为芯片奠定数学基础的那些大神们都是谁？

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在你看本文的这一刻，手机处理器中，正有千万门级别或者亿门的MOS管在关闭和打开，来实现不同的与，或，非的运算。tjTednc

这一切数学基础都要从数学最基础的原理来说起。tjTednc

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人类从呱呱坠地开始，耳濡目染最开始就是十进制，小朋友也很容易接受，并可以很快掌握十进制的加减法。tjTednc

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十进制非常通俗易懂，它成为了人类最早的数学基础。tjTednc

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世界上不同的文明都出现了类似十进制的描述，例如几千年前的苏美尔楔形文字，汉字，后来的阿拉伯数字都有十进制的描述。tjTednc

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这个不奇怪，因为人类进化了十根手指头，用手指头来数数再自然而不过了。tjTednc

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但是，有人觉得十进制并不完美。tjTednc

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1701年，德国数学家莱布尼茨法国科学院提交并宣读了他题为《数学科学新论》的研究论文。tjTednc

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在这篇文章中，他提出了用0和1两个数表示全部数字的方法。tjTednc

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如上表所示，每个数字都可以表示一个二进制数。二进制数中只有0和1，没有其他符号。tjTednc

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鲁道夫·奥古斯特公爵研究莱布尼茨的二进制数学后认为二进制数学符合《圣经·创世纪》中的记载，是“上帝的算法”。因为太简洁了，符合上帝从无到有创建世界的描述。tjTednc

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奥古斯特公爵很喜欢，但是科学院却把这篇论文给拒了。tjTednc

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当时的法国科学院院长单内认为，看不出这篇论文有什么用处。tjTednc

论文有什么用？tjTednc

解决了什么重大问题？tjTednc

有什么实际的价值？tjTednc

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这个思路就一直存在于审稿人的脑海里面，几百年未曾改变。tjTednc

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那么有个问题摆在莱布尼茨的面前，二进制有什么用？tjTednc

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作为我们300多年后的我们，对于二进制的作用是非常清楚的。但是在当时给二进制找个应用的确是个难题。tjTednc

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但是莱布尼茨作为大数学家，大哲学家。十七世纪的亚里士多德。可以和牛顿争夺微积分发明权的大佬，肯定不会被这个小问题难住。tjTednc

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但是用了两年时间，直到1703年莱布尼茨收到白晋所寄的伏羲八卦图，才发现自己的二进制体系与伏羲八卦图的一致性。tjTednc

白晋是法国神父，曾经担任康熙皇帝的数学老师，1697年奉康熙之名回法国招募人才，并与莱布尼茨相识，多次书信来往，白晋由此给莱布尼茨介绍了中国的易经和八卦。tjTednc

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于是，莱布尼茨补充了本项研究意义，并发表在法国皇家科学院院刊上。tjTednc

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标题是《二进制算术阐述-仅仅使用数字0和1兼论其效能及伏羲数字的意义》。tjTednc

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莱布尼茨大神，打算用二进制来描述中国的八卦。tjTednc

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对于卦象来说，一根长线代表阳爻（阳），两根短线代表阴爻（阴）。tjTednc

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也就是阴阳。tjTednc

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易经说：道生一，无极变太极；一生二，太极变两仪；二生三，两仪变四象；三生万物，四象旋转，世界诞生。八卦，正如其名，是对宇宙后续变化的推演。tjTednc

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从伏羲八卦中找到二进制的意义，这个不是什么戏说。tjTednc

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很多学者论述了，莱布尼茨不是根据八卦而发明二进制，而是发明了二进制才遇到了伏羲八卦，因此本文采用这个说法，而不牵强附会中国的八卦给了莱布尼茨发明二进制的灵感。tjTednc

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对于莱布尼茨这个大数学家来说，二进制只是其很小的数学成就，但是却是当今信息时代存在的数学基础。tjTednc

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我们这个信息时代都是建立在二进制上的，在这一点上，莱布尼茨居功至伟。tjTednc

但是，仅仅有二进制是不够的。tjTednc

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莱布尼茨发明了二进制。此时的二进制还只能像十进制那样运算，加减乘除。tjTednc

就是换个方式来运算，没有什么稀奇。tjTednc

而乔治·布尔则带了一种运算，那就是布尔代数。tjTednc

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乔治·布尔，1815年出生在英格兰的林肯，少时家贫，但敏而好学，自学成才，全部的志向都放在数学上。tjTednc

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1847年，布尔出版了《逻辑的数学分析，论演绎推理的演算法》。tjTednc

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从此在数学界名气大震，并且从“编外人员”到有了正式的教职。tjTednc

