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.
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.
Both students are correct in their understanding of multiplying by ten in their respective base systems.
To the first student, who is referring to the base-ten system, it is indeed true that multiplying a number by ten simply involves adding a zero at the end of the number. This works because each position in the base-ten system represents a power of ten, and moving a digit one place to the left (i.e., adding a zero at the end) increases its value by a factor of ten.
To the second student, who mentions base five and base eight systems, their statement is also correct. In base five, multiplying a number by five involves adding a zero at the end, just like in base ten. The same applies to base eight, where multiplying by eight requires placing a zero at the end of the number.
This method of multiplying by the base value itself and appending a zero to the end of the number works in all positional numeration systems. It is because each position represents a power of the base, and multiplying by the base simply moves the digits one position to the left, effectively increasing their value by a factor of the base.
The first student is correct. In our base-ten system, multiplying by ten is as simple as appending a zero to the end of a number. However, the second student's claim is not true for all base systems.
To explain why, let's consider the first student's claim about multiplying by ten in our base-ten system. In base ten, each digit's place value is ten times that of the digit to its right. So, when we multiply a number by ten, it essentially shifts all the digits one place to the left, and a zero is added at the rightmost position. This works because we have ten symbols (0-9) to represent numbers.
Now, let's compare this to base five and base eight systems, as mentioned by the second student. In base five, there are only five symbols (0-4) to represent numbers, and each digit's place value is five times that of the digit to its right. Similarly, in base eight, there are only eight symbols (0-7) to represent numbers, and each digit's place value is eight times that of the digit to its right.
When multiplying by the base itself (5 or 8), it does not simply involve adding a zero to the end of the number, as in base ten. Instead, it requires shifting the digits to the left and multiplying by the base value. For example, in base five, to multiply by five, we need to shift all digits one place to the left and multiply by five. In base eight, we need to shift all digits one place to the left and multiply by eight.
Therefore, the second student's claim is not universally true for all base systems, including base five or base eight. Each base system has its own rules for multiplication and how to shift digits based on the base value.
In summary, the first student is correct about the base-ten system, where multiplying by ten simply involves adding a zero at the end. However, the same logic does not work in other base systems like base five or base eight, where multiplying by the base value requires shifting digits and performing multiplication based on the given base.