# Review of past work  (Page 3/8)

 Page 3 / 8
$\begin{array}{c}\hfill a×b÷b=a\\ \hfill 5×4÷4=5\end{array}$

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 ( $-$ ).

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
Do somebody tell me a best nano engineering book for beginners?
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!