this post was submitted on 27 Jun 2024
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Sure, let's do it in base 3. 3 in base 3 is 10, and 3^(-1) is 10^(-1), so:
1/3 in base 10 = 1/10 in base 3
0.3... in base 10 = 0.1 in base 3
Multiply by 3 on both sides:
3 × 0.3... in base 10 = 10 × 0.1 in base 3
0.9... in base 10 = 1 in base 3.
But 1 in base 3 is also 1 in base 10, so:
0.9... in base 10 = 1 in base 10
You're having to use ... to make your conversion again. If you need to to an irrational number to make your equation correct, it isn't really correct.
Do you know what an irrational number is?