# 7.2 Imaginary concepts -- complex numbers

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This module introduces the concept of complex numbers in Algebra.

A “complex number” is the sum of two parts: a real number by itself, and a real number multiplied by $i$ . It can therefore be written as $a+\mathrm{bi}$ , where $a$ and $b$ are real numbers.

The first part, $a$ , is referred to as the real part . The second part, $\mathrm{bi}$ , is referred to as the imaginary part .

Examples of complex numbers $a+\mathrm{bi}$ ( $a$ is the “real part”; $\mathrm{bi}$ is the “imaginary part”)
$3+\mathrm{2i}$ $a=3$ , $b=2$
$\pi$ $a=\pi$ , $b=0$ (no imaginary part: a “pure real number”)
$-i$ $a=0$ , $b=-1$ (no real part: a “pure imaginary number”)

Some numbers are not obviously in the form $a+\mathrm{bi}$ . However, any number can be put in this form.

Example 1: Putting a fraction into $a+\mathrm{bi}$ form ( $i$ in the numerator)
$\frac{3-4i}{5}$ is a valid complex number. But it is not in the form $a+\mathrm{bi}$ , and we cannot immediately see what the real and imaginary parts are.
To see the parts, we rewrite it like this:
$\frac{3-4i}{5}$ $=$ $\frac{3}{5}$ $\frac{4}{5}i$
Why does that work? It’s just the ordinary rules of fractions, applied backward. (Try multiplying and then subtracting on the right to confirm this.) But now we have a form we can use:
$\frac{3-4i}{5}$ $a=$ $\frac{3}{5}$ , $b=–\frac{4}{5}$
So we see that fractions are very easy to break up, if the $i$ is in the numerator. An $i$ in the denominator is a bit trickier to deal with.
Example 2: Putting a fraction into $a+\mathrm{bi}$ form ( $i$ in the denominator)
$\frac{1}{i}$ $=$ $\frac{1\cdot i}{i\cdot i}$ Multiplying the top and bottom of a fraction by the same number never changes the value of the fraction: it just rewrites it in a different form.
$=$ $\frac{i}{-1}$ Because $i•i$ is ${i}^{2}$ , or –1.
$=\mathrm{-i}$ This is not a property of $i$ , but of –1. Similarly, $\frac{5}{-1}$ $=–5$ .
$\frac{1}{i}$ : $a=0$ , $b=-1$ since we rewrote it as $\mathrm{-i}$ , or $0-\mathrm{1i}$

Finally, what if the denominator is a more complicated complex number? The trick in this case is similar to the trick we used for rationalizing the denominator: we multiply by a quantity known as the complex conjugate of the denominator .

## Definition of complex conjugate

The complex conjugate of the number $a+\mathrm{bi}$ is $a-\mathrm{bi}$ . In words, you leave the real part alone, and change the sign of the imaginary part.

Here is how we can use the “complex conjugate” to simplify a fraction.

Example: Using the Complex Conjugate to put a fraction into $a+\mathrm{bi}$ form
$\frac{5}{3-4i}$ The fraction: a complex number not currently in the form $a+bi$
$=$ $\frac{5\left(3+4i\right)}{\left(3-4i\right)\left(3+4i\right)}$ Multiply the top and bottom by the complex conjugate of the denominator
$=$ $\frac{\text{15}+\text{20}i}{{3}^{2}-\left(4i{\right)}^{2}}$ Remember, $\left(x+a\right)\left(x–a\right)={x}^{2}{\mathrm{–a}}^{2}$
$=$ $\frac{\text{15}+\text{20}i}{9+\text{16}}$ ${\left(\mathrm{4i}\right)}^{2}={4}^{2}{i}^{2}=16\left(–1\right)=–16$ , which we are subtracting from 9
$=$ $\frac{\text{15}+\text{20}i}{\text{25}}$ Success! The top has $i$ , but the bottom doesn’t. This is easy to deal with.
$=$ $\frac{\text{15}}{\text{25}}$ $+$ $\frac{\text{20}i}{\text{25}}$ Break the fraction up, just as we did in a previous example.
$=$ $\frac{3}{5}$ $+$ $\frac{4}{5}i$ So we’re there! $a=$ $\frac{3}{5}$ and $b=\frac{4}{5}$

Any number of any kind can be written as $a+\mathrm{bi}$ . The above examples show how to rewrite fractions in this form. In the text, you go through a worksheet designed to rewrite $\sqrt{-1}$ as three different complex numbers. Once you understand this exercise, you can rewrite other radicals, such as $\sqrt{i}$ , in $a+\mathrm{bi}$ form.

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
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 .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
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 .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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