We love 1729

Algebra Level 2

$\lfloor 1729.999\ldots \rfloor = \, ?$

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

1729 1730

2 solutions

Sravanth C.
Feb 7, 2016

Let us assume $1729.999...=x$ .

So, $17299.999...=10x$ .

Subtracting the second equation from the first, we get: $10x-x=17299.999...-1729.999...=15570\\9x=15570\implies \boxed{x=1730}$

Hence, $\left\lfloor1730\right\rfloor=1730$

Zeros are missing in last second and third line in 1557..
Anyways Can't it be done this way: 1729.99...=1729+0.99.....=1729+1=1730 (since 0.99..=1).
Hence $\lfloor1730\rfloor$ =1730

Rishabh Jain - 5 years, 4 months ago

Both are same

Aman Rckstar - 5 years, 4 months ago

Thanks! I've edited it. $\ddot\smile$

Sravanth C. - 5 years, 4 months ago

Why is that so that today no one has dedicated any of his post to Dmitri Mendeleev’s, the greatest contributor in periodic table (Anyways it's his 182nd birthday :-} )

Rishabh Jain - 5 years, 4 months ago

@Rishabh Jain – Yeah, maybe I'll post one!

Sravanth C. - 5 years, 4 months ago

the answer should be 1729 because its a floor function the floor function rounds off the value to lowest possible integer value

vidhit chandra - 5 years, 4 months ago

You've got the right answer ! It's written on the curse !

Phil TAMA - 2 years, 8 months ago

Eli Ross Staff
Feb 8, 2016

This is a very simple counterexample to the common misconception that $\lim_{n\rightarrow\infty} f(a_n) = f\left(\lim_{n\rightarrow\infty} a_n\right).$

@Aareyan Manzoor : Do you see why? Would you like to add a common misconception wiki about this?

$\lim_{n \rightarrow \infty} f(a_{n}) = f \left (\lim_{n \rightarrow \infty} a_{n} \right)$ is true when f is a continuous function and the floor function is not a continuous function

Guillermo Templado - 5 years, 4 months ago

We love 1729

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