A brief history of numerical systems

數(shù)值系統(tǒng)簡史

來源視頻

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

只用這十個(gè)符號， 我們可以寫出任何有理數(shù)

但是為什么是這幾個(gè)符號呢？

為什么有十個(gè)？

而且為什么人們會(huì)按照這樣的方式排列它們呢？

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.

有史以來，數(shù)字一直是生活中必不可少的。

最早人們通常用身體的某部分或計(jì)數(shù)標(biāo)記

來表示一群動(dòng)物或部落的人的數(shù)量。

但是隨著生活越來越復(fù)雜 需要數(shù)的數(shù)量也不斷增加

這些方法不再夠用了。

隨著不同文明的發(fā)展，

人們想出了很多用了記錄更多數(shù)量的辦法。

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.

很多數(shù)字系統(tǒng)，

比如希臘數(shù)字

希伯來數(shù)字

以及埃及數(shù)字

只是原來計(jì)數(shù)標(biāo)記的加強(qiáng)版

加入了用來代表更高數(shù)量級的新符號。

每個(gè)符號都盡可能多次重復(fù)使用再把它們加起來。

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.

羅馬數(shù)字添加了另一種方式。

如果前面有一個(gè)值更大的數(shù)字

它們會(huì)被相減，而不會(huì)被相加。

但盡管有了這種創(chuàng)新

對較大的數(shù)字來說 這依舊是種累贅的方法。

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.

有一種更有用更優(yōu)雅的方式

稱為定位數(shù)系。

之前的數(shù)字系統(tǒng)需要不斷重復(fù)地畫很多符號

而且每一個(gè)更大的數(shù)量級都需要引入新的符號。

但是定位數(shù)系可以重復(fù)使用同樣的符號，

根據(jù)它們的位置賦予它們不同的值。

一些社會(huì)文明發(fā)展了自己的定位數(shù)系

其中包括巴比倫人

古中國人

還有阿芝特克人。

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.

到了第八世紀(jì)，印度數(shù)學(xué)家完善了一種記數(shù)制

它在接下來的幾個(gè)世紀(jì)中

被阿拉伯商人，學(xué)者和征服者傳到了歐洲。

這就是十進(jìn)制

一種可以只用十個(gè)獨(dú)特的圖像字符 就能表示出任何數(shù)字的方法。

這些字符的位置表明了10的不同次方，

從右開始，次方數(shù)向左不斷遞增。

比如數(shù)字316，

讀成?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.

這個(gè)方法的一個(gè)巨大突破是

同時(shí)也被瑪雅人發(fā)明了的數(shù)字0。

舊的定位數(shù)系沒有這個(gè)符號，

便會(huì)在那個(gè)位置留一個(gè)空格，

這讓63和603，12和120難以區(qū)分。

0這既是一個(gè)值又是一個(gè)占位符的特質(zhì)

讓它成為一個(gè)可靠，一致的符號。

當(dāng)然，也可以用任何十個(gè)符號

來代替數(shù)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.

很長一段時(shí)間 圖像字符在各地區(qū)不斷變化發(fā)展著

大多數(shù)學(xué)者認(rèn)為我們?nèi)缃竦臄?shù)字

是從北非阿拉伯王國馬格里布地區(qū)曾用過的符號

進(jìn)化而來的。

到十五世紀(jì) 我們現(xiàn)在日常所熟悉的阿拉伯?dāng)?shù)字體系

已經(jīng)取代了羅馬數(shù)字

變成了世界上最常用的數(shù)字系統(tǒng)。

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.

那為什么阿拉伯?dāng)?shù)字系統(tǒng)和其他的一些

都用十進(jìn)制呢？

最可能的答案是因?yàn)樗亲詈唵蔚摹?/p>

這也解釋了阿芝特克人使用二十進(jìn)制的原因

但是其他進(jìn)制也是可以用的

巴比倫數(shù)字是六十進(jìn)制。

很多人認(rèn)為十二進(jìn)制

也挺好的。

12和60都是因數(shù)很多的合數(shù)，它們可以被2，被3，被4，被6整除，

用這些數(shù)來表示共同因數(shù)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.

事實(shí)上，我們?nèi)粘Ｉ钪写嬖诤芏鄶?shù)字系統(tǒng)，

從測量角度和時(shí)間，到日常的計(jì)量單位，比如一打（a dozen意為12個(gè)，a gross意為144個(gè)）

當(dāng)然，二進(jìn)制

也被使用于所有的電子設(shè)備。

盡管程序員也將八進(jìn)制和十六進(jìn)制用于更精簡的表達(dá)。

所以下一次你使用一個(gè)很大的數(shù)字時(shí)，

想想你僅用了這幾個(gè)符號就獲得了一個(gè)如此大的量，

也試試看你是否能用不同的方式把它表達(dá)出來。