Who Is Dedekind?

Number Theory Level 1

True or false :

$\quad$ $\large 0.{111111\ldots}_2 = 1$ .

True False

2 solutions

Mayank Chaturvedi
May 15, 2016

The number in base 2 is nothing but : $\frac { 1 }{ 2 } +\frac { 1 }{ { 2 }^{ 2 } } +\frac { 1 }{ { 2 }^{ 3 } } +\frac { 1 }{ { 2 }^{ 4 } } .................$ in base 10 .That is 1

Notice that if we concatenate the digit $n-1$ into a repeating decimal in base $n$ as so: $0.\overline{(n-1)(n-1)\ldots}_{n}$ , we get that its value is simply $1$ .

Sharky Kesa - 5 years ago

Nice observation!

Mayank Chaturvedi - 5 years ago

Lance Fernando
May 25, 2016

The repeating decimal 1 in base 2 is similar to repeating decimal 9 in base 10 - so it somewhat equalizes to 1 because of rounding.

There is no rounding involved.

Pi Han Goh - 5 years ago

I'm 4 years late, but you can consider it having to be a number x between the two numbers, for the statement to be false. as we cannot find the number x (because it doesn't exist), the statement is true.

Antonije Mirkovic - 10 months ago

Who Is Dedekind?

2 solutions

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