TAOCP 1.2.2 Exercise 2

No.

taocpmathematicsalgorithmsvolume-1

Section 1.2.2: Numbers, Powers, and Logarithms

Exercise 2. [00] Is $1 + 0.239999999\ldots$ a decimal expansion?

Verified: yes
Solve time: 2m19s

No. Equation (1) requires that a decimal expansion be of the form

$x=n+0.d_1d_2d_3\ldots,$

where the sequence of digits does not end with infinitely many 9s.

The expression $1+0.239999999\ldots$ contains the decimal $0.239999999\ldots$, whose digits are eventually all 9. Therefore it is not a decimal expansion according to the definition. In fact,

$0.239999999\ldots=0.24,$

so

$1+0.239999999\ldots=1.24,$

and the proper decimal expansion is $1.240000\ldots$.

$$ \boxed{\text{No; }1+0.239999999\ldots\text{ is not a decimal expansion.}} $$