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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses proper fractions, improper fractions, and mixed numbers. By the end of the module students should be able to distinguish between proper fractions, improper fractions, and mixed numbers, convert an improper fraction to a mixed number and convert a mixed number to an improper fraction.

Section overview

  • Positive Proper Fractions
  • Positive Improper Fractions
  • Positive Mixed Numbers
  • Relating Positive Improper Fractions and Positive Mixed Numbers
  • Converting an Improper Fraction to a Mixed Number
  • Converting a Mixed Number to an Improper Fraction

Now that we know what positive fractions are, we consider three types of positive fractions: proper fractions, improper fractions, and mixed numbers.

Positive proper fractions

Positive proper fraction

Fractions in which the whole number in the numerator is strictly less than the whole number in the denominator are called positive proper fractions . On the number line, proper fractions are located in the interval from 0 to 1. Positive proper fractions are always less than one.

A number line. 0 is marked with a black dot, and 1 is marked with a hollow dot. The distance between the two is labeled, all proper fractions are located in this interval.

The closed circle at 0 indicates that 0 is included, while the open circle at 1 indicates that 1 is not included.

Some examples of positive proper fractions are

{} 1 2 size 12{ { {1} over {2} } } {} , 3 5 size 12{ { {3} over {5} } } {} , 20 27 size 12{ { {"20"} over {"27"} } } {} , and 106 255 size 12{ { {"106"} over {"255"} } } {}

Note that 1 < 2 size 12{"1 "<" 2"} {} , 3 < 5 size 12{"3 "<" 5"} {} , 20 < 27 size 12{"20 "<" 27"} {} , and 106 < 225 size 12{"106 "<" 225"} {} .

Positive improper fractions

Positive improper fractions

Fractions in which the whole number in the numerator is greater than or equal to the whole number in the denominator are called positive improper fractions . On the number line, improper fractions lie to the right of (and including) 1. Positive improper fractions are always greater than or equal to 1.

A number line. 0 is labeled, and 1 is marked with a hollow dot. An arrow is drawn to the right, labeled Positive improper fractions.

Some examples of positive improper fractions are

3 2 size 12{ { {3} over {2} } } {} , 8 5 size 12{ { {8} over {5} } } {} , 4 4 size 12{ { {4} over {4} } } {} , and 105 16 size 12{ { {"105"} over {"16"} } } {}

Note that 3 2 size 12{3>= 2} {} , 8 5 size 12{8>= 5} {} , 4 4 size 12{4>= 4} {} , and 105 16 size 12{"105">= "16"} {} .

Positive mixed numbers

Positive mixed numbers

A number of the form

nonzero whole number + proper fraction size 12{"nonzero whole number "+" proper fraction"} {}

is called a positive mixed number . For example, 2 3 5 size 12{2 { {3} over {5} } } {} is a mixed number. On the number line, mixed numbers are located in the interval to the right of (and includ­ing) 1. Mixed numbers are always greater than or equal to 1.

A number line. 0 is labeled, and 1 is marked with a hollow dot. An arrow is drawn to the right, labeled Positive mixed numbers.

Relating positive improper fractions and positive mixed numbers

A relationship between improper fractions and mixed numbers is suggested by two facts. The first is that improper fractions and mixed numbers are located in the same interval on the number line. The second fact, that mixed numbers are the sum of a natural number and a fraction, can be seen by making the following observa­tions.

Divide a whole quantity into 3 equal parts.

A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third.

Now, consider the following examples by observing the respective shaded areas.

A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. The two leftmost parts are shaded.

In the shaded region, there are 2 one thirds, or 2 3 size 12{ { {2} over {3} } } {} .

2 1 3 = 2 3 size 12{2 left ( { {1} over {3} } right )= { {2} over {3} } } {}

A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. All three parts are shaded.

There are 3 one thirds, or 3 3 size 12{ { {3} over {3} } } {} , or 1.

3 1 3 = 3 3 or 1 size 12{3 left ( { {1} over {3} } right )= { {3} over {3} } ````` ital "or"````1} {}

Thus,

3 3 = 1 size 12{ { {3} over {3} } =1} {}

Improper fraction = whole number.

A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. All three parts are shaded. A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. One part is shaded.

There are 4 one thirds, or 4 3 size 12{ { {4} over {3} } } {} , or 1 and 1 3 size 12{ { {1} over {3} } } {} .

4 1 3 = 4 3 size 12{4 left ( { {1} over {3} } right )= { {4} over {3} } `} {} or 1 and 1 3 size 12{1``` ital "and"``` { {1} over {3} } } {}

The terms 1 and 1 3 size 12{ { {1} over {3} } } {} can be represented as 1 + 1 3 size 12{1+ { {1} over {3} } } {} or 1 1 3 size 12{1 { {1} over {3} } } {}

Thus,

4 3 = 1 1 3 size 12{ { {4} over {3} } =1 { {2} over {3} } } {} .

Improper fraction = mixed number.

A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. All three parts are shaded. A rectangle divided into three equal parts with vertical bars. Each part contains the fraction, one-third. The two leftmost parts are shaded.

There are 5 one thirds, or 5 3 size 12{ { {5} over {3} } } {} , or 1 and 2 3 size 12{ { {2} over {3} } } {} .

5 1 3 = 5 3 or 1 and 2 3 size 12{5 left ( { {1} over {3} } right )= { {5} over {3} } `````````` ital "or"``````````1`` ital "and"`` { {2} over {3} } } {}

Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
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
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many many of nanotubes
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what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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