top of the cone BD as shown in the fig  FIG-1. And l is the lateral height of the cone AD as shown in FIG-1.

What is a cone

A cone is an object as shown in the FIG-1. A cone should have a point “A” as shown in the above fig. And this point is called the vertex of the cone. The cone should also have the height. From the fig. the height is “AB”. In the top there should be a circular shape as shown in the fig. The diameter of the circle is “CD”. The diameter and the height should be in perpendicular. Then only it can call as a right circular cone. the top shape also should be in perfect circle in order to call it “circular cone”. Now we can also observe from the fig., the curved surface of the cone is formed with a slant height. The lateral height in this case is AC and AD.

Curved surface area of a cone

As shown in the above fig. we can make the curved surface area of a cone. The curved surface area of the con in the fig-2a can be made of with circular paper in the fig-2b. Her the missing part of the fig-2a is because of the missing part if the fig-2b.

We can cut the circular paper in number of pieces shown in the fig-2b in such a way that we get a smaltrangle with base b1. If the number of the triangle is in hundreds, then the base b1,b2,b3,…… of the triangle can be considered straight line. And the height of the triangle is the lateral height of the cone. that is nothing but the radius OB  of the circle. And if we arrange all that small triangles we can get the curved surface area of the cone.

Now we know the area of each small triangle =   ½*b1*l

Area of all small triangle = ½*b1*l + ½*b1*l + ½*b1*l + ……….

½l(b1+b2+b3+…..)

Now if we add all the b1+b2+b3 ….. we will get the circumference of the circle that is nothing but 2πr.

so the curved surface area =½l*2πr 

=πrl

Now we have to consider the area of the top circle of the con . Which is πr².

total surface area of the cone = πrl+πr²  = πr (l+r)

so the total surface area of the right angle circular cone is  πr(l+r) 

Where r is the radius of the cone and l is the lateral height of the cone.

IMPORTANT notes

From the above fig-3, we can see the triangle ABC  in the cone. Here AB is the height of the cone, BD is the radius of the cone and AD is the slant height. So if we have only the radius and height of the cone we can find out the slant height. Because slant height is required to find out the surface are. By using pythagoras theorem we can find out the slant height of the cone by the below equation

AD² = AB²+BD²

AD = √(AB²+BD²)

Where AD  is the slant height of the cone, AB  is the height of the cone and BD is the radius of the cone.

please contact me in the comment or through the Forum

How can find the surface area of the cylinder

How can find surface area of cylinder

Surface area of the cylinder =2πr(H+r)

Dear friends,

In this post I will explain how we can find out the surface area of the cylinder.

Consider cylinder having height H and the diameter of the base circle D as shown in the figure FIG1-A. Now consider a rectangular paper having the height “H” as same as the height of the cylinder height “H”. Also assume that the length of the paper  “L” is just enough to cover the complete the curved surface area of the cylinder as shown in the FIG1-B.

So now we know the area of the paper is H*L.

We can also know that the length of the paper is equal to the circumference of the base circle of the cylinder.

So circumference of the circle is = 2πr

so area of the paper = H*2πr.

There for the

Total curved surface area of the cylinder =H*2πr.

Now we have to consider the area of the base and top circle, inorder to get the total surface area of the cylinder.

we know that the area of the circle = πr².

area of 2 circle =2πr².

So the total surface area of the cylinder = H*2πr +2πr²   = 2πr(H+r)

There for

Surface area of the cylinder =2πr(H+r)

Where H is the height of the cylinder and “r’ is the radius of the cylinder.

If some one need more explanation on this topic, please contact through the comment section. Or please ask your question on our forum.

Installing webmin in Ubuntu server

Dear friends,

In our last post we see that how we can access our home ubuntu server online. After installing the ubuntu server we have only the default webpage of the server is available to access from any browser. And we cannot access the command line of the server from other computers. If we need access the command line of the server from other computers, we have to use the putty software in the clint computer. And by using the username and password we can access the server. Otherwise  the command line of the server can directly access through the connected monitor of the server. But managing the server through only the command line is little difficult, util you are not expert in that. So in this post we will explain  how to Installing webmin in Ubuntu server. This will act as a interface between server and us. through this we can easily manage our server. And it can access from browser of any computers. So after the installation of webmin wecan remove  server monitor and other accessory.

Installing webmin in Ubuntu server

Please proceed the following steps to install webmin in you ubuntu server.

