大家好，今天我給大家?guī)?lái)《代數(shù)學(xué)入門》第一集的文稿及其翻譯。本集以故事的形式講述了十進(jìn)制在誕生前人們使用的計(jì)數(shù)系統(tǒng)。通過(guò)對(duì)比，我們直觀感受到十進(jìn)制的不僅可以表達(dá)任意大的數(shù)字，而且在寫法上十分簡(jiǎn)便。這位我們進(jìn)一步學(xué)習(xí)數(shù)字起了一個(gè)良好的開(kāi)頭。

大家好，我是教授馮·史莫霍,歡迎來(lái)到《數(shù)趣課堂》

Hello. I’m Professor Von Schmohawk and welcome to Why U.

這個(gè)系列的課程以介紹代數(shù)學(xué)為主。但在我們開(kāi)始之前，我們首先來(lái)談一下什么是“數(shù)字”。

This series of lectures is an introduction to Algebra. But before we discuss Algebra, we should start by taking a closer look at the things we call numbers.

在很早的文化中，數(shù)字已成為日常理解和交流數(shù)量必不可少的工具

In even the earliest cultures, understanding and communicating quantities has been essential to everyday life.

人類學(xué)家告訴我們，即便是在最原始的石器時(shí)代，數(shù)字的概念已然成形

Anthropologists tell us that even the most primitive stone-age cultures had some concept of “number”.

但是比起現(xiàn)在，早期的的計(jì)數(shù)系統(tǒng)十分簡(jiǎn)易。一切都很簡(jiǎn)單。

However, early number systems were much more limited than today's base-10 number system. Life was simple back then.

我們并不需要計(jì)量精確的數(shù)目。如果你知道附近有一群羚羊。精確測(cè)定數(shù)目和距離并非明智之舉。

We didn't need names for exact quantities.If you knew that there was a herd of gazelles nearby, it didn't matter exactly how many gazelles were in the herd or exactly how many miles away they were.

重要的是那群羚羊是否能一網(wǎng)打進(jìn)

What was important was that there were a lot of gazelles and they were just over the hill.

即便最近，一些澳洲原住民只能數(shù)到“二”，再大的數(shù)只好稱“許多”

Even recently, certain Australian aboriginal tribes counted only to two, with any number larger than two called "much" or "many".

生活在亞馬遜的美洲印第安人可以數(shù)到“六”，盡管他們稱“三”為“二一”，稱“四”為“二二”，以此類推

South American Indians along the Amazon had names for numbers up to six, although three was called "two-one", four was "two-two" and so on.

生活在南非的布須曼人也有類似的命名方法，但因?yàn)檫^(guò)于冗長(zhǎng)而止步于“十”

Bushmen of South Africa had a similar way of naming quantities, but stopped at ten because the names became too long.

雙方在交易物品的時(shí)候，不允許一方的數(shù)目大于“二”。

This tribe would not do "financial" transactions involving numbers greater than two.?

比如說(shuō)，他們不會(huì)直接用兩頭牛換四只豬。而是拆分成兩次交易，每次用一頭牛交易兩只豬

For example, they would not trade two cows for four pigs. Instead, they would trade one cow for two pigs and then in a second transaction trade another cow for another two pigs.

如果你沒(méi)有學(xué)過(guò)數(shù)字，但是想向別人描述你看到了多少只羚羊，你的第一選擇是用手指表述羚羊的數(shù)目

If you had never heard of numbers and you wanted to describe to someone exactly how many gazelles you had seen just over the hill, you might use your fingers to represent how many gazelles there were.?

當(dāng)然這種方法對(duì)超過(guò)十只羚羊的情況便不那么奏效了。

Of course this would become difficult if there were more than ten gazelles.?

或許你可以在木頭上刻一道刻痕、用一顆石子代表一只羚羊。但是這種方法在面對(duì)眾多羚羊時(shí)只會(huì)越來(lái)越繁瑣。

You could cut marks on a stick or maybe gather a group of pebbles to represent the number of gazelles, but then, this would also become cumbersome if there were a lot of gazelles.

另一種方法是用不同的名字或符號(hào)來(lái)指代每一個(gè)數(shù)字。這種方法看上去簡(jiǎn)單直接，但是仍有問(wèn)題。

Another option would be to invent different names and symbols for each possible quantity. This seems like a simple solution but there is still a problem.

要知道此時(shí)的你對(duì)今天通行的十進(jìn)制一無(wú)所知，你需要發(fā)明一套全新的符號(hào)體系來(lái)給每個(gè)數(shù)字一一命名。

Remember that you know nothing about our modern base-10 number system which uses only ten symbols in different combinations, so you will have to invent a new word and symbol for every possible quantity.

