Two students were discussing their lesson on different number bases for positional numeration systems. One student said there is an advantage in using our base-ten system over other systems because to multiply by ten, just put a zero at the end of the number. The other student said the same thing was true when multiplying by five in base five or eight in base eight. Which student is correct? What would you say to the students in this situ-ation? If it works the same in different base systems explain why. If not, explain why not.

Answer 1

The first student is correct in the context of our base-ten system, but their statement does not hold true for all positional numeration systems.

In our base-ten system, multiplying a number by ten is simply achieved by appending a zero to the end of the number. For example, 5 * 10 = 50. This is a unique property of the base-ten system because the value of ten is equal to the base itself.

However, the second student's statement is not entirely accurate. In base five, multiplying a number by five does not involve appending a zero. Rather, it involves multiplying the number by five and placing it in the "tens" place. For example, 2 * 5 = 10 in base five, not 20. Similarly, in base eight, multiplying a number by eight does not result in simply appending a zero. It involves multiplying the number by eight and placing it in the "eights" place.

So, while the idea of multiplying by the base itself to achieve a similar effect does hold true in our base-ten system, it is not universal across all positional numeration systems. Different bases have different rules for multiplication, which depend on the values of the digits in that base.