# 3.8 Summary of key concepts

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The basic operations with real numbers are presented in this chapter. The concept of absolute value is discussed both geometrically and symbolically. The geometric presentation offers a visual understanding of the meaning of |x|. The symbolic presentation includes a literal explanation of how to use the definition. Negative exponents are developed, using reciprocals and the rules of exponents the student has already learned. Scientific notation is also included, using unique and real-life examples.This module contains a summary of the key concepts in the chapter "Basic Operations with Real Numbers".

## Positive and negative numbers ( [link] )

A number is denoted as positive if it is directly preceded by a " $+$ " sign or no sign at all. A number is denoted as negative if it is directly preceded by a " $-$ " sign.

Opposites are numbers that are the same distance from zero on the number line but have opposite signs.

## Double-negative property ( [link] )

$-\left(-a\right)=a$

## Absolute value (geometric) ( [link] )

The absolute value of a number $a$ , denoted $|a|$ , is the distance from $a$ to 0 on the number line.

## Absolute value (algebraic) ( [link] )

$|a|=\left\{\begin{array}{cc}a& \text{if}\text{\hspace{0.17em}}a\ge 0\\ -a& \text{if}\text{\hspace{0.17em}}a<0\end{array}$

like signs , add the absolute values of the numbers and associate the common sign with the sum.
unlike signs , subtract the smaller absolute value from the larger absolute value and associate the sign of the larger absolute value with the difference.

$0+\text{any}\text{\hspace{0.17em}}\text{number}=\text{that}\text{\hspace{0.17em}}\text{particular}\text{\hspace{0.17em}}\text{number}$ , that is, $0+a=a$ for any real number $a$ .

Since adding 0 to a real number leaves that number unchanged, 0 is called the additive identity.

## Definition of subtraction ( [link] )

$a-b=a+\left(-b\right)$

## Subtraction of signed numbers ( [link] )

To perform the subtraction $a-b$ , add the opposite of $b$ to $a$ , that is, change the sign of $b$ and add.

## Multiplication and division of signed numbers ( [link] )

$\begin{array}{lll}\left(+\right)\text{\hspace{0.17em}}\left(+\right)=+\hfill & \frac{\left(+\right)}{\left(+\right)}=+\hfill & \frac{\left(+\right)}{\left(-\right)}=-\hfill \\ \left(-\right)\text{\hspace{0.17em}}\left(-\right)=+\hfill & \hfill & \hfill \\ \left(+\right)\text{\hspace{0.17em}}\left(-\right)=-\hfill & \hfill & \hfill \\ \left(-\right)\text{\hspace{0.17em}}\left(+\right)=-\hfill & \frac{\left(-\right)}{\left(-\right)}=+\hfill & \frac{\left(-\right)}{\left(+\right)}=-\hfill \end{array}$

Two numbers are reciprocals of each other if their product is 1. The numbers 4 and $\frac{1}{4}$ are reciprocals since $\left(4\right)\text{\hspace{0.17em}}\left(\frac{1}{4}\right)=1$ .

## Negative exponents ( [link] )

If $n$ is any natural number and $x$ is any nonzero real number, then ${x}^{-n}=\frac{1}{{x}^{n}}$ .

## Writing a number in scientific notation ( [link] )

To write a number in scientific notation:

1. Move the decimal point so that there is one nonzero digit to its left.
2. Multiply the result by a power of 10 using an exponent whose absolute value is the number of places the decimal point was moved. Make the exponent positive if the decimal point was moved to the left and negative if the decimal point was moved to the right.

## Converting from scientific notation: Positive exponent ( [link] )

To convert a number written in scientific notation to a number in standard form when there is a positive exponent as the power of 10, move the decimal point to the right the number of places prescribed by the exponent on the 10.

## Negative exponent ( [link] )

To convert a number written in scientific notation to a number in standard form when there is a negative exponent as the power of 10, move the decimal point to the left the number of places prescribed by the exponent on the 10.

where we get a research paper on Nano chemistry....?
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
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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