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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 ( $-$ ) 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" .

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:

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
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
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