比乘法更大的是乘方,比乘方更大的是什么?

    小学时,老师说,由于生活中经常需要把同一个数加很多很多次,因此人们发明了乘法。 a × b 就表示 b 个 a 相加。初中时,老师说,由于生活中经常需要把同一个数乘很多很多次,因此人们发明了乘方。 a ^ b 就表示 b 个 a 相乘。令人失望的是,到了高中时,我们并没有学到更牛 B 的运算符号;大学都快学完了,似乎也没见到乘方升级的苗头。乘方之上究竟是什么?下面,有请今天的主角——超级幂——登场!

    很容易想到,比乘方更大一级的运算就是把 b 个 “a 次方” 重叠起来。不过,这里我们却遇到了一个之前不曾遇到的问题: a ^ a ^ a 究竟应该等于 (a ^ a) ^ a ,还是 a ^ (a ^ a) ?。我们不妨来算一算,不同算法得到的结果相差多远:

(2 ^ 2) ^ 2 = 4 ^ 2 = 16
2 ^ (2 ^ 2) = 2 ^ 4 = 16

    难道两种不同的计算顺序,得到的结果总是相同的吗?让我们换 a = 3 试试:

(3 ^ 3) ^ 3 = 27 ^ 3 = 19683
3 ^ (3 ^ 3) = 3 ^ 27 = 7625597484987

    哇,这下可就差远了。可以想象,如果把 “a 次方” 再多迭代几次,从右往左算和从左往右算会差得更多。恐怖的是,当有多重指数时,运算正是按照从右往左算的顺序进行的。试想,若有一种运算专门用来表示 b 个 a 构成的指数塔,这种运算的威力会多大。


    1947 年,数学家 Goodstein 发现,不管初始时选取哪个自然数,按照某种预先定义好的规则进行迭代,数列最终将变成 0 。但是,数列收敛到 0 的速度极其缓慢,以至于 Goodstein 需要处理一些连乘方也无法表达出来的大数。于是, Goodstein 便正式提出了这种超越乘方的运算。他把 b 个指数 a 迭代的结果记为 ba ,也就是把 b 放在 a 的左上角。在国外的一些论坛上,有时也能看见 a^^b 的表示方法,便于在纯文本格式下的传播。不过,当时 Goodstein 并没有用超级幂 (superexponentiation) 一词,而是用的 tetration 一词。这是由前缀“四” (tetra-) 和迭代 (iteration) 一词合成的,意即排在加法、乘法、乘方之后的第四级运算。事实上, tetration 比 superexponentiation 更常用一些。网上甚至有一个 tetration 论坛,论坛里活跃着一群热爱 tetration 的数学 geek 。

  

    超级幂是一个极为厉害的运算,它的增长速度非常惊人。在很小的数之间进行超级幂运算,就有可能得到一个巨大的天文数字。 32 等于 2 ^ (2 ^ 2) = 16 ,而 42 就等于 2 ^ (2 ^ (2 ^ 2)) = 65536 。那么, 52 等于多少呢?它应当等于 2 的 65536 次方,其结果是一个上万位的数。那 62 呢? 100100 呢?大家自己去想象吧。

    我们能轻松定义出超级幂的概念,但为什么这个东西却如此“小众”呢?当然,超级幂缺乏很多加减乘除和乘方运算具有的性质,这是一个重要的原因;不过,我想应该还有一个最基本的原因吧——超级幂本身没有什么实用价值。重复对折纸张、增长率的叠加、赌博游戏中的翻番,它们都可以用乘方来描述。实际生活中有什么事情正好能用超级幂来描述的吗?我想应该不会有吧。

 
 

  

    人类的想象力是无止境的。即使超级幂已经大到无法用言语描述的地步,大家还是会问,再把 “a 次超级幂” 迭代 b 层(注意运算顺序仍是从最深那一层开始),又会得到什么?是否就得到了第五级的运算呢?或许你马上就意识到了,这样扩展上去是没有尽头的,每一级运算迭代之后都能产生更高一级的运算。虽然此时脑子已经有点乱了,但是数学语言的严格性和理想性告诉我们,利用某种清晰的数学符号和递归法则,我们一定有办法定义出等级越来越高的运算来。
    Goodstein 牛就牛在这儿。他定义了 Goodstein 记号 G(n, a, b) ,来表示 a 与 b 之间的第 n 级运算。当 n = 0 时,规定 G(0, a, b) = b + 1 。也就是说,第零级运算是一个一元运算——自然数的后继。当 n = 1 时,规定边界值 G(1, a, 0) = a ,并规定 G(1, a, b) 表示对 G(1, a, 0) 的值进行上一级操作(后继操作),并重复迭代 b 次,其结果也就是 a 加上 b 。⼀般地,有:

