# Review of past work  (Page 8/8)

 Page 8 / 8

You can add and multiply rational numbers and still get a rational number at the end, which is very useful. If we have 4 integers $a,b,c$ and $d$ , then the rules for adding and multiplying rational numbers are

$\begin{array}{ccc}\hfill \frac{a}{b}+\frac{c}{d}& =& \frac{ad+bc}{bd}\hfill \\ \hfill \frac{a}{b}×\frac{c}{d}& =& \frac{ac}{bd}\hfill \end{array}$

## Notation tip

The statement "4 integers $a,b,c$ and $d$ " can be written formally as $\left\{a,b,c,d\right\}\in \mathbb{Z}$ because the $\in$ symbol means in and we say that $a,b,c$ and $d$ are in the set of integers.

Two rational numbers ( $\frac{a}{b}$ and $\frac{c}{d}$ ) represent the same number if $ad=bc$ . It is always best to simplify any rational number, so that the denominator is as small as possible. This can be achieved by dividing both the numerator and the denominator by the same integer. For example, the rational number $1000/10000$ can be divided by 1000 on the top and the bottom, which gives $1/10$ . $\frac{2}{3}$ of a pizza is the same as $\frac{8}{12}$ ( [link] ).

You can also add rational numbers together by finding the lowest common denominator and then adding the numerators. Finding a lowest common denominator means finding the lowest number that both denominators are a factor Some people say divisor instead of factor. of. A factor of a number is an integer which evenly divides that number without leaving a remainder. The following numbers all have a factor of 3

$3,6,9,12,15,18,21,24,...$

and the following all have factors of 4

$4,8,12,16,20,24,28,...$

The common denominators between 3 and 4 are all the numbers that appear in both of these lists, like 12 and 24. The lowest common denominator of 3 and 4 is the smallest number that has both 3 and 4 as factors, which is 12.

For example, if we wish to add $\frac{3}{4}+\frac{2}{3}$ , we first need to write both fractions so that their denominators are the same by finding the lowest common denominator, which we know is 12. We can do this by multiplying $\frac{3}{4}$ by $\frac{3}{3}$ and $\frac{2}{3}$ by $\frac{4}{4}$ . $\frac{3}{3}$ and $\frac{4}{4}$ are really just complicated ways of writing 1. Multiplying a number by 1 doesn't change the number.

$\begin{array}{ccc}\hfill \frac{3}{4}+\frac{2}{3}& =& \frac{3}{4}×\frac{3}{3}+\frac{2}{3}×\frac{4}{4}\hfill \\ \hfill & =& \frac{3×3}{4×3}+\frac{2×4}{3×4}\hfill \\ \hfill & =& \frac{9}{12}+\frac{8}{12}\hfill \\ \hfill & =& \frac{9+8}{12}\hfill \\ \hfill & =& \frac{17}{12}\hfill \end{array}$

Dividing by a rational number is the same as multiplying by its reciprocal, as long as neither the numerator nor the denominator is zero:

$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}.\frac{d}{c}=\frac{ad}{bc}$

A rational number may be a proper or improper fraction.

Proper fractions have a numerator that is smaller than the denominator. For example,

$\frac{-1}{2},\frac{3}{15},\frac{-5}{-20}$

are proper fractions.

Improper fractions have a numerator that is larger than the denominator. For example,

$\frac{-10}{2},\frac{15}{13},\frac{-53}{-20}$

are improper fractions. Improper fractions can always be written as the sum of an integer and a proper fraction.

## Converting rationals into decimal numbers

Converting rationals into decimal numbers is very easy.

If you use a calculator, you can simply divide the numerator by the denominator.

If you do not have a calculator, then you have to use long division.

Since long division was first taught in primary school, it will not be discussed here. If you have trouble with long division, then please ask your friends or your teacher to explain it to you.

## Irrational numbers

An irrational number is any real number that is not a rational number. When expressed as decimals, these numbers can never be fully written out as they have an infinite number of decimal places which never fall into a repeating pattern. For example, $\sqrt{2}=1,41421356...$ , $\pi =3,14159265...$ . $\pi$ is a Greek letter and is pronounced “pie”.

## Real numbers

1. Identify the number type (rational, irrational, real, integer) of each of the following numbers:
1. $\frac{c}{d}$ if $c$ is an integer and if $d$ is irrational.
2. $\frac{3}{2}$
3. -25
4. 1,525
5. $\sqrt{10}$
2. Is the following pair of numbers real and rational or real and irrational? Explain. $\sqrt{4}$ ; $\frac{1}{8}$

## Mathematical symbols

The following is a table of the meanings of some mathematical signs and symbols that you should have come across in earlier grades.

 Sign or Symbol Meaning $>$ greater than $<$ less than $\ge$ greater than or equal to $\le$ less than or equal to

So if we write $x>5$ , we say that $x$ is greater than 5 and if we write $x\ge y$ , we mean that $x$ can be greater than or equal to $y$ . Similarly, $<$ means is less than' and $\le$ means is less than or equal to'. Instead of saying that $x$ is between 6 and 10, we often write $6 . This directly means `six is less than $x$ which in turn is less than ten'.

## Mathematical symbols

1. Write the following in symbols:
1. $x$ is greater than 1
2. $y$ is less than or equal to $z$
3. $a$ is greater than or equal to 21
4. $p$ is greater than or equal to 21 and $p$ is less than or equal to 25

## Infinity

Infinity (symbol $\infty$ ) is usually thought of as something like “the largest possible number" or “the furthest possible distance". In mathematics, infinity is often treated as if it were a number, but it is clearly a very different type of “number" than integers or reals.

When talking about recurring decimals and irrational numbers, the term infinite was used to describe never-ending digits.

## End of chapter exercises

1. Calculate
1. $18-6×2$
2. $10+3\left(2+6\right)$
3. $50-10\left(4-2\right)+6$
4. $2×9-3\left(6-1\right)+1$
5. $8+24÷4×2$
6. $30-3×4+2$
7. $36÷4\left(5-2\right)+6$
8. $20-4×2+3$
9. $4+6\left(8+2\right)-3$
10. $100-10\left(2+3\right)+4$
2. If $p=q+4r$ , then $r=.....$
3. Solve $\frac{x-2}{3}=x-3$

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
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
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
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
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?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!