# 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.

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Adin Reply
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research.net
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sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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there is no specific books for beginners but there is book called principle of nanotechnology
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s.
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Tarell
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Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
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Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
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CYNTHIA
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s. Reply
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for screen printed electrodes ?
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of graphene you mean?
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Graphene has a hexagonal structure
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Source:  OpenStax, Math 1508 (lecture) readings in precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11354/1.1
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