A brief history of numerical systems

数值系统简史

来源视频

One, two, three, four, five, six, seven, eight, nine, and zero.

With just these ten symbols, we can write any rational number imaginable.

But why these particular symbols?

Why ten of them?

And why do we arrange them the way we do?

1、2、3......8、9、10

只用这十个符号， 我们可以写出任何有理数

但是为什么是这几个符号呢？

为什么有十个？

而且为什么人们会按照这样的方式排列它们呢？

Numbers have been a fact of life throughout recorded history.

Early humans likely counted animals in a flock or members in a tribe

using body parts or tally marks.

But as the complexity of life increased, along with the number of things to count,

these methods were no longer sufficient.

So as they developed,

different civilizations came up  with ways of recording higher numbers.

有史以来，数字一直是生活中必不可少的。

最早人们通常用身体的某部分或计数标记

来表示一群动物或部落的人的数量。

但是随着生活越来越复杂 需要数的数量也不断增加

这些方法不再够用了。

随着不同文明的发展，

人们想出了很多用了记录更多数量的办法。

Many of these systems,

like Greek,

Hebrew,

and Egyptian numerals,

were just extensions of tally marks

with new symbols added to represent larger magnitudes of value.

Each symbol was repeated as many times as necessary and all were added together.

很多数字系统，

比如希腊数字

希伯来数字

以及埃及数字

只是原来计数标记的加强版

加入了用来代表更高数量级的新符号。

每个符号都尽可能多次重复使用再把它们加起来。

Roman numerals added another twist.

If a numeral appeared before one with a higher value,

it would be subtracted rather than added.

But even with this innovation,

it was still a cumbersome method for writing large numbers.

罗马数字添加了另一种方式。

如果前面有一个值更大的数字

它们会被相减，而不会被相加。

但尽管有了这种创新

对较大的数字来说 这依旧是种累赘的方法。

The way to a more useful  and elegant system

lay in something called  positional notation.

Previous number systems needed to draw many symbols repeatedly

and invent a new symbol  for each larger magnitude.

But a positional system could reuse the same symbols,

assigning them different values based on their position in the sequence.

Several civilizations developed positional notation independently,

including the Babylonians,

the Ancient Chinese,

and the Aztecs.

有一种更有用更优雅的方式

称为定位数系。

之前的数字系统需要不断重复地画很多符号

而且每一个更大的数量级都需要引入新的符号。

但是定位数系可以重复使用同样的符号，

根据它们的位置赋予它们不同的值。

一些社会文明发展了自己的定位数系

其中包括巴比伦人

古中国人

还有阿芝特克人。

By the 8th century, Indian mathematicians had perfected such a system

and over the next several centuries,

Arab merchants, scholars, and conquerors began to spread it into Europe.

This was a decimal, or base ten, system,

which could represent any number using only ten unique glyphs.

The positions of these symbols indicate different powers of ten,

starting on the right  and increasing as we move left.

For example, the number 316

reads as 6 x 10^0

plus 1 x 10^1

plus 3 x 10^2.

到了第八世纪，印度数学家完善了一种记数制

它在接下来的几个世纪中

被阿拉伯商人，学者和征服者传到了欧洲。

这就是十进制

一种可以只用十个独特的图像字符 就能表示出任何数字的方法。

这些字符的位置表明了10的不同次方，

从右开始，次方数向左不断递增。

比如数字316，

读成 6乘以10的0次方

加上 1乘以10的1次方

加上 3乘以10的2次方。

A key breakthrough of this system,

which was also independently  developed by the Mayans,

was the number zero.

Older positional notation systems  that lacked this symbol

would leave a blank in its place,

making it hard to distinguish  between 63 and 603,

or 12 and 120.

The understanding of zero as both a value and a placeholder

made for reliable and consistent notation.

Of course, it's possible  to use any ten symbols

to represent the numerals  zero through nine.

这个方法的一个巨大突破是

同时也被玛雅人发明了的数字0。

旧的定位数系没有这个符号，

便会在那个位置留一个空格，

这让63和603，12和120难以区分。

0这既是一个值又是一个占位符的特质

让它成为一个可靠，一致的符号。

当然，也可以用任何十个符号

来代替数字0到9。

For a long time,  the glyphs varied regionally.

Most scholars agree  that our current digits

evolved from those used in the North African Maghreb region

of the Arab Empire.

And by the 15th century, what we now know as the Hindu-Arabic numeral system

had replaced Roman numerals in everyday life

to become the most commonly  used number system in the world.

很长一段时间 图像字符在各地区不断变化发展着

大多数学者认为我们如今的数字

是从北非阿拉伯王国马格里布地区曾用过的符号

进化而来的。

到十五世纪 我们现在日常所熟悉的阿拉伯数字体系

已经取代了罗马数字

变成了世界上最常用的数字系统。

So why did the Hindu-Arabic system, along with so many others,

use base ten?

The most likely answer is the simplest.

That also explains why the Aztecs used a base , or vigesimal system.

But other bases are possible, too.

Babylonian numerals were sexigesimal, or base 60.

Any many people think that a base 12 , or duodecimal system,

would be a good idea.

Like , is a highly composite number that can be divided by two,three,four,and six,

making it much better for representing common fractions.

那为什么阿拉伯数字系统和其他的一些

都用十进制呢？

最可能的答案是因为它是最简单的。

这也解释了阿芝特克人使用二十进制的原因

但是其他进制也是可以用的

巴比伦数字是六十进制。

很多人认为十二进制

也挺好的。

12和60都是因数很多的合数，它们可以被2，被3，被4，被6整除，

用这些数来表示共同因数更好一些.

In fact, both systems appear in our everyday lives,

from how we measure degrees and time,

to common measurements, like a dozen or a gross.

And, of course, the base two,  or binary system,

is used in all of our digital devices,

though programmers also use base eight and base  for more compact notation.

So the next time you use a large number,

think of the massive quantity captured in just these few symbols,

and see if you can come up  with a different way to represent it.

事实上，我们日常生活中存在很多数字系统，

从测量角度和时间，到日常的计量单位，比如一打（a dozen意为12个，a gross意为144个）

当然，二进制

也被使用于所有的电子设备。

尽管程序员也将八进制和十六进制用于更精简的表达。

所以下一次你使用一个很大的数字时，

想想你仅用了这几个符号就获得了一个如此大的量，

也试试看你是否能用不同的方式把它表达出来。