G(n, a, b) = G(n – 1, a, G(n, a, b – 1))

    其中边界值为

G(1, a, 0) = a
G(2, a, 0) = 0
G(3, a, 0) = 1
G(4, a, 0) = 1
G(5, a, 0) = 1

    这就形式化地给出了第 n 级运算的意思。

    类似的东西不止一次地被提出过。两年前给大家介绍过世界上最大的数,当时就用到了 Knuth 箭头记号。这也是一种表示大数的方法,其思想与 Goodstein 记号几乎完全一样。 Ackermann 函数也是一个神速增长的函数,它的定义也有异曲同工之处。很多外文数学论坛则用 a [n] b 来表示 a 与 b 之间的第 n 级运算,是我比较喜欢的一种符号。

    当然,有 a [n] b ,必然会有 a [a [n] b] b ,从而又会有 a [a [a [n] b] b] b ⋯⋯没有最大的数,只有更大的数。人脑和数学是两个神奇的东西,没有什么数大到人脑想不出来,也没有什么数大到数学表示不出来。仅仅在脑中试想一下 100 [100] 100 ,你的思想就已经超越了整个宇宙的大小了。

81 条评论

  • 魔神翼

    第一次抢到沙发?

  • 魔神翼

    再顶一下

  • sss正和

    M牛是否要开讲Goodstein定理了~

  • 国标37

    坐板凳欣赏。。。

  • biohu

    果然够大。。。

  • ykzls

    好大!

  • BLDT

    呵呵,还能说点啥呢?真够牛B!

  • shilcare

    那个Goodstein记号真牛B!

  • lmhjf

    G(0, a, b) = b + 1
    应该是
    G(0, a, b) = a + 1

  • cat.s

    哇,这下可就差远了。

    我每次都没有看内容。。我每次都能看到这种句子。。
    越来越sunjj了

  • cartman

    曾经想过一件事情:log ab=log a+log b
    那么乘方应该定义成a^b=exp(ln a*ln b)
    然后可以继续往上搞。。

  • cartman

    曾经想过一件事情:log ab=log a+log b
    那么乘方应该定义成a^b=exp(ln a*ln b)
    然后可以继续往上搞。。。

  • wuzhengkai

    这个纯粹是思维游戏吧。。。

  • lxm

    G(n, 2, 2)=4
    是这样吧

  • morrowind

    大脑结构真是神奇

  • et27

    实际生活中有什么事情正好能用超级幂来描述的吗?//分形数学是不是就是这样的迭代的?

  • b4283

    那個 Goodstein 和 Ackermann 函數有相似的地方呢

  • swj05652

    这些都只是可计算的数(Turing Machine 意义下可计算)
    还有一些不可计算数,比如busy beaver function 它的增长速度超过任何一种可计算的数

  • Ernest

    仿照Sigma和Pi創造出“連乘方”或者“連超乘方”……?

  • 邯郸单身

    支持一下!不错!

  • clover

    noip复赛到了,每当烦躁的时候都会看看M牛的blog!虽然看的一头雾水吧!

  • 欺天

    人的认识是无限的。

  • BY

    我还以为这个就是ackermann函数,这个名字还在当年学并查集的时候见过。话说你现在在干什么?还在写书吗?

  • BY

    我还以为这个就是ackermann函数也。你现在在干什么?还在写书吗?
    出来吃饭我请客,都没得联系了

  • cgy4ever

    说来也真是巧妙,这几个星期我也在想同样主题的东西。
    今年IMO的Problem5(http://www.imo-official.org/problems.aspx)
    里面的那个简单的装置,从右向左数,每一个box似乎就相当于一个运算级。

  • 爱好者

    顶一下楼主

  • exile

    第0级运算会有一些矛盾。

  • Code之行人

    这个一般是理论上的探索研究,不过日常可用不到这么庞大的数据。

  • 夜弓

    “不管初始时选取哪个自然数,按照某种预先定义好的规则进行迭代,数列最终将变成 0”
    是在说Goodstein定理么?
    那个数列确实很慢,记得好像证明要用到集合论

