# Complex numbers

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An introduction to complex numbers.

While the fundamental signal used in electrical engineering is the sinusoid, it can be expressed mathematically in terms of an even more fundamental signal: the complex exponential . Representing sinusoids in terms of complex exponentials is not a mathematical oddity. Fluency with complex numbers and rational functions of complex variables is a critical skill all engineers master.Understanding information and power system designs and developing new systems all hinge on using complex numbers. In short, they are critical to modern electrical engineering, a realization made over a century ago.

## Definitions

The notion of the square root of $-1$ originated with the quadratic formula: the solution of certain quadratic equations mathematically exists only if the so-called imaginary quantity $\sqrt{-1}$ could be defined. Euler first used $i$ for the imaginary unit but that notation did not take hold untilroughly Ampère's time. Ampère used the symbol $i$ to denote current (intensité de current).It wasn't until the twentieth century that the importance of complex numbers to circuit theory became evident. By then, using $i$ for current was entrenched and electrical engineers chose $i$ for writing complex numbers.

An imaginary number has the form $ib=\sqrt{-b^{2}}$ . A complex number , $z$ , consists of the ordered pair ( $a$ , $b$ ), $a$ is the real component and $b$ is the imaginary component (the $i$ is suppressed because the imaginary component of the pair is always in the second position). The imaginary number $ib$ equals ( $0$ , $b$ ). Note that $a$ and $b$ are real-valued numbers.

[link] shows that we can locate a complex number in what we call the complex plane . Here, $a$ , the real part, is the $x$ -coordinate and $b$ , the imaginary part, is the $y$ -coordinate. A complex number is an ordered pair ( a , b ) that can be regarded as coordinates in the plane. Complex numbers can also be expressed in polar coordinates as r ∠ θ . From analytic geometry, we know that locations in the plane can be expressed as the sum of vectors, with the vectors corresponding to the $x$ and $y$ directions. Consequently, a complex number $z$ can be expressed as the (vector) sum $z=a+ib$ where $i$ indicates the $y$ -coordinate. This representation is known as the Cartesian form of $z$ . An imaginary number can't be numerically added to a real number;rather, this notation for a complex number represents vector addition, but it provides a convenient notation when we perform arithmetic manipulations.

Some obvious terminology. The real part of the complex number $z=a+ib$ , written as $\Re (z)$ , equals $a$ . We consider the real part as a function that works by selecting that componentof a complex number not multiplied by $i$ . The imaginary part of $z$ , $\Im (z)$ , equals $b$ : that part of a complex number that is multiplied by $i$ . Again, both the real and imaginary parts of a complex number are real-valued.

The complex conjugate of $z$ , written as $\overline{z}$ , has the same real part as $z$ but an imaginary part of the opposite sign.

$\begin{array}{c}z=\Re (z)+i\Im (z)\\ \overline{z}=\Re (z)-i\Im (z)\end{array}$

Using Cartesian notation, the following properties easily follow.

• If we add two complex numbers, the real part of the result equals the sum of the real parts and the imaginary part equals the sum of the imaginary parts. This property follows from the laws of vector addition. ${a}_{1}+i{b}_{1}+{a}_{2}+i{b}_{2}={a}_{1}+{a}_{2}+i({b}_{1}+{b}_{2})$ In this way, the real and imaginary parts remain separate.
• The product of $i$ and a real number is an imaginary number: $ia$ . The product of $i$ and an imaginary number is a real number: $iib=-b$ because $i^{2}=-1$ . Consequently, multiplying a complex number by $i$ rotates the number's position by $90$ degrees.

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There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
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Application of nanotechnology in medicine
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I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
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LITNING
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
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what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
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Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
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what is Nano technology ?
write examples of Nano molecule?
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The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
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What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
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