this post was submitted on 27 Jun 2024
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If you can't do it without fractions or a ... then it can't be done.
1/3=0.333...
2/3=0.666...
3/3=0.999...=1
Fractions and base 10 are two different systems. You're only approximating what 1/3 is when you write out 0.3333...
The ... is because you can't actually make it correct in base 10.
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?