# Review of past work  (Page 3/8)

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$\begin{array}{c}\hfill a×b÷b=a\\ \hfill 5×4÷4=5\end{array}$

Sometimes you will see a multiplication of letters as a dot or without any symbol. Don't worry, its exactly the same thing. Mathematicians areefficient and like to write things in the shortest, neatest way possible.

$\begin{array}{ccc}\hfill abc& =& a×b×c\hfill \\ \hfill a·b·c& =& a×b×c\hfill \end{array}$

It is usually neater to write known numbers to the left, and letters to the right. So although $4x$ and $x4$ are the same thing, it looks better to write $4x$ . In this case, the “4” is a constant that is referred to as the coefficient of $x$ .

## Commutativity for multiplication

The fact that $ab=ba$ is known as the commutative property of multiplication. Therefore, both addition and multiplication are described as commutative operations.

## Brackets

Brackets Sometimes people say “parentheses” instead of “brackets”. in mathematics are used to show the order in which you must do things. This is important as you can get different answers depending on the order in which youdo things. For example:

$\left(5×5\right)+20=45$

whereas

$5×\left(5+20\right)=125$

If there are no brackets, you should always do multiplications and divisions first and then additions and subtractions Multiplying and dividing can be performed in any order as it doesn't matter. Likewise itdoesn't matter which order you do addition and subtraction. Just as long as you do any $×÷$ before any $+-$ . . You can always put your own brackets into equations using this rule to make things easier for yourself, for example:

$\begin{array}{ccc}\hfill a×b+c÷d& =& \left(a×b\right)+\left(c÷d\right)\hfill \\ \hfill 5×5+20÷4& =& \left(5×5\right)+\left(20÷4\right)\hfill \end{array}$

If you see a multiplication outside a bracket like this

$\begin{array}{c}\hfill a\left(b+c\right)\\ \hfill 3\left(4-3\right)\end{array}$

then it means you have to multiply each part inside the bracket by the number outside

$\begin{array}{ccc}\hfill a\left(b+c\right)& =& ab+ac\hfill \\ \hfill 3\left(4-3\right)& =& 3×4-3×3=12-9=3\hfill \end{array}$

unless you can simplify everything inside the bracket into a single term. In fact, in the above example, it would have been smarter to have done this

$3\left(4-3\right)=3×\left(1\right)=3$

It can happen with letters too

$3\left(4a-3a\right)=3×\left(a\right)=3a$

## Distributivity

The fact that $a\left(b+c\right)=ab+ac$ is known as the distributive property.

If there are two brackets multiplied by each other, then you can do it one stepat a time:

$\begin{array}{ccc}\hfill \left(a+b\right)\left(c+d\right)& =& a\left(c+d\right)+b\left(c+d\right)\hfill \\ \hfill & =& ac+ad+bc+bd\hfill \\ \hfill \left(a+3\right)\left(4+d\right)& =& a\left(4+d\right)+3\left(4+d\right)\hfill \\ \hfill & =& 4a+ad+12+3d\hfill \end{array}$

## What is a negative number?

Negative numbers can be very confusing to begin with, but there is nothing to be afraid of. The numbers that are used most often are greater than zero. Thesenumbers are known as positive numbers .

A negative number is a number that is less than zero. So, if we were to take a positive number $a$ and subtract it from zero, the answer would be the negative of $a$ .

$0-a=-a$

On a number line, a negative number appears to the left of zero and a positive number appears to the right of zero.

## Working with negative numbers

When you are adding a negative number, it is the same as subtracting that number if it were positive. Likewise, if you subtract a negative number, it is the sameas adding the number if it were positive. Numbers are either positive or negative and we call this their s ign. A positive number has a positive sign ( $+$ ) and a negative number has a negative sign ( $-$ ).

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
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
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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