Next message: Leneice divinity: "[Channel-talknumber] INFINITIVE NUMBER SYSTEM"
Both finite systems of numbers like the system described in Lesson 1 (units digit arithmetic--or remainders when dividing by 10) and infinite systems (like the set of integers) are important to understand. One reason is that they model situations that we want to understand in our world.
For example, units digit arithmetic can be useful in determining whether a number that is arrived at after several operations like addition, subtraction or multiplication is odd or even (something that is determined by the units digit). On the other hand a finite system (at least small ones) are quite unsuitable for counting or for finding the product of numbers--here an infinite system might be better. How would you enumerate the terms of a sequence like 1/2, 1/4, 1/8, 1/16, ... if you didn't have an idea of infinity?
As for the distributive property, I think of it as a compatibility rule that tells me how addition and multiplication "interact". For example this might be useful if I wanted to know if for any integer X, the quantity 3X + 15 was divisible by 3? Using the distributive property I can write 3X + 15 = 3(x + 5). I can now "see" that this is a number that is divisible by 3. That was sort of obvious, maybe.
A more difficult problem might look like this: If I charge $X for a ticket plus a $2 handling fee, then 500 - 2X people will come to the event. How much revenue will I make if I charge $X?
The revenue may be written as (X + 2)(500 - 2X). This can be written as a quadratic expression, but how? The distributive law gives the result:
(X+2)(500-2X) = (X+2)*500 - (X+2)*2X = 500X + 2*500 - X*2X - 2*2X = 1000 + 496X - 2X^2.
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