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 rootof , . 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 the Wolfram Language 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. For example, John Wallis wrote, "These Imaginary Quantities (as they are commonly called) arising from the Supposed Root of a Negative Square (when they happen) are reputed to imply that the Case proposed is Impossible" (Wells 1986, p. 22).

Through the Euler formula, a complex number

(1)

may be written in "phasor" form

(2)

Here,  is known as the complex modulus (or sometimes the complex norm) and  is known as the complex argument or phase. The plot above shows what is known as an Argand diagram of the point , where the dashed circle represents the complex modulus  of  and the angle  represents itscomplex argument. Historically, the geometric representation of a complex number as simply a point in the plane was important because it made the whole idea of a complex number more acceptable. In particular, "imaginary" numbers became accepted partly through their visualization.

Unlike real numbers, complex numbers do not have a natural ordering, so there is no analog of complex-valued inequalities. This property is not so surprising however when they are viewed as being elements in the complex plane, since points in a plane also lack a natural ordering.

The absolute square of  is defined by , with  the complex conjugate, and the argument may be computed from

(3)

The real  and imaginary parts  are given by

			(4)
			(5)
			(6)
			(7)

de Moivre's identity relates powers of complex numbers for real  by

(8)

A power of complex number  to a positive integer exponent  can be written in closed form as

(9)

The first few are explicitly

			(10)
			(11)
			(12)
			(13)

(Abramowitz and Stegun 1972).

Complex addition

(14)

complex subtraction

(15)

complex multiplication

(16)

and complex division

(17)

can also be defined for complex numbers. Complex numbers may also be taken to complex powers. For example, complex exponentiation obeys

(18)

where  is the complex argument.

It is easy to hit but LOVE is more powerful :)

COMPUTER NETWORK AND ITS TYPES

  COMPUTER NETWORK AND ITS  TYPES

Contents

Network: 2

Benefits of Networking. 2

LOCAL AREA NETWORKING.. 3

Example of linear topology. 3

Example of star topology. 4

Example of tree or Expanded star topology. 5

METROPOLITAN AREA NETWORKING.. 5

WIDE AREA NETWORKING.. 6

PERSONAL AREA NETWORKING.. 7

CAMPUS AREA NETWORKING.. 7

INTERNET. 7

INTRANET. 7

EXTRANET. 8

Bibliography. 8

Network:

A network works like a bridge between the devices with the help of which it exchange the data. It may exchange data between computers and sometimes other devices as well. It connects computer to computer and computer to other devices like printer, scanner etc. also. In networking, we may use wires to connect different computers together or we can connect them through wireless source as well.

Depending upon the area, we categorize the network in different classes to study them in easy and classified manner.

Benefits of Networking:

There are so many advantages of networking but here I will mention few ones that are most important in today’s world:

FILE SHARING

From sharing files you can view, modify, and copy files stored on a different computer on the network just as easily as if they were stored on your computer.

RESOURCE SHARING

Resources such as printers, fax machines, Storage Devices (HDD, FDD and CD Drives), Webcam, Scanners, Modem and many more devices can be shared.

PROGRAM SHARING

Just as you can share files on a network, you can often also share program on a network. For example, if you have the right type of software license, you can have a shared copy of Microsoft Office, or some other program, and keep it on the network server, from where it is also run

Depending upon the region and area we have divided the types of network in the following categories:

LOCAL AREA NETWORKING:

A Local Area Network (LAN) is a network that is confined to a relatively small area. It is generally limited to a geographic area such as a writing lab, schools, buildings, offices etc.

In this type of networking, we have a one main Server computer that works as a head and have all the data in it. We prefers to have a best powerful system having all best devices for a server because it accesses the other computers connected with it.

Mainly it is connected through a wire that is known as optical fiber but we can also connect them with the help of wireless devices also. Different types of ways are there to physically connect devices and is known as physical topology. There are several types of physical topology by which we connect computers and devices know as linear bus, star and tree (expended star) topology.

Example of linear bus topology

 [1]

Example of star topology

 [2]

Example of tree or Expanded star topology

 [3]

So, it was all about the Local area networking, now we will further proceed to the new types of networking that is:

METROPOLITAN AREA NETWORKING:

A metropolitan area network (MAN) is a network that interconnects users with computer resources in a geographic area or region larger than that covered by even a large local area network (LAN) but smaller than the area covered by a wide area network (WAN).

Its area is nearly equal to the city and it can work in the entire city. For example If we have different campuses of our factory with in the specific city or a large area, we can connect them through the metropolitan area networking with the help of which we can easily share the information or data.

It mainly uses optical fiber and sometimes wireless source for sharing.

 [4]

Main advantage of MAN is to extent the LAN connectivity of buildings /offices that are within the same city or urban area.

WIDE AREA NETWORKING

Wide Area Networks are used to connect server machines and computers across continents are countries for constant information updates. Wide Area Networks, are used across the globe, many networks connect with one another across continents to create one giant Wide Area Network. Wide Area Networks use optic fibre as their communication medium. The largest example of a Wide Area Network is the internet itself, which connects all users to the information and data that is available on the internet.

PERSONAL AREA NETWORKING

It is the network in which computer and its parts are connected that belongs to the one specific person only that is why it is known as personal area network. Example is personal computer, fax machine, PDAs, scanner etc. it usually works with in the limit of 10 meters.

CAMPUS AREA NETWORKING

It is actually the interconnection of local area networking within the limited geographical region of 1 – 5 KM. For Example We have a local area networking in different departments of a university including library, gymnasium, presentation hall etc and we interconnect them through an optical fiber.

INTERNET:

The Internet is the largest example of an internetwork. It is a global system of interconnected governmental, academic, corporate, public, and private computer networks. In simple and short words, it is the network of networks.

INTRANET

An intranet is a private network that is contained within an enterprise or a management or a company.  The administrative entity limits use of the intranet to its authorized users. Most commonly, an intranet is the internal LAN of an organization. A large intranet typically has at least one web server to provide users with organizational information. It is in simple words, the internal network of your organization or secured version if internet.

EXTRANET

An extranet is a network that is also under the administrative control of a single organization, but supports a limited connection to a specific external network. For example, an organization may provide access to some aspects of its intranet to share data with its business partners or customers. These other entities are not necessarily trusted from a security standpoint. Network connection to an extranet is often, but not always, implemented via WAN technology.

So, here is the picture that actually differentiate between the LAN, MAN and WAN clearly.

 [5]

Bibliography

[1]	[Online]. Available: http://fcit.usf.edu/network/chap5/chap5.htm.
[2]	[Online]. Available: http://fcit.usf.edu/network/chap5/chap5.htm.
[3]	[Online]. Available: http://fcit.usf.edu/network/chap5/chap5.htm.
[4]	[Online]. Available: https://www.google.com.pk/search?q=USES+OF+metropolitan+area+network+(man)&biw=1366&bih=643&source=lnms&tbm=isch&sa=X&ved=0ahUKEwiy2ZLe9dDJAhUDXBoKHX68A_YQ_AUIBigB.
[5]	[Online]. Available: http://www.erg.abdn.ac.uk/users/gorry/course/intro-pages/man.html.