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Rational Numbers



Rational Number: A number that can be expressed as a fraction in which the denominator is not 0.

A rational number can be any whole number, fraction, mixed number or decimal.

It can be expressed as a fraction or decimal that has two integers and the denominator does not equal zero.
Terminating or repeating decimals must be found when the fraction is divided.
There are two numbers in a fraction; a numerator and a denominator. The number on the top is the numerator and the number on the bottom is the denominator.

Terminating and Repeating Decimals.

+ Terminating - a number in which the decimals end (or terminate) (For example : 5/2 = 1.5)
+ Repeating - a decimal with endlessly repeating digits. (for example: 88/33 = 2.6666666666666)

Rational Numbers are important! They are used in the real world EVERYDAY!


Even though we are not thinking about if the number is rational or not, we still use them in our everyday lives. At school or in the kitchen. We even see them on T.V!

EXAMPLES:
1)Baking: Ingredients in recipes are often listed as fractions to show the measurements. For example, a 1/2 cup of flour going into a batch of cookie dough. 1/2 is a rational number.

2)Commercials: Many commercials use rational numbers as statistics to get you to buy their products. For example, 4/5 dentists approve this toothpaste, or 9/10 women like this lipstick best.

3)Medical Field: Medical journals use
statistics to inform people about the risks of certain things. Such as 1/5 deaths in America are related to smoking or 1/4 Americans are overweight.


4) Math Class (of course): We use Rational numbers in math class absolutely EVERYDAY.


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Some Rational History
There was once an ancient Greek mathematician named Pythagoras. He believed that all numbers were rational (could be written as a fraction). However, a very clever student of his, Hippasus, proved him wrong. He said that you could not represent the square root of 2 as a fraction and therefore, not all numbers were rational, but rather irrational.
Pythagoras had a hard time accepting this idea of irrational numbers. He stuck with his theory that all numbers had perfect values. In the end he could not prove Hippasus' theory of irrational numbers wrong, so he killed him.


Rational Vs. Irrational


An Irrational Number is a number that cannot be written as a simple fraction (the decimal goes on forever).
For example, π (Pi) is an Irrational Number. The value of π (Pi) is "
3.1415926535897932384626433832795 (and more...)" Rational numbers can be written as a simple fraction (hence the term rational).
You cannot write down a
simple fraction to equal Pi. It is impossible. Therefore, Pi is an Irrational Number. Here is a chart to show you the difference:



Rational Numbers
Irrational Numbers
8
9.33333333333333333->
4
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6
12.66666666666->



Fractions!

Multiplying Fractions:
1) Multiply the numerators
2) Multiply the denominators
3) *Reduce to lowest terms*

Examples:
(2/5)(6/10)
^the numerators are '2' and '6'
the denominators are '5' and '10'
multiply them together
=12/50
=6/25
^the fraction 12/50 can be reduced into lower terms which is 6/25

(6/5)(3/4)
^the numerators are '6' and '3'
the denominators are '5' and '4'
multiply them together
=18/20
=9/10
^the fraction 18/20 can be reduced into lower terms which is 9/10




Dividing Fractions:
1) Flip the second fraction to it's reciprocal. (Remember:
Divvy means Flippy)
2) Multiply the numerators
3) Multiply the denominators
4) *Reduce to lowest terms*
Examples:
(4/5)(2/3)
= (4/5)(3/2)
^when dividing fractions, you must flip the second fraction
than multiply the denominators and numerators together
=12/10
=6/5 (improper)

^the fraction 12/10 can be reduced into lower terms which is 6/5
this type of fraction is called an improper fraction when the numerator is larger than the denominator
you can change them into a proper fraction or keep it improper.
=1 1/5 (proper)

(2/3)(1/4)
=(2/3)(4/1)
^flip the second fraction
multiply the denominators and numerators together
=8/3 (improper)
=2 2/3 (proper)




Adding Fractions:
1) Find a common denominator
2) Add the numerators (*NOT THE DENOMINATORS)
3) Reduce to lowest terms
Examples:
3/4 + 2/3
= 9/12 + 8/12 (whatever you do to the denominator, you must do to the numerator)
= 17/12 (improper)
= 1 5/12 (proper)

- 4/7 + 1/3 (adding with integers is the same, you do not change anything)
= -12/21 + 7/21
= - 5/21 ( -5/12 is lowest terms, they cannot both be divided evenly anymore)






Subtracting Fractions:
1) Find a common denominator
2) Subtract the numerators (*NOT THE DENOMINATORS)
3) Reduced to lowest terms
Examples:
5/6 - 1/2
= 10/12 - 6/12 (whatever you do to the denominator, you must do to the numerator)
= 4/12
= 1/3 (lowest terms)

- 2/3 - (- 3/4)
= - 8/12 - ( - 9/12)
= - 8/12 + 9/12 (
*NOTE: when there are there is a subtraction sign and a negative, change the signs to positive)
= 1/12



Adding AND Subtracting Mixed Numbers

*Convert both or the one mixed number to an improper fraction for BOTH adding and subtracting
Examples:
2 1/4 + 1/2
= 9/4 + 1/2
= 9/4 + 2/4
= 11/4 (improper)
= 2 3/4 (proper)

4 1/3 - 2 1/2
= 13/3 - 5/2
= 26/6 - 15/6
= 11/6 (improper)
= 1 5/6 (proper)



By; Rhys, Pat, Lilly and Brooke N.



CITATIONS:

PICTURES-
http://www.karlscalculus.org/rationals.gif
http://en.wikipedia.org/wiki/File:Fracciones.gif

INFORMATION-
Rational Number. 1 Dec. 2008 <http://en.wikipedia.org/wiki/Rational_number>.
Math Page. 2001. 6 Dec. 2008 <http://www.themathpage.com/aPrecalc/rational-irrational-numbers.htm>.
Pierce, Rod. Maths Fun. Ed. Rod Pierce. 2007. 10 Dec. 2008 <http://www.mathsisfun.com/rational-numbers.html>.
Spector, L. 2007. Multiplying Fractions Dividing Fractions. 16 Dec. 2008 <http://www.themathpage.com/Arith/multiply-fractions-divide-fractions.htm>.
Math Power 9 Ontario Edition. (1999). Toronto: McGraw-Hill Ryerson.