# 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. On the number line, numbers increase towards the right and decrease towards the left. Positive numbers appear to the right of zero and negativenumbers appear to the left 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 ( $-$ ).

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
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Source:  OpenStax, Siyavula textbooks: grade 10 maths [ncs]. OpenStax CNX. Aug 05, 2011 Download for free at http://cnx.org/content/col11239/1.2
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