Does point 9 repeating equal 1?

Answer 1

Many say YES, but only using a parlor trick called "limits"... limits arbitrarily stop a repeating number (using theorems (just theory)) so it no longer repeats infinitely.

Take 0.333... (repeating)... by definition, it's an infinite repetition of the number 9 behind the decimal point, and will never be a finite number without the use of "limits" to stop the repetition (rounding the number to the nearest and stopping).

Think of these numbers as distances, and it becomes easier to understand... 1/3, as an expression distance, is a finite point (by definition) on a line representing a distance between two points, however, 0.333 repeating cannot be a point on a line because the 3 repeats to infinity... adding another "3" to the repeating number never gets you 1/3rd of the way to the next number... remember that adding another "3" to the infinite series is only adding "0.3" and is NOT the same as adding "0.333 repeating". So adding 0.3 infinitely will never get you to a "finite" point... kinda the "point" to the concept of infinity.

"Infinity" melts mathematicians brains, so they had to invent theories incorporating "limits" to keep them from burning out their little brain cells.

I have no problem admitting that for all "practical" purposes, 0.999... might as well be equal to 1, however, the reality is that on the scale of "the infinite", .999 repeating, by definition, is infinitely far from ever being 1 :o)