Are there infinitely many numbers in the interval [0,1]?

This is part of a series on common misconceptions.

True or False?

There are infinitely many numbers in the interval $[0, 1].$

Why some people say it's true: Every new combination of digits after "0." leads to a new number between 0 and 1. Since there are infinitely many possible combinations, there are infinitely many numbers in $[0, 1]$.

Why some people say it's false: Since the length of the interval is finite, there can't possibly be infinitely many numbers in $[0, 1]$.

The statement is $\color{#20A900}{\textbf{true}}$.

Proof: The fraction $\frac{ 1 } { n }$ will always be in the interval $[0, 1 ]$ for all positive integers $n$. Since there are infinitely many positive integers, there are infinitely many numbers in the given interval. $_\square$

Rebuttal: (Address any concerns that people have)
Reply: (Explain why the argument is not vald)

---

Rebuttal:
Reply:

(If relevant, add in an example problem that demonstates understanding of this misconception.)

See Also

• List of Common Misconceptions