  • 小卡尔维诺

    M牛,我昨天看到关于0乘以任何数是否为0的问题,希望您能帮忙说明下,里面牵扯到无穷大以及无穷大的级别,那么无穷小有级别吗?谢谢

  • 白左

    相比之下无穷反而亲切多了……

  • 白左

    @34楼 无穷也是有势的。但是无论多么大的势的无穷,被0乘都为0.可以说0的势是无穷小——所以0最牛逼

  • L

    tetration的一些性质很有意思。比如它的对数一般可以用Lambert W表示。
    我很奇怪…这些“四次以上“的运算为什么不能有实际作用?好像也没有多少人考虑过这个问题。

  • 小卡尔维诺

    谢谢楼上的指教

  • yh

    显然“第一个用20个汉字无法表达出来的数”就大到人脑想不出来。。。

  • M

    这个零级运算不对吧

  • hjss06

    10^80就已经超越了整个宇宙的大小了。

    宇宙中大约有原子10^80个

    • 哈喾球Rikka

      原子不是最小的,原子半径最小大概是10⁻¹¹米(10皮米)级别。如果到原子核,质子,中子等这些亚原子粒子的话可以到10⁻¹⁵米(1飞米)级别。而这还不够,目前人类所能测量的最小的长度大约在10⁻¹⁸m(1阿米)级别,比如最近的探测引力波的工作就需要达到这种精度的探测。不过目前有物理意义的最小长度叫做普朗克长度,大概是1.6×10⁻³⁵米(10皮幺米)。以这个作为单位计算:可观测宇宙体积 = 8.71×10¹⁸⁴ 立方普朗克长度。相比之下10⁸⁰就太小了。

  • suifeng

    曾经多次思考此问题, 看了楼主的文章才清晰的理清逻辑; 在理清逻辑过程中, 发现一点小bug;
    当 n = 0 时,有 G(0, a, b) = b + 1。
    个人认为是 :
    当 n = 0 时,有 G(0, a, b) = a + 1。
    也就是 a[0]b = a + 1; 在 a [n] b 中, 当n = 0时, 忽略b, 对应的 a[0]b = a + 1, 即是 [0]b == + 1
    a[1]b = a[0]a[0]a[0]…[0]a (b次运算), 根据 [0]b == + 1, 即是
    a[1]b = a + 1 + 1 + … + a (b次+1运算), 即a[1]b = a + b
    a[2]b = a [1] a [1] a [1] … [1] a = a + a + a + … + a = a * b

  • dandan02468

    还有不定级运算,非整数级常数运算,这是我的论文稿件中的,但论文或还未写完。

  • dandan02468

    超级幂在实际生活中有运用,例如:有a个人,要去a个地方,且无任何限制,则结果为2^^a,即a^a;若有a个人,可以选择另外a个人与他们结伴,到a个不同的地方,则结果为3^^a,即a^a^a,如果把a个人可以与另外a个人结伴而行这句话重复n次,则结果为(n+2)^^a,即a^a^a^……^a^a,共有n+1个^。所以作者说实际生活中用不到超级幂是错误的。

    • 哈喾球Rikka

      你这只是理论条件,根据你的说法,别说超级幂,不管多高级的运算都能用到。因为我可以无限往上加变数。a个人找a个人结伴在a个不同的日子去a个不同的地方这句话重复n次,或重复n次:a个人找a个人结伴在a个不同的日子带a数量的现金去a个不同的地方与a个人聚会讨论了a件不同的事……这么说不管几级运算都能在现实生活中用上?

  • Lamdy

    R_3{x=3;x=3^x}=3^^4

  • orbea jersey

    呵呵 猜一下 是除方吗

  • LenenTom

    (a ^ a) ^ a ,还是 a ^ (a ^ a) ?
    这是个问题……应该是第一种吧。

  • Keyskyer

    我一直想用这个来计算宇宙的发展概念,一直苦于值数不明确,无法进行测量。空间维度也可以用这个算量。

  • Keyskyer

    我一直想用这个算宇宙的发展,只是苦于无法得到准确的值数,用这个还可以算维度空间。用这个算量是非常精确的,比玛雅算法更准。

  • doubiemaker

    超乘方可不可以用来表示大数字,就像科学记数法一样?

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