Copy and paste below cmand in your command line of the server and hit enter. This command is use to down load the latest version of the webmin package to our server.

wget http://www.webmin.com/download/deb/webmin-current.deb

Run the below command. It is used to update the apt-get index file for latest updates of the packages. And this step is very important.

sudo apt-get update

Run the below command to get required dependency for the webmin package.

sudo apt-get install perl libnet-ssleay-perl openssl libauthen-pam-perl libpam-runtime libio-pty-perl apt-sh

Now we can run the below command to install the webmin package.

sudo dpkg --install webmin-current.deb

After the successful installation ww can see the output saying that,

Webmin install complete. You can now login to https://ip-XXX-XX-XX-XXX:10000/
as root with your root password, or as any user who can use sudo.

Now at this point we can access the webmin  package of our server through any browser in our home network. For that we have to type https:our server ip:10000.

now we can see our webmin control panel as shown below.

webmin control panel

By this control panel we can do almost everything to our ubuntu server. If you need any help and support regarding this matter you are welcome to contact us through our forum.

thanks and regars

 

How can access our home router from outside home (port forwarding)

Introduction

Dear friends In my last post I wrote about how we can setup an ubuntu server. And also we see that how we can access our server in the browser of any devices connected in our home network. In this post we will study how we can access the server from outside our home and how we can do the port forwarding , that means anywhere from the world. for that purpose we have to study two things. One is remote management of router and other is port forwarding. That I will explain below one by one.

Remote management of the router

In this section we will study how we can enable the remote management of the router. Through this we can enable or disable the remote management of the router. After enabling this option we can access the router from outside our local network. To access the remote management enabled router from outside, we have to type the public IP address of our router in any browser of the device which is connected with the internet. Then  we can see the default gateway page and we can make any changes of the setting of the router from outside. In order to connect the router from inside our home we can directly type the default gateway IP  address in the browser of any computer connected in our home network.

Practical example

In this Example we will see how we can enable the remote management of the TPLINK router. Once you understand this setting in the TP LINK router we can do with other company model also.

Consider your home network is connected with a TP LINK router. Open your favorite browser like google chrome, and type ( 192.168.0.1), this is the default gateway ip address of your router. By default all the company is normally set this IP address as the default ip address of the device. If your computer and router is connected properly and you are not getting the default page of the router, then please check the default IP address of your particular router in its operating manual. The default username and password can also see in the manual. For TP LINK the default user name is admin and password is adminadmin.  For D-LINK the username is admin and password is blank.

TP link router port forwarding
TP link router port forwarding
CHANGE setting
  1. Once you connected properly you can see the page as same as above. Now we can go to the security option, remote management. In that page we can see two option,  first one is web management port. the default value of this is 80. We can keep this value as it is. But some internet service provider block this port or incoming traffic, That case you can change this port number (0_65000). If you change this port other than 80, then we have to type the public ip address like “publicipaddress:new_port_number”. In order to get your public IP address you can search “public ip” in google ad you can see that. Or you can see the ip address under status option of the router, in than under WAN we can see the IP address.
  2.  the second option is Remote Management IP Address:. The default value is 0.0.0.0. we have to change this value 255.255.255.255. So that we can access the router from any computer from anywhere. If you want only allow the access from a particular IP you can mention that IP here. Please also remember to change the default password of the router under the system  tools menu, password option.  
Port forwarding

In the above section we studied how we can access the router from anywhere. In this section we will see how we can access our or office ubuntu server from anywhere. For that we have to do port forwarding function. In order to do that please go to the forwarding menu of the router.

Here we can enter the service port number example 1213. Internal port  example 80. IP address, this is our ubuntu server IP address which we note it down in a paper during the installation. So now someone browse our “public ip address: 1213” the he/she will get our ubuntu server default page. So at this point anyone can get our server content like audio, video and images if we kept that in our server. for public use.

Making server IP as static

In the above section we see that how can we access home  server from outside. But please remember the router will forward the incoming traffic to the server always in the server IP we mentioned in the port forwarding section. But we have to know that the router may assign different Ip address to all devices connected to it through it DHCP  server at different time. This is called dynamic IP address allocation of the router. So we have to keep our ubuntu server have a constant IP.  For that we have to go to the DHPC menu of the router. From that menu  go to Address Reservation option and click on add new. Then enter the mac address  and the IP address of our ubuntu server, which we noted down in a paper during the installation of the server. the s=ave the setting and re boot the router. That it.  Now we can always access our home server from anywhere through our public ip address and the port number we assigned to it.

If you require any support regarding this mater always you can contact through our forum

Thank you for reading this article.

