Convert 2620.123 base 10 to base 5

Question

I understand the integer part, keep dividing by 5 and I get 40440, but for the fraction part I did a calculation, but it does not match with the answer given.

I do the fraction part like this:

$$0.123 \times 5=0.615$$ $$0.615 \times 5=3.075$$ $$0.075 \times 5=0.375$$ $$0.375 \times 5=1.875$$ $$0.875 \times 5=4.375$$

Shouldn't I continue to do it until the fraction part becomes 0? The final answer given by my tutor is 40440.0301. He didn't continue the calculation even though the fraction part has not reached 0 yet. Or is the calculation so long that he just skipped the rest?

Answer 1

You can easily verify that his answer is not exact:

$$0.0301_{\text{five}}=\frac3{25}+\frac1{625}=\frac{76}{625}\;,$$

while $$0.123=\frac{123}{1000}\;.$$

If the two were equal, we’d have $1000\cdot76=625\cdot123$, which is obviously false even without carrying out the full computation: the former ends in $0$, the latter in $5$.

Answer 2

You did it correctly. After the $.0301$ it repeats $141414...$ so the tutor may have known that and just stopped before that.

Answer 3

After reaching a certain point, say up to 5 digits of fraction, the teacher may treat the resulting base-n number as if it were much probable that the base n number is an infinite fraction : such goes the convention. However, when you find some part repeating, it is a iterating fraction for sure. Otherwise, it is likely to be a non-iterating non-terminating fraction - as goes the drill of saying.

You are not bound to follow what your teacher did. You are free to your will. Your teacher's act suffices the permit to cease your calculation at the exam hall after reaching four decimal or base-n points. Add three dots an we are done. Again I say, it does not necessarily mean that the fraction is non-iterating and non-terminating. You are just disposed of the toll of continuing.