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在网上找到的0乘以无穷大的直观解释
送交者: pzzdm.com 2010年04月05日19:33:13 于 [教育学术] 发送悄悄话

MathematicsPerpendicular lines – zero times infinity

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Mathematics

Perpendicular lines – zero times infinity

PROBLEM

The equation for the x-axis is y=0; slope: 0
The equation for the y-axis is x=0; slope: infinity

If the x-axis and y-axis are perpendicular, the product of the slopes must be –1, but in this case it is not true. How can we explain this? Are the x-axis and y-axis not perpendicular?
SOLUTIONS

Seed Expert Claude Baudoin writes:
The product of zero times infinity is not defined, therefore it "can" be -1.

One way of looking at this is to consider two perpendicular lines, one very close to the x-axis, the other one very close to the y-axis, and to make them "tend" together toward these axes. For instance, the slope of the first one may be 0.01 and the slope of the other -100, so the product is -1. Keep rotating them closer to (respectively) horizontal and vertical. The slopes might now be 0.001 and -1000, and the product is still -1. As you keep rotating the two lines closer to the axes (the limits), the slopes become 0 and infinity. Since the product has been constant at -1, the limit of that product remains equal to -1.

Laurent van Roy adds:
In theory, zero multiplied by infinity is undetermined (as is zero divided by zero) so you cannot prove that the x-axis and y-axis are not perpendicular, but neither can you prove that they are not!

If a line is y=ax+b, then a perpendicular line will be y=(-1/a)x+c. 
If you take the x-axis as the first line, its slope (a) is 0. The perpendicular one is y=(-1/0)x+c which has a slope of infinity.

I prefer to explain it this way:
If a line is ay = bx, then a perpendicular line will be by = -ax.
If you take the x-axis as the first line, you get b=0, and every perpendicular line is 0y=0=ax. Now you get the y-axis. This means that both are indeed perpendicular.

0%(0)
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  再给你个机会想想清楚。。。 - algtrd 04/06/10 (91)
    一天过去了,你用泰勒级数算出3.14和1.414?  /无内容 - 逻辑 04/06/10 (123)
        你算根号2,用泰勒级数在0展开?  /无内容 - 逻辑 04/06/10 (136)
          拜托你仔细看贴好不好?用sqrt(1+x)展开。  /无内容 - algtrd 04/06/10 (110)
            你原来就是以为没人知道sqrt(x)在0点不是解析的呀? - algtrd 04/06/10 (113)
              我当然知道可以开,但是他A没说出收敛半径,所以我打他这点。  /无内容 - 逻辑 04/06/10 (98)
            x=1。当初是留下这点打A,被你横生枝节,不过先前说你民科, - 逻辑 04/06/10 (110)
              打A固然是好事,但既然是有关数学,就要说得严密。 - algtrd 04/06/10 (105)
            1+x也无法展开,因为要求x<1。不过atan可以,因为可以  /无内容 - 逻辑 04/06/10 (114)
              x=1处级数可以收敛,只不过不是“绝对收敛”。  /无内容 - algtrd 04/06/10 (109)
                那是,看得出你是科班。我忽略一点就是,atan在x=1也适用 - 逻辑 04/06/10 (124)
        我本来用根号2打他,被你用pi搅和了。  /无内容 - 逻辑 04/06/10 (126)
        同意,pi可以,根号2不行!你怎么用根号2?  /无内容 - 逻辑 04/06/10 (105)
    你不认为你在对牛弹琴?  /无内容 - aa881 04/05/10 (113)
      他本来就是牛。他还没算出3.14和1.414。  /无内容 - 逻辑 04/06/10 (116)
    你还是没有看懂上面的一贴! - pzzdm.com 04/05/10 (116)
  再用另外方式改一改: - algtrd 04/05/10 (108)
    对啊,你已经把垂直改为不垂直了,答案当然不同  /无内容 - pzzdm.com 04/05/10 (107)
        你根本没有读懂我贴的。改,可以,但要保持x,y轴的垂直 - pzzdm.com 04/05/10 (95)
          就是说,横线趋近与X轴的速率应该跟竖线趋近Y轴的速率相等 - pzzdm.com 04/05/10 (115)
            你原来说“0*无穷大”,没有提到过要以同样速率趋近0和无穷大 - algtrd 04/05/10 (118)
              你好好看看我上面的帖子好吗?要保持X,Y或 - pzzdm.com 04/05/10 (118)
      你们总是把我问的问题改一改,然后说我的答案是错的,这太荒谬了  /无内容 - pzzdm.com 04/05/10 (91)
        不是要改你的问题, - algtrd 04/05/10 (90)
          0乘以无穷大,就相当于X和Y轴相乘 - pzzdm.com 04/05/10 (138)
            我问的是0乘以无穷大,不是无穷小乘以无穷大!!!!!!!!!  /无内容 - pzzdm.com 04/05/10 (142)
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