 

Uses Of Ubuntu Server In Our Home or Small Business

How to install ubuntu server in our home computer

Dear friends

In this post  I like to explain the uses of a Ubuntu server in our home. And also I will explain How to install ubuntu server. Before that I like to tell you what is a server . A server can made up with any computer by installing the server software. A server can provide lot of services to  all other  computers or mobile phone and other devices  in our home. as well as it can be access from our outside home. Ubuntu server is a open source free licensed server software from Linux. So everyone can use it freely without violating any privacy policy.

How to install ubuntu server

For a home application we can select any old or new computer to make a Ubuntu home server. The monitor, keyboard, mouse are only required for the installation only. For the normal working of the server only the cpu is enough. Installation is very simple and all the step by step instruction is given in this article , if you have any doubt for setting up the Ubuntu server at any point you can contact me through  our Technology forum. After installing the server as explained in the article we can make the use of the server as in the following ways. You can also see the released version of ubuntu server here

Uses of ubuntu server
  1. File storage: The ubuntu server can use as a files hub for everybody in the home. everyone can use to store their files and access it from any devices connected int the home. It can also be access from outside home. we can also upload our huge number valid document to it. And can be very easily accessed by search with in seconds. And this type of data can be save permanently and cannot be deleted, so it will be available in our whole life. As people are accessing the server through browsers for there file, they cannot make any damage to the server so it will run long time without any error.
  2. Samba file sharing: By this we can make common folders in the server. And that folder can access from any device in our home, as one of the folder in that device itself.
  3. Office management: By  use of this server we can manage our office functionality like accounts, Invoice, customer , supplier register and much more as required. Please see the demo software for our hardware shop.

These are some of the uses only we can also make use of lot of things like Email Server,webserver, database server, CCTV camera system, intercom IP phone network and much much more.

How to put the server in functionality
How to install ubuntu
How to install ubuntu

After completing the installation  as mentioned inthe how to install ubuntu server ,  wecan log in to the server by giving the username and password. Then the server will show the command prompt in the screen. Please remember to connect the internet or in the home network from the beginning of installation and keep it always. Now we can run the below command in the cond line.

ifconfig

It will show your network card along with it mac address and connected IP address, please note them in a paper. Remember that, the IP address may change time to time. This is because the server, other computers in the home network is connected to the home router. This router have a dynamic DHCP server and it will change the  IP address to the connected devices time to time. Please make the server IP address static in the router so the Ip address will be same always. Now you can access your new server from any connected device in your home network. For that we have to type the ip address of the server in the browser. So it will display the default webpage in the server.

Please we remember to ask any doubt at any time through the forum above mentioned.

Thank you

How can show PHP,HTML AND JAVASCRIPT code in wordpress post

PHP tutorial part-02

showing codes in wordpress

Dear friends,

In this post I like to write about how we can write PHP, HTML and JavaScript code in a wordpress post. If you write the code in plane text it will not appear in the  browser properly.  And if somebody copy and paste that code in their program it will not work properly so we had to write the codes in a different method in the wordpress post. in this post I will explain the two different method of how we can write the PHP, HTML and JavaScript code in the wordpress post.

Method-1

In the first method  click the post edit window, at the point where we want to insert the code. Then click to the text editor, that is available in the right side top corner of the editor window. There we can change the editor mode in Visual or text mode. In the text mode you can click on the code tab from the top menu, then it will open a code opening tag in the writing place. Then we can manually type <pre>tag. Now you can type or paste your actual code which one you want to show to your reader. Then close the </pre> tag first and click on the code tab on the top menu, then  code tab will be closed automatically. Now you can change  the editor mode in visual, and you can see your code in a coloured box. Now you can change the color of the text also. Her the code tag will say the browser to execute the thing inside the code tabe as a html code and the pre tag is the code to execute the things inside it as a plane text as it is without any modification or change to it. So the reader can copy and paste this code in there programs.

Method – 2

In this method you have to use one wordpress plugin. The name of the plugin is “tiny MCE” advanced. For checking weather this plugin is installed in your WordPress, you have to log in your wordpress account and in the left side navigation bar go to the plugin option.  If it is not installed click on the add new plugin tab at the top side. after installing the plug-in, activate and go to the setting of the “tiny MCE advaced”. We can add lot of the editing tools here.Please add the code tab. Now click on the save changes button and can go to the post option in which post we want to show the PHP or HTML codes. Inthe post editor window paste the code, which you want to show your reader,  and select them with mouse and click on the code tab in the toolbar. That is it, now your code will be shown to your reader as it is, and the reader can copy and paste the code  in their application. In this case that box background  will not be made ask like the first method. I hope this article is helpful for you. If you have any doubt please give me a comment surely I will give a reply.

Thank you for reading