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Sums of Irrational Numbers
Name: Jehanzeb
Status: student
Age: N/A
Location: N/A
Country: N/A
Date: 9/25/2005
Question:
Is the sum of two irrational numbers equal to a rational
number?
Replies:
The sum of two rational numbers, p/q + n/m = (pm +qn)/qm. If either p/q or
n/m = X or Y, respectively, and either X or Y cannot be expressed as a
ratio, then the sum cannot be expressed as a ratio. However, an irrational
number can be expressed as an INFINITE sum of rational numbers. This was
first noted by Newton (1676), but Euler gave the first proof (1774). So
the sum of two IRRATIONAL numbers can be expressed as an INFINITE SUM of
rational numbers.
Vince Calder
Jehanzeb,
Two irrational numbers very RARELY add up to a rational number.
Any integer, finite decimal, or repeating decimal is a rational number. A
rational number can be represented as a fraction. An irrational number
cannot. It IS true that two RATIONAL numbers add up to a rational number:
two fractions always add up to a fraction. Here are two examples:
(1/2)+(1/3)=(5/6), 1+3=4. The latter applies because you can represent it
as (1/1)+(3/1)=(4/1).
However, sqrt(2) is not rational because there is no fraction, no ratio of
integers, that will equal sqrt(2). It calculates to be a decimal that never
repeats and never ends. The same can be said for sqrt(3). Also, there is
no way to write sqrt(2)+sqrt(3) as a fraction. In fact, the representation
is already in its simplest form. To get two irrational numbers to add up to
a rational number, you need to add irrational numbers such as [1+sqrt(2)]
and [1-sqrt(2)]. In this case, the irrational portions just happen to
cancel out, leaving:
[1+sqrt(2)]+[1-sqrt(2)]=2. 2 is a rational number (i.e. 2/1).
Dr. Ken Mellendorf
Physics Instructor
Illinois Central College
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Update: June 2012
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