# Review of past work  (Page 2/8)

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## Letters and arithmetic

The simplest thing that can be done with numbers is adding, subtracting, multiplying or dividing them. When two numbers are added, subtracted, multiplied or divided, you are performing arithmetic Arithmetic is derived from the Greek word arithmos meaning number . . These four basic operations can be performed on any two real numbers.

Mathematics as a language uses special notation to write things down. So instead of:

$\mathrm{one}\mathrm{plus}\mathrm{one}\mathrm{is}\mathrm{equal}\mathrm{to}\mathrm{two}$

mathematicians write

$1+1=2$

In earlier grades, place holders were used to indicate missing numbers in an equation.

$\begin{array}{c}\hfill 1+\square =2\\ \hfill 4-\square =2\\ \hfill \square +3-2\square =2\end{array}$

However, place holders only work well for simple equations. For more advanced mathematical workings, letters are usually used to represent numbers.

$\begin{array}{c}\hfill 1+x=2\\ \hfill 4-y=2\\ \hfill z+3-2z=2\end{array}$

These letters are referred to as variables , since they can take on any value depending on what is required. For example, $x=1$ in [link] , but $x=26$ in $2+x=28$ .

A constant has a fixed value. The number 1 is a constant. The speed of light in a vacuum is also a constant which has been defined to be exactly 299 792 458 m $·$ s ${}^{-1}$ (read metres per second). The speed of light is a big number and it takes up space to always write down the entire number. Therefore, letters are also used to represent some constants. In the case of the speed of light, it is accepted that the letter $c$ represents the speed of light. Such constants represented by letters occur most often in physics and chemistry.

Additionally, letters can be used to describe a situation mathematically. For example, the following equation

$x+y=z$

can be used to describe the situation of finding how much change can be expected for buying an item. In this equation, $y$ represents the price of the item you are buying, $x$ represents the amount of change you should get back and $z$ is the amount of money given to the cashier. So, if the price is R10 and you gave the cashier R15, then write R15 instead of $z$ and R10 instead of $y$ and the change is then $x$ .

$x+10=15$

We will learn how to “solve” this equation towards the end of this chapter.

## Addition and subtraction

Addition ( $+$ ) and subtraction ( $-$ ) are the most basic operations between numbers but they are very closely related to each other. You can think of subtracting as being the opposite of adding since adding a number and then subtracting the same number will not change what you started with. For example, if we start with $a$ and add $b$ , then subtract $b$ , we will just get back to $a$ again:

$\begin{array}{c}\hfill a+b-b=a\\ \hfill 5+2-2=5\end{array}$

If we look at a number line, then addition means that we move to the right and subtraction means that we move to the left.

The order in which numbers are added does not matter, but the order in which numbers are subtracted does matter. This means that:

$\begin{array}{ccc}\hfill a+b& =& b+a\hfill \\ \hfill a-b& \ne & b-a\phantom{\rule{1.em}{0ex}}\mathrm{if}\mathrm{a}\ne \mathrm{b}\hfill \end{array}$

The sign $\ne$ means “is not equal to”. For example, $2+3=5$ and $3+2=5$ , but $5-3=2$ and $3-5=-2$ . $-2$ is a negative number, which is explained in detail in "Negative Numbers" .

## Commutativity for addition

The fact that $a+b=b+a$ , is known as the commutative property for addition.

## Multiplication and division

Just like addition and subtraction, multiplication ( $×$ , $·$ ) and division ( $÷$ , /) are opposites of each other. Multiplying by a number and then dividing by the same number gets us back to the start again:

#### 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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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