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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

so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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
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Cied
what is biological synthesis of nanoparticles
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what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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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.
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what is nano technology
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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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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