比方說(shuō)，我曾到訪過(guò)一個(gè)叫“可可洛克島”的地方。當(dāng)?shù)厝耸沁@樣稱呼前三十個(gè)數(shù)字的：

For instance, in my primitive tribe on Cocoloco island Here are the first thirty numbers we use.

zoop(1), floop(2), dop(3), trom(4), mim(5), zap(6), weeny(7), glumby(8), bimpy(9), neechy(10), frump(11), wackit(12), trimble(13), walaki(14), kravitz(15), jolo(16), ponzo(17), kolob(18), krub(19), wallop(20), zoomy(21), mombozo(22),balleemi(23), toramoo(24), fallazit(25), smip(26), bazooloo(27), eekeena(28), eechiwa(29), and ZORTAN(30).

過(guò)了“三十”，我們就只能說(shuō)“許多”

After Zortan(30) we just say "a whole bunch".

這種計(jì)數(shù)方法有諸多問(wèn)題。

There were several problems with this number system.

首先，光是記憶這三十個(gè)符號(hào)就讓人難受。

First of all, since we had over thirty symbols to memorize, math was really difficult.

比如說(shuō),如果你有“十四”個(gè)椰子，然后你吃掉了“九”個(gè)椰子，還剩下多少個(gè)椰子？

For instance, if you have “walaki(14)” coconuts, and then you eat “mim(5)” coconuts, how many are left?

答案顯然是“九”個(gè)椰子。

The answer is obviously “bimpy(9)” coconuts

但是要記住所有的組合需要經(jīng)過(guò)漫長(zhǎng)的學(xué)習(xí)。

but it takes many years of school to memorize all the combinations.

此外，如果你有“二十七”個(gè)椰子，然后別人給了你“四”個(gè)，一共有多少椰子呢？

Also, what happens if you have “bazooloo(27)” coconuts and then someone gives you “trom(4)” more?

這個(gè)嘛，你便只能說(shuō)“許多”椰子。

Well, then you’d have “a whole bunch of” coconuts.

其實(shí)在以前，有的本土數(shù)學(xué)家通過(guò)發(fā)明一系列新的稱謂符號(hào)來(lái)拓展現(xiàn)有的計(jì)數(shù)體系。

In the past, some of the mathematicians on Cocoloco Island actually invented an "advanced" number system with several thousand names and symbols to handle problems like this.

但是這些體系都不能解決往最大的數(shù)字加“一”的難題。

But as soon as they did, someone always came up with a problem which they did not have a number for by adding “zoop” to the biggest number.

有一天，一位聰明的本土數(shù)學(xué)家用“組合符號(hào)”的主意解決這個(gè)問(wèn)題。

Then one especially brilliant Cocoloconian mathematician came up with the idea of combining symbols.

在“三十”之后，下一個(gè)數(shù)是“三十·一”，以此類推。

After zortan(30), the next number would be zortan-zoop, then zortan-floop(30-1), and so on.

當(dāng)你數(shù)到“三十·三十”，我們?cè)儆萌齻€(gè)符號(hào)來(lái)接下去，比如“三十·三十·一”，“三十·三十·二”。

When you got to zortan-zortan(30-30), you then would go to zortan-zortan-zoop(30-30-1) and zortan-zortan-floop(30-03-2).

這種方法雖然可行，但越到后面越加冗長(zhǎng)。

This worked well, but numbers could get quite long.

比如說(shuō)，如果我們要用這種方法表示一千個(gè)椰子，就得說(shuō)三十三遍“三十”還要加上“十”。

For instance, our number "one-thousand" in Cocoloconian would be zortan*30-neechy

如果要寫出來(lái)更加麻煩。

which we would write as ...

許多年后，人們意識(shí)到一起一遍又一遍的寫“三十”。

After many years, people realized that it wasn’t necessary to write zortan over and over.

我們可以直接用“三十”出現(xiàn)的次數(shù)代替。

The first symbol could just represent the number of zortans.

舉個(gè)例子，“三十·三十·三十·一”可以簡(jiǎn)寫成“三·一”。

For instance, the number zortan-zortan-zortan-zoop would be dop-zoop, three zortans plus a zoop.

“三十·三十·三十·三十·一”可以簡(jiǎn)寫成“四·一”。

Zortan-zortan-zortan-zortzan-zoop would be trom-zoop, four zortans plus a zoop.

第一個(gè)位置最多可以表示三十個(gè)“三十”的個(gè)數(shù)，即我們的九百。

The first symbol could represent up to zortan zortans, or 900.

因此這種兩位制計(jì)數(shù)方法只能數(shù)到我們的一千，盡管當(dāng)時(shí)這是個(gè)大數(shù)目

This two-symbol system still couldn't quite get up to 1000 , but nobody had that many coconuts anyway.

有一天，從遙遠(yuǎn)的棒果多果到駛來(lái)了一艘船

Then one day a boat arrived from the distant island of Bongopongo.

