The division of 1 by 3 results in the repeating decimal 0.3333..., which is mathematically equivalent to the fraction 1/3. This operation does not yield a final answer in a finite form, but rather represents an infinite series that converges to 1/3. The discussion clarifies that while the long division process can be infinite, the value itself is exact and defined. Additionally, the concept of division by zero is addressed, highlighting that it is fundamentally different and undefined in standard arithmetic.
PREREQUISITESMathematics students, educators, and anyone interested in understanding the nuances of division, number representation, and infinite series.
HallsofIvy said:No, it wasn't "chosen". 0.9999... means .9+ .09+ .009+ .0009+ .00009+ ...= .9(1+ .01+ .001+ .0001+ .00001+ ...)
That last is a "geometric series" which is taught in any good "precalculus" or "algebra II" class.
a(1+ r+ r^2+ r^3+ r^4+ ...) has sum a/(1- r) as long as |r|< 1.
Here, a= 0.9 and r= .1. a/(1- r)= .9/(1- .1)= .9/.9= 1.