A brief history of numerical systems - Alessandra King

譯者: Cissy Yun 審譯者: 庭芝 梁

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

1 2 3 4 5 6 7 8 9 和 0

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

用這十個符號

我們可以寫出任何想像得到的有理數

But why these particular symbols?

但為什麼是這些特殊的符號呢？

Why ten of them?

為什麼會有十個？

And why do we arrange them the way we do?

又為什麼，我們會按照 這個順序來排列它們呢？

Numbers have been a fact of life throughout recorded history.

有史以來，數字都是 人類生活中不可缺少的一部分

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

早期的人類在計算動物 或是部落中的人數時

using body parts or tally marks.

會使用身體的某個部位 或是畫記符號來算

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

但隨著生活的複雜化 計算的數量也不斷增加

these methods were no longer sufficient.

這些方法便不夠用了

So as they developed,

隨著人類的發展

different civilizations came up with ways of recording higher numbers.

不同的文明發展出 更多記錄數量的方法

Many of these systems,

許多記數系統

例如希臘數字

like Greek,

Hebrew,

希伯來數字

and Egyptian numerals,

以及埃及數字

were just extensions of tally marks

這些只是舊有計數符號的延伸

with new symbols added to represent larger magnitudes of value.

並加入了新的符號 用來代表更高的數量級

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

需要時，符號可以重複出現 並且用相加的總和來表示數量

而羅馬數字還有另一種表示方式

Roman numerals added another twist.

If a numeral appeared before one with a higher value,

如果一個數字 擺在另一個更大的數字之前

it would be subtracted rather than added.

它們會相減，而不是相加

But even with this innovation,

但是，儘管有了創新的記數方式

it was still a cumbersome method for writing large numbers.

要記錄比較大的數字時 這種方式仍然很麻煩

The way to a more useful and elegant system

另一種更方便、更優雅的記數方式

lay in something called positional notation.

稱為「位置記法」

先前的記數系統 需要重複畫記許多符號

Previous number systems needed to draw many symbols repeatedly

and invent a new symbol for each larger magnitude.

而且表示更大的數量級時 需要創造新的符號

But a positional system could reuse the same symbols,

但是位置記法 可以重複使用相同的數字符號

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

而且根據它們的位置 來代表不同的數值

Several civilizations developed positional notation independently,

有些文明發展出自己的位置記法

including the Babylonians,

巴比倫文明

the Ancient Chinese,

古中國文明

and the Aztecs.

還有阿茲特克文明

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

第八世紀時，印度的數學家 提出了一種更完善的記數系統

and over the next several centuries,

在之後的幾個世紀中

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

由阿拉伯商人、學者 和征服者傳到了歐洲

This was a decimal, or base ten, system,

這個方式稱為「十進位制」

which could represent any number using only ten unique glyphs.

就是用十個不同的數字符號 來表示任何的數字

The positions of these symbols indicate different powers of ten,

這些符號的位置 代表 10 的不同次方

starting on the right and increasing as we move left.

從數字的右邊開始 向左不斷遞增次方數

For example, the number 316

比如說數字 316

reads as 6x10^0

可以讀成 6 乘以 10 的 0 次方

plus 1x10^1

加上 1 乘以 10 的 1 次方

plus 3x10^2.

加上 3 乘以 10 的 2 次方

A key breakthrough of this system,

這個記數系統的其中一項重要突破

which was also independently developed by the Mayans,

是由馬雅人所發展出來的

was the number zero.

就是數字 0 的概念

Older positional notation systems that lacked this symbol

以往的位置記法 並沒有代表 0 的符號

would leave a blank in its place,

所以人們會在那個位置上空一格

making it hard to distinguish between 63 and 603,

這使得某些數字難以區分 例如 63 和 603

or 12 and 120.

或是 12 和 120

The understanding of zero as both a value and a placeholder

數字 0，不但被視為一個數值 同時也是一個「佔位符號」

made for reliable and consistent notation.

這使得位置記法 能夠更為可靠而且一致

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

當然，也可以用任何十個符號

to represent the numerals zero through nine.

來代替 0 到 9 的數字

For a long time, the glyphs varied regionally.

有很長一段時間 各地區採用不同的數字符號

許多學者認為 我們現今使用的數字

Most scholars agree that our current digits

evolved from those used in the North African Maghreb region

是來自當年的阿拉伯王國 位於非洲北部的馬格里布地區

of the Arab Empire.

當地人們使用過的符號 進化而來的

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

到了十五世紀 我們熟悉的阿拉伯數字

had replaced Roman numerals in everyday life

取代了羅馬數字

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

成為世界上最多人使用的記數系統

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

為什麼阿拉伯數字 和其他的記數系統

use base ten?

都是採用十進位制呢？

The most likely answer is the simplest.

最可能的答案是： 十進位制是最簡單的

That also explains why the Aztecs used a base 20, or vigesimal system.

這也解釋了阿茲特克人 使用二十進位制的原因

But other bases are possible, too.

但是還有其他幾種進位制可以使用

Babylonian numerals were sexigesimal, or base 60.

巴比倫數字是六十進位制

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

而很多人認為十二進位制

would be a good idea.

也是不錯的方式

Like 60, 12 is a highly composite number that can be divided by two,

和 60 一樣，12 也是一個高合成數

可以被 2、3、4

three,

four,

and six,

和 6 整除

making it much better for representing common fractions.

所以用 12 進位制 表示普通分數會更為方便

In fact, both systems appear in our everyday lives,

事實上，12 和 60 進位制 也存在我們的日常生活中

from how we measure degrees and time,

從測量角度、時間到計量單位

to common measurements, like a dozen or a gross.

例如一打 (12 個) 和一蘿 (12 打)

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

當然，還有二進位制

is used in all of our digital devices,

也被用在所有的數位裝置上

雖然程式設計師也會使用 8 進位制和 16 進位制

though programmers also use base eight and base 16 for more compact notation.

作為更簡潔的表達方式

So the next time you use a large number,

所以，當你下次要用到 一個很大的數字時

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

想想你只用幾個符號 就可以表達很大的數量

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

另外也可以試試看： 是否能用不同的方式來表達數字