Department of Energy Argonne National Laboratory Office of Science NEWTON's Homepage NEWTON's Homepage
NEWTON, Ask A Scientist!
NEWTON Home Page NEWTON Teachers Visit Our Archives Ask A Question How To Ask A Question Question of the Week Our Expert Scientists Volunteer at NEWTON! Frequently Asked Questions Referencing NEWTON About NEWTON About Ask A Scientist Education At Argonne Irrational Multiples
Name:  Andrew M.
Status: student	
Age:  N/A
Location: N/A
Country: N/A
Date: N/A 


Question:
If 'N' is an irrational number. Is '2N' rational or irrational? why?



Replies:
Andrew,
If 'N' is irrational, then so is '2N'.

'N' is irrational. If '2N' were rational, then '2N' can be expressed as a ratio of integers: 2N=i/j. By dividing both sides by 2, N=i/(2j). Since 'j' is an integer, so is 2j. That would mean i/(2j)=N is a ratio of integers, a rational number. We already know N is not rational. As a result, our original assumption that 2N is rational cannot be true: 2N is irrational.

Dr. Mellendorf
Illinois Central College


Irrational. Here is why: N is irrational. Assume, 2*N is rational. This means: R= 2*N=p/q where 'p' and 'q' are integers. But then R/2 = p/2q is also rational. But R/2 = N would have to be rational, which contradicts the stipulation that N is irrational. QED by contradiction.

Vince Calder



Click here to return to the Mathematics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (help@newton.dep.anl.gov), or at Argonne's Educational Programs

NEWTON AND ASK A SCIENTIST
Educational Programs
Building 360
9700 S. Cass Ave.
Argonne, Illinois
60439-4845, USA
Update: June 2012
Weclome To Newton

Argonne National Laboratory