Expert: Josh - 4/6/2004

Question
Hello:

I want to thank you for the reply; hoverver, I have a follow-up question.

I have tried to decipher an understandable answer from your first reply but was unabe to do so.  Can you provide a simplified version?

I thank you for your reply.
-------------------------
Followup To
Question -
Hello:

Why is it necessary to change or to convert from different types of units to units that are the same in order to determine the answer?


A three-foot wide shelf is used to hold a set of notebooks. Each notebook has a 3/4 inch wide binding.
How many notebooks can fit on the shelf?

Three feet is changed to 36 inches, and the units  are the same, inches.

36 inches / (3/4) inches = 36/1 X 4/3 = 48 (48 notebooks)

Or, 3/4 inches can be changed to feet:

3 feet / (1/16) feet = 3/1 X 16/1 = 48/1 = (48 notebooks)

Once again, I thank you for any helpful reply that you may provide.

Answer -
Anonymous,

For a given problem, the magnitude of each quantity in an expression must be measured relative to a reference scale. That's why we have a dimensionless metric system in the first place. Provided that the units in a mathematical equation are compatible, we can do everything with numbers, without using a single word.

For instance, we cannot treat one meter and one foot as being the same thing. If we are to discard the units, the numbers that we make reference to must be relative to the one another and measured with respect to some underlying scale. You cannot compare 3 feet with 3/4 inch. But you can divide 36 inches by 3/4 inches, without even thinking about "inches". Similarly, 36 meters divided by 3/4 meters also yields the same answer. Both give you the same number of partitions. That is all mathematics is concerned with. How you interpret the equation, what it means is irrelevant. The person who sets up the problem knows what he/she is dealing with.

Consistency is the key. For instance, a UK and a US accountant can do all the calculations exclusively in pounds or dollars to determine the return on an investment. The mathematical ideas extend across national boundaries. The concepts and mathematical laws are universal truth. They should not depend on the context or situation. For instance, just because the initial deposit represents different monetary values in their respective countries, as long as the interest rate and investment period remain the same, the final answer will be the same, apart from a scaling factor (given by the US/UK exchange rate).

Answer
Numbers are meant to suggest the size of something.
For example, 10 is ten times larger than 1.
In the metric system, whatever number you think of, it is measured relative to the base unit "1". For example, 100 means one hundred multiples of "1".

eg., 10 miles is compatible with 1 mile.
The mathematical statement "10 > 1" is true.
You can make a straight-forward comparison, because the number "10" and "1" are both measured using the same distance rule.

But mile and kilometer are incompatible, for instance. we cannot suggest that 10 miles equal ten times 10 kilometer. Even though that 10=10 is true.

Because we are measuring them against different two different distance metrics.

In order for these two quantities to have real meaning, we need to perform a conversion, taking into consideration the ratio between miles and kilometers. Then, the two quantities are measured with respect to the same distance rule.

Let's say that 1 mi = 1.6 km.
Then, we can say that 10 [miles] x 1.6 [km/miles] > 10 [km].
The part we should concentrate on is 16 > 10.