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For example, the rational number 5 6 can be written in decimal notation as 0 , 8 3 ˙ and similarly, the decimal number 0,25 can be written as a rational number as 1 4 .

Notation for repeating decimals

You can use a bar over the repeated numbers to indicate that the decimal is a repeating decimal.

Converting terminating decimals into rational numbers

A decimal number has an integer part and a fractional part. For example 10 , 589 has an integer part of 10 and a fractional part of 0 , 589 because 10 + 0 , 589 = 10 , 589 . The fractional part can be written as a rational number, i.e. with a numerator and a denominator that are integers.

Each digit after the decimal point is a fraction with a denominator in increasing powers of ten. For example:

  • 1 10 is 0 , 1
  • 1 100 is 0 , 01

This means that:

10 , 589 = 10 + 5 10 + 8 100 + 9 1000 = 10 589 1000 = 10589 1000

Fractions

  1. Write the following as fractions:
    1. 0 , 1
    2. 0 , 12
    3. 0 , 58
    4. 0 , 2589

Converting repeating decimals into rational numbers

When the decimal is a repeating decimal, a bit more work is needed to write the fractional part of the decimal number as a fraction. We will explain by means of an example.

If we wish to write 0 , 3 ˙ in the form a b (where a and b are integers) then we would proceed as follows

x = 0 , 33333 ... 10 x = 3 , 33333 ... multiply by 10 on both sides 9 x = 3 ( subtracting the second equation from the first equation ) x = 3 9 = 1 3

And another example would be to write 5 , 4 ˙ 3 ˙ 2 ˙ as a rational fraction.

x = 5 , 432432432 ... 1000 x = 5432 , 432432432 ... multiply by 1000 on both sides 999 x = 5427 ( subtracting the second equation from the first equation ) x = 5427 999 = 201 37

For the first example, the decimal was multiplied by 10 and for the second example, the decimal was multiplied by 1000. This is because for the first example there was only one digit (i.e. 3) recurring, while for the second example there were three digits (i.e. 432) recurring.

In general, if you have one digit recurring, then multiply by 10. If you have two digits recurring, then multiply by 100. If you have three digits recurring, then multiply by 1000. Can you spot the pattern yet?

The number of zeros is the same as the number of recurring digits.

Not all decimal numbers can be written as rational numbers. Why? Irrational decimal numbers like 2 = 1 , 4142135 . . . cannot be written with an integer numerator and denominator, because they do not have a pattern of recurring digits. However, when possible, you should try to use rational numbers or fractions instead of decimals.

Repeated decimal notation

  1. Write the following using the repeated decimal notation:
    1. 0 , 11111111 ...
    2. 0 , 1212121212 ...
    3. 0 , 123123123123 ...
    4. 0 , 11414541454145 ...
  2. Write the following in decimal form, using the repeated decimal notation:
    1. 2 3
    2. 1 3 11
    3. 4 5 6
    4. 2 1 9
  3. Write the following decimals in fractional form:
    1. 0 , 633 3 ˙
    2. 5 , 3131 31 ¯
    3. 0 , 99999 9 ˙

Summary

  • Real numbers can be either rational or irrational.
  • A rational number is any number which can be written as a b where a and b are integers and b 0
  • The following are rational numbers:
    1. Fractions with both denominator and numerator as integers.
    2. Integers.
    3. Decimal numbers that end.
    4. Decimal numbers that repeat.

End of chapter exercises

  1. If a is an integer, b is an integer and c is irrational, which of the following are rational numbers?
    1. 5 6
    2. a 3
    3. b 2
    4. 1 c
  2. Write each decimal as a simple fraction:
    1. 0 , 5
    2. 0 , 12
    3. 0 , 6
    4. 1 , 59
    5. 12 , 27 7 ˙
  3. Show that the decimal 3 , 21 1 ˙ 8 ˙ is a rational number.
  4. Express 0 , 7 8 ˙ as a fraction a b where a , b Z (show all working).

Questions & Answers

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
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
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.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
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 [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
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