> (1 + .999...)/2 = .5 + .499... = .999.. Why is .999.../2 = .499... ?

peterhalburt33 · on April 27, 2022

You can compute it digit by digit:

.999…/2 = .45 + .045 + .0045 + …

jonsen · on April 27, 2022

Yes, but your argument goes circular with this. And someone could argue that

  .999.../2 = .499...  <=>  .999... = 2*.499... = .499...+.499...

which "surely must" have 8 as the rightmost digit. Moving it further away from 1.

peterhalburt33 · on April 27, 2022

Agreed this isn’t airtight, but if you accept the premise that real numbers are specified by their sequence decimal digits: .499…+.499… has 9 as its first digit after the decimal, 9 as its second and so on - the only other number that has this decimal expansion is .999… infinitely repeating, so these must be the same numbers.