Download 11_Rationalizing and Fraction Bar_Nelms

Why Question # 11 (2-Part Question)
Part 1: When and why do we rationalize denominators?
Part 2: Why is the "bar" used to separate a numerator from a denominator in a fraction?
Part 1
What is it?
Definition of Rational Numbers: Rational numbers are numbers that can be expressed as the ratio of two
whole numbers. Numbers that end in a terminating decimal can be expressed as a ratio of two numbers. Also,
numbers that have a decimal part that begins to repeat, our considered to be rational. Numbers that fit into this
category are �25=5=5/1, �16=4=4/1, 2/3=.6666666667, 1/3=.3333333333, 3/4=.75, 1/2=.5
Definition of Irrational Numbers: Irrational numbers are numbers that cannot be expressed as a ratio of two
numbers. Numbers that fit into this category are pi, e, and the square root of two. The decimal place will never
stop or repeat, so to mark it with a ratio of two numbers would be impossible. (reductio ad abdurdum )
(incommensurable)
Definition of Rationalizing the Denominator: When the square or cube root that is applied to the denominator
of a fraction, is removed.
Ex.
When do we do this?
When the denominator of a fraction is expressed as a radical, that is not a perfect root. Perfect roots are
number that are the product of a whole number squared. The goal is to make a fraction that is equal to the first
one.
Why do we do this?
Theory: The reason that people are asked to rationalize the denominator of a fraction when it contains a square
root of a non-perfect root, is to create a standard form of notation that allows people to check answers correctly,
and simply by combining or canceling like terms.
Real Reason: To put the denominator into its simplest form. A fraction with a radical denominator is not
considered to be in simplest form. No matter how long and hard one searches, they will never find a ratio of two
whole numbers that represents the square root of a non perfect square number. You cannot find a common
number to simply a fraction containing the square root of a non-perfect root number as a denominator. How big
is 1/�2? About 1/1.41.......... How far will you go?
How do we do this?
There are two ways to rationalize the denominator.
1. Multiply the numerator and denominator by the root.
Ex.
2. Multiply the numerator and denominator by the conjugate. A conjugate is when you change the sign in the
middle of the two terms.
Ex.
Part 2
What is the bar called?
The bar or line can be called two different things:
1. When it is slanting forward it is called a solidus, virgule, or forward slash. Ex. 3/4
2. When it is horizontal it is called a vinculum or informally a "fraction bar". Ex.
3
4
When was it first used?
Fractions were used as early as (2800 BC) in the ancient Indus Valley civilization. (Not backed up with a strong
source.
Brahmagupta (c. 628) and Bhaskara (c. 1150) wrote fractions as we do today but without the bar.
The horizontal fraction bar was introduced by the Arabs and several sources attribute the horizontal fraction bar
to al-Hassar around 1200.
Fibonacci (c.1175-1250) was the first European mathematician to use the fraction bar as it is used today.
Why is the bar used to separate the numerator from the denominator?
To do just that, separate the numerator from the denominator. The bottom number represents a whole being cut up into an
equal number of pieces. The top number defines what part of the whole one has. If we didn't use the bar to separate the
two numbers, we might think the top number is an exponent, or that it has nothing to do with the bottom number. By
including the bar we are grouping the two numbers together to make them represent a fractional part of a whole.
Resources
1 <http://www.mathsisfun.com/algebra/rationalize-denominator.html> (Viewed on 6/21/10)
2 Livio, Mario. The Golden Ratio. (2003)
3 <http://www.physicsforums.com/showthread.php?t=130776> (Viewed on 6/21/10)
4 <http://mathforum.org/library/drmath/view/68610.html> (Viewed on 6/21/10)
5 Cajori, Florian. A History of Mathema0cal Nota0ons. 1928-­‐1929
6 <http://jeff560.tripod.com/fractions.html> (Viewed on 6/21/10)
7 <http://www.cut-the-knot.org/WhatIs/WhatIsFraction.shtml> (Viewed on 6/21/10)
8 <http://en.wikipedia.org/wiki/Vinculum_(symbol)> (Viewed on 6/21/10)
9 <http://en.wikipedia.org/wiki/fraction_(mathematics)> (Viewed on 6/21/10)