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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

a b + c d = a d + b c b d a b × c d = a c b d

Notation tip

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

Two rational numbers ( a b and c d ) represent the same number if a d = b c . 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 . 2 3 of a pizza is the same as 8 12 ( [link] ).

8 12 of the pizza is the same as 2 3 of the pizza.

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 3 4 + 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 3 4 by 3 3 and 2 3 by 4 4 . 3 3 and 4 4 are really just complicated ways of writing 1. Multiplying a number by 1 doesn't change the number.

3 4 + 2 3 = 3 4 × 3 3 + 2 3 × 4 4 = 3 × 3 4 × 3 + 2 × 4 3 × 4 = 9 12 + 8 12 = 9 + 8 12 = 17 12

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

a b ÷ c d = a b . d c = a d b c

A rational number may be a proper or improper fraction.

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

- 1 2 , 3 15 , - 5 - 20

are proper fractions.

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

- 10 2 , 15 13 , - 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, 2 = 1 , 41421356 ... , π = 3 , 14159265 ... . π 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. c d if c is an integer and if d is irrational.
    2. 3 2
    3. -25
    4. 1,525
    5. 10
  2. Is the following pair of numbers real and rational or real and irrational? Explain. 4 ; 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
greater than or equal to
less than or equal to

So if we write x > 5 , we say that x is greater than 5 and if we write x y , we mean that x can be greater than or equal to y . Similarly, < means `is less than' and means `is less than or equal to'. Instead of saying that x is between 6 and 10, we often write 6 < x < 10 . 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 (symbol ) 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 ( 2 + 6 )
    3. 50 - 10 ( 4 - 2 ) + 6
    4. 2 × 9 - 3 ( 6 - 1 ) + 1
    5. 8 + 24 ÷ 4 × 2
    6. 30 - 3 × 4 + 2
    7. 36 ÷ 4 ( 5 - 2 ) + 6
    8. 20 - 4 × 2 + 3
    9. 4 + 6 ( 8 + 2 ) - 3
    10. 100 - 10 ( 2 + 3 ) + 4
  2. If p = q + 4 r , then r = . . . . .
  3. Solve x - 2 3 = x - 3

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
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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