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 Infinite Cuts
Name: Johnnie
Status: student	
Age:  N/A
Location: N/A
Country: N/A
Date: N/A 


Question:
I have cut a cube into two, then again, again, and again. Finally, I'm down to a google-plex sub-atomic level. So, in theory, I should be able to divide something forever and always have a half remaining. Which brings me to my idea that there is no such thing as nothing! Can math prove me wrong?



Replies:
There is an entire branch of mathematics devoted to problems of the type you mention -- that branch is infinite series, and as the name suggests it has to do with infinite sums. Some sums converge to a finite number [convergent]. Others just keep getting bigger and bigger as you add more terms. The sum you used as an example: 1+1/2+1/4 + ... + 1/2n +... has no finite limit [divergent]. It just keeps getting bigger and bigger.

In contrast, it is possible to prove mathematically that the sum of the series converges: 1 + 1/4 + 1/16 + ... + 1/n^2 + ... = (pi^2)/6.

Unterman


Within the realm of mathematics, many things are possible. What you have found is that tere is no such thing as a smallest positive number. You have discovered that numbers are "continuous": between any two numbers there is another number. Nothing, the value zero, does exist. However, you can always get closer to it without actually reaching it. This is also the basis of limits: getting closer and closer to something without ever actually reaching it.

Kenneth Mellendorf



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