無限∞的問題

infinity + r = r + infinity = infinity
(-infinity) + r = r + (-infinity) = -infinity
infinity + infinity = infinity
(-infinity) + (-infinity) = -infinity
infinity - r = infinity
(-infinity) - r = -infinity
r - infinity = -infinity
r - (-infinity) = infinity
infinity - (-infinity) = infinity
(-infinity) - infinity = -infinity
infinity * r = r * infinity = infinity for r > 0
(-infinity) * r = r * (-infinity) = -infinity for r > 0
infinity * r = r * infinity = -infinity for r < 0
(-infinity) * r = r * (-infinity) = infinity for r < 0
infinity * infinity = (-infinity) * (-infinity) = infinity
infinity * (-infinity) = (-infinity) * infinity = -infinity
infinity / r = infinity for r > 0
(-infinity) / r = -infinity for r > 0
infinity / r = -infinity for r < 0
(-infinity) / r = infinity for r < 0
r / infinity = 0
r / (-infinity) = 0

Where we get into trouble is with defining the following:

infinity + (-infinity)
(-infinity) + infinity
infinity - infinity
(-infinity) - (-infinity)
0 * infinity
infinity * 0
0 * (-infinity)
(-infinity) * 0
infinity / infinity
infinity / (-infinity)
(-infinity) / infinity
(-infinity) / (-infinity)
infinity / 0 = infinity
(-infinity) / 0 = -infinity

The above datas explain the answers of your first 4 questions
but since infinity - infinity cannot be defined, your last question cannot be answered.

this information is quoted from an experienced mathematician. It is highly believable.

其實∞+
　　∞ -
　　∞×
　　∞÷都是等於∞
因為∞只是一個符號

∞是一個無限的概念，不是一個實數。
我嘗試用一個比喻來回答這個問題：把∞比喻作這個宇宙。宇宙有多大，任何人都不知道。
∞+1=∞──在宇宙內有一顆新星誕生，這仍是一個宇宙，我們仍然不知道祂有多大。
∞-1=∞──在宇宙內有一顆恆星滅亡，這仍是一個宇宙，我們仍然不知道祂有多大。
∞x2=∞──假如上帝要擴大這個宇宙，我們只會知道宇宙大了，但我們不會知道祂有多大。
∞/2=∞──假如上帝要把這個宇宙一分為二，我們只會知道宇宙小了，但我們仍然不會知道祂有多大。

1111111......跟999999......不是一樣大，但因為我們根本不知道它們有多大，所以唯有都用∞來代表它們好了。

2006-11-08 18:20:46 補充：
Copying without answering the question!

所有正整數有∞個
所有雙數有∞個
所有正整數和雙數的數目都是無限

∞+1=∞
∞-1=∞
∞x2=∞
∞/2=∞

11111111111................(無限個1)
99999999999................(無限個9)
都係無限
因為∞x1=∞
以及∞x9=∞

唔係,∞只係一個符號,唔係實數