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1854年，他又出版了《思维规律的研究，作为逻辑与概率的数学理论的基础》，其中完满地讨论了这个主题并奠定了所谓的数理逻辑的基础。为这一学科的发展铺平了道路。tjTednc

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布尔代数讲述内容很多，但其最典型的就是两种值和三种运算。tjTednc

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布尔在整个计算中，定义两种值，也就是真和假。1（true），0（false）。tjTednc

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三种基本运算，就是与（AND），或（OR），非（NOT）。tjTednc

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与或非也可以用符号表示：   &   |    ！  三种符号表示。tjTednc

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与就是两个值都为真才为真。tjTednc

或就是两个值有一个为真就为真。tjTednc

真的非就是假，假的非就是真。tjTednc

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这个就是两种值，三种运算的含义。tjTednc

有点哲学的意思。tjTednc

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布尔通过研究发现：所有的数字算数运算，都可以用布尔代数化简成为 0和1的与或非操作。tjTednc

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例如：加法a+b tjTednc

进位就是a&b，而加完后0位是（a&！b）|（b&！a）tjTednc

也就是可以用与、或、非来表示加减。tjTednc

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 二进制配合布尔代数，能够表示所有的数学运算。tjTednc

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1864年，布尔去世。但是他发明的这些计算方式，将在100年后得到应用，成为计算机体系中的基本运算。tjTednc

用与、或、非的运算来等价表示加减乘除的运算，这个就是布尔留给后世的财富。tjTednc

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克劳德·艾尔伍德·香农是美国数学家，也是信息论的创始人。tjTednc

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但是，在成为信息论奠基者之前，学生时代的他就发表了一篇划时代的论文。tjTednc

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1936年，正在读硕士的香农发表了著名论文《继电器和开关电路的符号分析》。tjTednc

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在当时能够实现开关电路的技术方案就是继电器。tjTednc

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通过磁体通断打开或者关闭。tjTednc

如果继电器打开，电流带来磁场，继电器合上，电路闭合。tjTednc

如果继电器关闭，磁场消失，继电器关闭，电路断开。tjTednc

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这个开关电路给了香农灵感，配合布尔代数。tjTednc

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香农给出了布尔代数一种物理实现方式。tjTednc

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串联的开关电路就是布尔运算的与操作，两个开关a和b都关闭，灯才能亮。tjTednc

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这个电路就是a&btjTednc

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而并联的开关电路就是布尔运算的或操作。两个开关有一个关闭，灯就会亮。tjTednc

这个电路就是a|btjTednc

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是不是看起来简单极了。tjTednc

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还没有硕士毕业的香农，奠定了现在数字电路的基础。tjTednc

布尔计算可以将任何运算化简为与、或、非。tjTednc

所以不断重复这两种电路，就可以通过电路实现任意运算。tjTednc

此时，从莱布尼茨提出了二进制，已经过去了两百多年，离布尔发表布尔运算也过去接近100年。tjTednc

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 终于，香农说，你们的数学表达，我可以用物理电路来实现了。tjTednc

这个就是现代计算机的基石。tjTednc

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哈佛大学的霍华德·加德纳（Howard Gardner）教授评价：“这可能是本世纪最重要、最著名的一篇硕士论文。”tjTednc

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1947年，美国物理学家肖克利、巴丁和布拉顿三人合作发明了晶体管。tjTednc

晶体管的诞生，替代了原来的继电器，电子管等设备。tjTednc

成为了整个信息时代的基础。tjTednc

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在晶体管发明十年后的1958年，34岁的基尔比加入德州仪器公司。然后有了把多个晶体管放到一起的想法。于是，集成电路诞生了。tjTednc

又过了十二年，1970年，第一个CPU，英特尔4004诞生了。tjTednc

这些基础都变成如下MOS管的开关电路，下图就是一个开关电路（gate）。通过控制栅极电压，实现Source和Drain的通断。tjTednc

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时至今日，芯片的集成度不断的增加。tjTednc

最新的CPU，GPU，手机处理器等等, 其晶体管的数量都已经超过百亿级别。tjTednc

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晶体管数量tjTednc

制程tjTednc

M1tjTednc

160亿tjTednc

5nmtjTednc

安培A100tjTednc

540亿tjTednc

7nmtjTednc

A14tjTednc

125亿tjTednc

5nmtjTednc

麒麟9000tjTednc

153亿tjTednc

5nmtjTednc

但是，其内部运行的数学逻辑，仍然是莱布尼茨提出的二进制，布尔发明的布尔计算，和香农描述的开关电路。tjTednc

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一直没有改变！tjTednc

欢迎关注：歪睿老哥，如果你觉得本文还不错，欢迎点赞，在看，分享。tjTednc

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