大伙驚呆了，因?yàn)樗麄儼l(fā)明了一種更簡(jiǎn)潔的計(jì)數(shù)方法來(lái)表示很大的數(shù)字

We were all amazed. The Bongoponganians had invented a much better system which could represent really big numbers with much fewer symbols.

顯然，早期的棒果多果人用指頭計(jì)數(shù)，因?yàn)橐还仓挥惺畟€(gè)符號(hào)。

Apparently, in early pre-history of Bongopongo they must have counted on their fingers because they only used ten different symbols. which they called “fingers” or “digits”.

他們先是用一位數(shù)從“一(1)”數(shù)到“九(9)”，接著用兩位數(shù)繼續(xù)下去

They counted up to bimpy, which they called “nine”, using a single digit. After “nine”, they used two digits.

第一位表示“十(10)”的次數(shù)，第二位則是在此基礎(chǔ)上累加

The first digit represented some number of tens, and the second digit would add from nothing to nine to that.

比如，數(shù)字“32”代表三個(gè)“十(10)”以及兩個(gè)“一(1)”

For instance, the number 32 would represent three tens and two ones.

數(shù)字“47”代表四個(gè)“十(10)”以及七個(gè)“一(1)”

The number 47 would be four tens and seven ones.

類似的，數(shù)字“80”代表八個(gè)“十(10)”以及零個(gè)“一(1)”

Likewise, the number 80 would be eight tens and zero ones.

但是這種計(jì)數(shù)法的厲害之處在于，當(dāng)數(shù)到“99”時(shí)，所有人都覺(jué)得到頭了。

But here is where their number system was really amazing. After getting up to 99, you would think that they would have to stop.

但是，奇跡發(fā)生了！他們只需要在前面加上一個(gè)數(shù)字，來(lái)表示“百(100)”的個(gè)數(shù)，這樣便能一直數(shù)到“九百·九十·九(999)”了

But no! They would just tack on another symbol in front which would now be used to represent hundreds.This could take you all the way to 999, which meant nine hundreds plus nine tens plus nine ones.

每一次到所謂最大的數(shù)，他們只需要在前面加上一位，便能繼續(xù)數(shù)下去

Every time they got to the biggest number they could represent, they would just tack on another digit which represented the next bigger number and kept going!

這個(gè)計(jì)數(shù)方法有幾大優(yōu)勢(shì)，首先我們只需要記住十個(gè)數(shù)字

This system had some big advantages. First of all there were only ten different symbols to remember.

其次，這種計(jì)數(shù)法每加一位，計(jì)數(shù)范圍便能擴(kuò)大原來(lái)的十倍。所以只需六位數(shù)便能統(tǒng)計(jì)最多一百萬(wàn)個(gè)椰子，那真夠多的！

Secondly, every time they added a digit, they could represent a number ten times bigger. So with six digits they could represent almost a million. That’s a lot of coconuts!

回顧之前的計(jì)數(shù)方法，我們?nèi)菀卓闯龅谝惶追椒ǖ膯?wèn)題所在：

Looking back, it is easy to see the problem with the first Cocoloconian number system.

如果給每一個(gè)數(shù)取一個(gè)名字，名字肯定不夠用。

Each quantity required a new symbol, so when you ran out of symbols, you ran out of numbers.

如果有三十個(gè)符號(hào)，我們最大只能數(shù)到三十。

With thirty symbols, the largest quantity that could be represented was thirty.

第二種方法采用了組合符號(hào)的方法，用多個(gè)符號(hào)表示一個(gè)數(shù)字。

The second system allowed these symbols to be combined in a single number and their values added.

但是因?yàn)闆](méi)有進(jìn)位的機(jī)制，所以數(shù)字越大表示越冗長(zhǎng)。

But since each symbol could add no more than thirty to a number, numbers got big really quick.

第三種方法用兩個(gè)符號(hào)簡(jiǎn)化了第二種方法的寫法，使得計(jì)數(shù)更加簡(jiǎn)便。

The third Cocoloconian system could combine two symbols one of which was multiplied by a factor of thirty. This allowed larger quantities to be created with only two digits.

然而，這種方法只能表示1000以內(nèi)的數(shù)。

However, the largest number was still less than 1000.

與此形成鮮明對(duì)比的是，棒果多果的方法只使用了十個(gè)符號(hào)

On the other hand, the Bongoponganian system used only ten symbols.

但是只需這十個(gè)符號(hào)，配上便捷的進(jìn)位制，便能以很少的數(shù)字表示很大的數(shù)目。

However, with just these ten symbols, very large quantities could be represented with just a few digits since each additional digit in a number represented a quantity ten times bigger than the previous digit.

這種計(jì)數(shù)方法簡(jiǎn)單有效，并且表示的數(shù)字沒(méi)有上限

This system was simple and efficient and the quantities that could be represented were unlimited.

這便是現(xiàn)在通行全球的“十進(jìn)位”計(jì)數(shù)方法

That is why this is the number system that the world uses today.