Why use different systems?

To explain this issue, I shall bring up the example of the fraction ⅓ . One third is not possible to be expressed in the decimal system - 0,3333… is just an approximation. And three times ⅓ should be 1, however in the decimal system it equals 0,9999…, which makes further calculations inaccurate. So, the main criterion on when to use a certain system is the accuracy. 

Accuracy of approximation of a fraction depends on two things, namely the fraction itself and the factors of the system base. Back to the example ⅓. When dividing 1 by 3 in the senary system, it equals 0,2 while in the decimal system it is a circulating fraction. 

To answer this question, I will mention the fraction ⅕. The previous paragraph seems to suggest that the senary system is more convenient than the decimal system. While it is true for fractions that have 3 or 6 in the denominator, one fifth in senary equals 0,(1) as opposed to the decimal system, where ⅕ is finite and equals 0,2. In other words, numeral systems can't be defined as "good" or "bad" - their usefulness depends on what we want to count.

Come to think about it, if we created a system base that can be divided by all the integers 1-10, it would eliminate most infinite fractions. Therefore, more numbers could be divided with accuracy. This number is 2520 = 7 • 8 • 9 • 5. As visible, it's quite a large number, which is an inconvenience. 

Another number could be used in order to provide better accuracy - a product of the prime numbers 2, 3, 5 and 7 = 210. Having it as a base of a system would also lead to an improvement in accuracy.