What is wrong with this proof?

Number Theory Level 1

Consider this "proof" that $0.99 = 1$ :

We know that $\frac{1}{3} = 0.33$ .
Multiplying both sides by 3, we have $\frac{3}{3} = 3(0.33) = 0.99$ .
Since $\frac{3}{3} =1,$ we have that $1 = 0.99.$

What is wrong with the proof?

The proof is correct.

3\left(\frac{1}{3}\right) \text{ is not equal to }1.

\frac{1}{3} \text{ is not really equal to } 0.33.

Decimals are not equal to fractions.

5 solutions

Robert De Guzman
Dec 10, 2014

The fraction is a rational number and $\frac{1}{3}$ is not really 0.33. But it is $\frac{1}{3} = 0.333\ldots \square$

$\frac{1}{3}$ is not an irrational number. Its is a rational number with a repeating decimal expansion.

Siddhartha Srivastava - 6 years, 6 months ago

Am I missing something, Robert? You asked,"What is wrong with the proof?".

What proof are you referring to? I see a statement only.

Guiseppi Butel - 6 years, 6 months ago

This is a very interesting comment, as I can see what he means. Just because you state something is a proof does not mean the reader will take it that way. As you wrote: Given 1/3 = 0.33. If it is Given then the reader may take that to mean you cannot contradict this fact, as you gave it willingly. On the other hand we see many proofs that begin with a statement many know is false and use it to prove a contradiction. The confusion is not in knowing what 1/3 is in a decimal representation it is in letting the reader know what you are doing is a proof. At least from my point of view.

Peter Michael - 5 years, 7 months ago

yes you missed that in above problem statement Robert proved 1=0.99

Rama Choudhary - 5 years, 6 months ago

This fraction is a rational number, but is not a finite decimal

Fernando Sánchez - 6 years, 6 months ago

Dear Robert......... 1/3 is not an irrational number...... when we convert it into fraction, it is a never ending (non-terminating) decimals.... but the digits are recurring.......... and recurring non-terminating decimals are rational numbers.... though this is not going to affect our question, but for your information only i wrote this...

Ambrish Rathore - 6 years, 6 months ago

I didn't think that there was anything wrong with the "proof" or "statement" or whatever, what was incorrect was the fact that 1/3 is not equal to 0.33. If it were though, and the proof states "Given that", so let's pretend, the proof makes sense. A logically sound proof can still give incorrect results if the assumptions are wrong

Andy Wright - 5 years, 8 months ago

Trevor Arashiro
Dec 11, 2014

Set $x=0.\overline{9}$

$10x=9.\overline{9}$

Subtract the first equation from the second

$10x-x=9.\overline{9}-0.\overline{9}$

$9x=9$

$x=1$

$\therefore 1=0.\overline{9}$

Alex Vistyazh
Dec 12, 2014

Easiest solution:

$0.33 = \frac{33}{100}$

$\frac{33}{100} \neq \frac{1}{3}$

Wonderful solution!

Avinash Kamath - 5 years, 6 months ago

Vishal S
Dec 11, 2014

1/3 is a rational number.A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers. 1/3=0.3333333333333333333333333333333333333333333333333333333 3333333333333333333333333333333333333333333..............................

The fact that $\frac{1}{3}$ is rational is not essential to the solution.

Jake Lai - 6 years, 6 months ago

Christopher Unrau
Dec 19, 2015

1/3 would be equal to the number 0.33333... Three times this is 0.9999... which is equal to one, so it was close, just didn't use repeating numbers.

What is wrong with this proof?

5 solutions

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