perplexus.info :: Just Math : Prove irrationality

Prove irrationality (Posted on 2011-08-07)

Prove that lg(3) is irrational.

Rem: lg(m)=t implies 10^t=m

See The Solution Submitted by Ady TZIDON

Rating: 5.0000 (3 votes)

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Puzzle Solution

Comment 4 of 4 |

Let lg(3) be rational. Then, we must have:

lg(3) = a/b

=> b* lg(3) = a

=> 3^b = 10 ^a

So, the lhs of the above equation is always odd, and the rhs of the above equation is always even. This leads to a contradiction.

Thereforem our earlier supposition that lg(3) was rational was erroneous. Therefore, lg(3) must be irrational.

Posted by K Sengupta on 2023-07-14 00:30:10

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