Today's abstract
Let's talk about impossible multiplication. An impossible multiplication is just it doesn't make any sense.
``How much is 6 Euro pasta for 5 people?''
is 6 * 5 = 30 Euro. But,
``How much is 6 Euro pasta for 7 Euro salad?''
doesn't make any sense. If someone does 6 * 7 = 42. What does it mean? But it seems some people teach these stuffs to children. They make the students to hate mathematics. That's pity.

Main part
Here multiplication is only with a number. If we have some variables x and y, we can do 3 * x + 5 * y, but not x * y. You might wonder what kind of mathematics we can not do x * y, that is such a simple operation. But I would like to emphasize, we have not so much freedom to compute numbers in real world with meaning. We all learned that we sometimes can not multiply numbers, again in real world.
Let's think about in a restaurant as an example. For instance, x or y are a menu A and B's price, respectively. As more vivid example, this restaurant has only two menus, a pasta and a salad. We are five people and went to this restaurant. The pasta costed 6 Euro and the salad costed 7 Euro. The three ordered pasta and two ordered salad. Then total amount is:
3(persons) * 6 (price of pasta) + 2 (persons) * 7 (price of salad)
This is basic of linear operation since there are additions and multiplications. You can see that the multiplications of the price of pasta and price of salad, 6 * 7 = 42 has no meaning here. What is price times price? Namely,
``How much is 6 Euro pasta for 5 people?''
I could understand this question, but,
``How much is 6 Euro pasta for 7 Euro salad?''
has no meaning. At least I can not explain what it is. There are two things I could not explain in this question. The first one is this multiplication itself, and the second is the question ``how much?'' Multiplication can change the unit, this question is that. But the question stays the same unit.
For instance, we have ten strings, each is five meter long, then total length are 5m * 10 = 50m. But if we have a land 5m * 10m = 50 square meters. The question here is:
``How long (meters) is the length of land that has five meter width and 10 meter height?''
If the anser is 50 meters length land, I got a trouble. Let me high light the unit of the first question.
``How much (Euro) is 6 Euro pasta for 7 Euro salad?''
I do not get the meaning of the question. If someone sells me 50m (length) land, I have a problem. It could be 50m long and 1mm width, and the price is the same as the 50m^2. I recommended to know the basic mathematics not to be cheated like this.
We do not do such multiplication since this is nonsense. We can always multiply the numbers themselves, i.e., 6 * 7 = 42. But, there is a case we can not do that if we think about the meaning. I once read a book explaining 1 * 1 = 1 as debt times debt becomes money. This is a good example that the calculation is right, but the explanation is wrong. It is simply we can not debt * debt (what does it mean?). If you were explained like this, I understand you hate mathematics. It is better to not to give a chance to cheaters. However, some cheaters learn mathematics to cheat people, oh well. By the way, I like a movie, Paper moon (http://www.imdb.com/title/tt0070510). In this movie, there is a cheat like here, exchanging nonsense to calculating the numbers.
したたり算法
1 week ago