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Introduction

As described in the chapter on review of past work, a number is a way of representing quantity. The numbers that will be used in high school are all real numbers, but there are many different ways of writing any single real number.

This chapter describes rational numbers .

Khan academy video on integers and rational numbers

The big picture of numbers

The term whole number does not have a consistent definition. Various authors use it in many different ways. We use the following definitions:

  • natural numbers are (1, 2, 3, ...)
  • whole numbers are (0, 1, 2, 3, ...)
  • integers are (... -3, -2, -1, 0, 1, 2, 3, ....)

Definition

The following numbers are all rational numbers.

10 1 , 21 7 , - 1 - 3 , 10 20 , - 3 6

You can see that all denominators and all numerators are integers.

Rational Number

A rational number is any number which can be written as:

a b

where a and b are integers and b 0 .

Only fractions which have a numerator and a denominator (that is not 0) that are integers are rational numbers.

This means that all integers are rational numbers, because they can be written with a denominator of 1.

Therefore

2 7 , π 20

are not examples of rational numbers, because in each case, either the numerator or the denominator is not an integer.

A number may not be written as an integer divided by another integer, but may still be a rational number. This is because the results may be expressed as an integer divided by an integer. The rule is, if a number can be writtenas a fraction of integers, it is rational even if it can also be written in another way as well. Here are two examples that might not look like rational numbersat first glance but are because there are equivalent forms that are expressed as an integer divided by another integer:

- 1 , 33 - 3 = 133 300 , - 3 6 , 39 = - 300 639 = - 100 213

Rational numbers

  1. If a is an integer, b is an integer and c is irrational, which of the following are rational numbers?
    (i) 5 6 (ii) a 3 (iii) b 2 (iv) 1 c
  2. If a 1 is a rational number, which of the following are valid values for a ?
    (i) 1 (ii) - 10 (iii) 2 (iv) 2 , 1

Forms of rational numbers

All integers and fractions with integer numerators and denominators are rational numbers. There are two more forms of rational numbers.

Investigation : decimal numbers

You can write the rational number 1 2 as the decimal number 0,5. Write the following numbers as decimals:

  1. 1 4
  2. 1 10
  3. 2 5
  4. 1 100
  5. 2 3

Do the numbers after the decimal comma end or do they continue? If they continue, is there a repeating pattern to the numbers?

You can write a rational number as a decimal number. Two types of decimal numbers can be written as rational numbers:

  1. decimal numbers that end or terminate , for example the fraction 4 10 can be written as 0,4.
  2. decimal numbers that have a repeating pattern of numbers, for example the fraction 1 3 can be written as 0 , 3 ˙ . The dot represents recurring 3 's i.e., 0 , 333 ... = 0 , 3 ˙ .

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:
    (a) 0 , 1 (b) 0 , 12 (c) 0 , 58 (d) 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

  1. Real numbers can be either rational or irrational.
  2. A rational number is any number which can be written as a b where a and b are integers and b 0
  3. 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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris 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
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
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
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?
bamidele Reply
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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