Basics

Algebra Level 3

Statement 1 : $0.99999\ldots = 1$ .

Statement 2 : $\lfloor 0.99999\ldots\rfloor = 1$ .

Which of the statements above is(are) correct?

Notation : $\lfloor \cdot \rfloor$ denotes the floor function .

Both are correct Both are wrong Statement 1 is correct; statement 2 is wrong Statement 1 is wrong; statement 2 is correct

1 solution

Hung Woei Neoh
Apr 21, 2016

$0.99999\ldots = 0.9 + 0.09 + 0.009 + 0.0009 + 0.00009 + \ldots$

This number is the sum of a geometric progression to infinity.

$a=0.9, r=\dfrac{0.09}{0.9} = 0.1, n=\infty$

Therefore, the sum is given as:

$S_{\infty} = \dfrac{a}{1-r} = \dfrac{0.9}{1-0.1} = \dfrac{0.9}{0.9} = 1$

From here, we know that $0.99999\ldots = 1$ , and statement 1 is correct.

For statement 2, $\lfloor 0.99999\ldots \rfloor = \lfloor 1 \rfloor = 1$ , therefore it is also correct.

The final answer: $\boxed{\text{Both are correct.}}$

isnt 0.99999999999999......... infinitesimally small than 1

then is statemant 2 correct??

mmmmm mmmmm - 4 years, 4 months ago