Scale Factor (computer Science) - Operations On Scaled Values

Operations On Scaled Values

Once the scaled representation of a real value is stored, the scaling can often be ignored until the value needs to come back into the "real world". For instance, adding two scaled values is just as valid as unscaling the values, adding the real values, and then scaling the result, and the former is much easier and faster. For other operations, however, the scaling is very important.

Multiplication, for instance, needs to take account of the fact that both numbers are scaled. As an example, consider two real world values A and B. The real world multiplication of these real world values is:

A * B = P

Now suppose we're storing the values with a scale factor of Z. If we simply multiply the stored representations we'll get the following:

AZ * BZ = Q

Note how AZ is the scaled real world value or A or simply the product of A * Z, and likewise, BZ is the scaled representation of B. Also note that we didn't write PZ as the answer, the reason is simple: PZ is not the answer. You can see this by rearranging the statement, where each line in the following are equivalent:

AZ * BZ = Q A * Z * B * Z = Q (A * B) * Z * Z = Q P * Z * Z = Q PZ * Z = Q

Note how we substituted P for A * B on line 4. You can now see that the result of AZ * BZ (which is Q) is NOT PZ, it's PZ * Z. If PZ were the answer, we could simply store it directly, since it has the scale factor built in, as is the case with addition and subtraction. For multiplication, however, you can see that the product of two scaled values has an extra scaling built in. As long as this is taken into account, there's still no need to convert AZ and BZ into A and B before performing the operation, you just need to divide the result by Z before storing it back. You will then have PZ stored as the result of the multiplication, which is fine because you weren't storing the result of AZ * BZ, you were storing the scaled representation of the result of A * B.