They say the pen is mightier than the sword, and authorities have often agreed.
人们认为文字比武器更有力，统治者们通常也这么认为。
From outlawed religious tracts and revolutionary manifestos to censored and burned books,
从非法宗教传单及革命宣言，到被删减和烧毁的书籍，
we know the potential power of words to overturn the social order.
我们了解了文字足以推翻社会秩序的潜能。
But as strange as it may seem, some numbers have also been considered dangerous enough to ban.
尽管这看起来很奇怪，但一些数字也被认为是危险的而被禁止。
Our distant ancestors long counted objects using simple tally marks.
我们的祖先们长期使用简单的计数符号来计数。
But as they developed agriculture and began living together in larger groups, this was no longer enough.
但当他们发展农业并开始群居时，这种方法便不再适用。
As numbers grew more complex, people began not just using them, but thinking about what they are and how they work.
当数字变得越来越复杂时，人们不再只是使用它们而开始思考它们是什么以及如何被使用。
And by 600 B.C.E. in Ancient Greece, the study of numbers was well-developed.
公元前600年的古希腊对数字的研究已很发达。
The mathematician Pythagoras and his school of followers found numerical patterns in shapes, music, and the stars.
数学家毕达哥拉斯和他的学生们发现了图形、音乐及星体的数字规律。
For them, mathematics held the deepest secrets of the universe.
对于古希腊人而言，数学家们掌握着宇宙中最深奥的秘密。
But one Pythagorean named Hippasus discovered something disturbing.
但是毕达哥拉斯学派中的希帕苏斯发现了一些令人不安的事情。
Some quantities, like the diagonal of a square with sides of length one
一些数字，比如边长为一的正方形的对角线长度
couldn't be expressed by any combination of whole numbers or fractions, no matter how small.
是不能以整数或小数形式来表示的，不管这些数字有多小。
These numbers, which we call irrational numbers, were perceived as a threat to the Pythagorean's notion of a perfect universe.
这些被我们称为无理数的数字，在当时被视为毕达哥拉斯完美宇宙学论中的一个威胁。
They imagined a reality that could be described with rational, numerical patterns.
他们想象出一个由有理数和数学规律组成的现实世界。
Historians write that Hippasus was exhiled for publicizing his findings,
历史学家写道，希帕苏斯由于公开这一发现而被流放，
while legends claim he was drowned as punishment from the gods.
但传说却是他被众神惩罚溺水而亡。
While irrational numbers upset philosophers,
当哲学家们为无理数所烦恼不安时，
later mathematical inventions would draw attention from political and religious authorities, as well.
后来关于它的数学研究发现也引起了政治及宗教两界的极大关注。
In the Middle Ages, while Europe was still using Roman numerals,
中世纪时期，当欧洲仍在使用罗马数字时，
other cultures had developed positional systems that included a symbol for zero.
其他国家已经发展出一套包含了零这一标志的数字系统。
When Arab travelers brought this system to the bustling maritime cities of Italy, its advantages for merchants and bankers was clear.
当阿拉伯旅行者将这套系统带到意大利繁华的海港城市，它带给商人和银行家的便利不言而喻。

But the authorities were more wary.
但统治阶级对此很谨慎。
Hindu-Arabic numerals were considered easier to forge or alter,
人们认为阿拉伯数字更容易被伪造或改变，
especially since they were less familiar to customers than to merchants.
尤其是顾客不如商人对其熟悉时。
And the concept of zero opened the door to negative numbers and the recording of debt at a time when moneylending was regarded with suspicion.
在人们对于金钱借贷很谨慎的时代，零的这一数字概念打开了负数和债务记录的大门。
In the 13th century, Florence banned the use of Hindu-Arabic numerals for record keeping.
13世纪时，佛罗伦萨禁止使用阿拉伯数字来进行记录。
And though they soon proved too useful to ignore, controversies over zero and negative numbers continued for a long time.
尽管由于阿拉伯数字太过方便使用，人们不久便忽视了这一禁令，但对零和负数的争议仍持续了很长时间。
Negative numbers were dismissed as absurd well into the 19th century.
直到19世纪负数仍被认为是不可理喻的存在。
And prominent mathematicians, like Gerolamo Cardano, avoided using zero,
声名显赫的数学家，比如吉罗拉莫·卡尔达诺，也避免使用数字零，
even though it would have made it much easier to find solutions to cubic and the quartic equations.
尽管零的使用会使解决立方和四次方程式变得简单很多。
Even today it's illegal to use some numbers for different reasons.
即便在今天，一些数字仍由于各种原因而被禁止使用。
Some are banned because of what they represent.
一些数字由于其代表含义而被禁。
For example, governments have prohibited the display of numbers that have symbolic meaning,
比如，政府会禁止那些含有标志性意义的数字排列组合，
such as the date of a revolution or connections to oppositional political figures or parties.
例如一场革命的日期或与反政府分子或政党有关的数字。
Other numbers are potentially illegal because of the information they carry.
其余的数字则由于它们所传达的信息而具有潜在非法性。
Just about any information, whether text, image, video, or executable programs can be translated into a string of numbers.
任何信息，无论是文本、图像、视频、还是可执行程序，都可以转译为一串数字。
But this means that protected information,
但这意味着机密信息，
whether copyrights, proprietary materials, or state secrets can also be represented as numbers,
无论是版权、专利材料、或是国家机密，都可以以数字的形式呈现，
so possessing or publishing these numbers may be treated as a criminal offense.
所以占有或公开这些数字也许会被视为犯罪。
This idea gathered attention in 2001
这种观点在2001年引起了公众注意，
when code that could be used to decrypt DVDs was widely shared and distributed in the form of a large prime number.
当时用于解密DVD的编码以大质数的形式被广泛分享和散布。
The idea of illegal numbers may sound absurd, but like words, written numbers are a way of expressing concepts and information.
非法数字的概念也许听起来很荒诞，但和文字一样，数字也是表达概念和信息的一种方式。
And in a world where calculations and algorithms shape more and more of our lives,
在数字计算和运算法则对我们生活产生越来越大影响的今天，
the mathematician's pencil grows stronger by the day.
数学家们的力量正日渐强大。