arrival in the Caribbean; that is, what’s 1492 minus 1066 – but you hate long subtraction

之間相格了多久；便是1492減去1066 - 但你討厭冗長的減法

with its borrowing and such.

其實我們可以用加法來做減法：第一我們需要把每一位數字換成它和9的差

Well, we can subtract by adding: first we just need to replace each digit of the smaller

除了最後一個數字要被換成它和10的差

number with 9 minus that digit, except the final digit gets replaced with 10 minus that

所以1066變成9減1，9減0，9減6，10減6

digit, so 1066 becomes nine minus 1, nine minus 0, nine minus six, ten minus 6.

把它和1492相加便得到10426，忽略第一個數字，便得到了我們的結果：

Adding that to 1492 gives 10426, and if we ignore the first digit, we get the answer:

1492減1066等於426. 你可以逕行檢驗運算來確定其正確性。

1492 minus 1066 is 426.

巧合的是，葡萄牙來往西印度群島的距離正是法國到英格蘭距離的426倍

You can check just to be sure.

不過無所謂啦，用加法來做減法的技巧可以用於所有的正數！8減6，

Coincidentally, it’s roughly 426 times farther from Portugal to the West Indies and back

既2，等於8加4，忽略第一個數字

than from France to England.

100減1？既是100加999，忽略第一個數字後便是99

But anyway, the subtraction-by-adding trick works for any positive numbers!

424,242減333,333就是424,242加666,667，忽略第一個數字後，既90,909

8 minus 6, or 2, is the same as 8 plus 4, ignoring the first digit.

這個就交給你來檢驗了

100 minus 1? is 100 plus 999, which is 99, ignoring the first digit.

這個技巧看似畫蛇添足，但假如你建造了一台能把數字相加的機器

And 424,242 minus 333,333 is 424,242 plus 666,667, or, 90,909 ignoring the first digit.

而你又想它能夠做減法？那麼在這種情況下

I’ll let you check that one.

用加法來做減法會容易得多：舉個例子，我一個叫Hank Green的好基友參與打造了一台加法運算機

This trick might seem useless, but suppose you built a machine that can add numbers together,

而這正是它是如而加法正是那台機器做減法的方式

and you wanted to make it subtract?

基本上，機器通過轉動數字輪來使數字相加，但是結果並不能無限大

Well, in that case it might be easier to make it subtract by adding: and matter of fact,

如果你相加的數字的和大於最大可能出現的數字，結果便會歸零

my friend Hank Green took apart an adding machine and that’s exactly how it subtracts

這種現象在計算機領域被稱為“上溢”，在數學裡成為“模算法“

- by adding!

但最重要的是，通過相加來使一個數字歸零意味著正的負數是可能存在的

Basically, the machine adds numbers by rotating numbered wheels, but there aren’t infinitely

因為，例如，那個與3相加等於0的數正是－3

many wheels so if you add up past the maximum possible number, you get back to zero – this

而在Hank的加法運算機上如果你把3和9997相加，你得到0

is called “overflow” in computing and “modular arithmetic” in mathematics.

所以可以說9997就等於－3

But most importantly, getting back to zero by adding means it’s possible to have positive

不幸的是，當你在做正常的運算時，數字的數量是無限大的，

negative numbers, since, for example, negative three is just what you add to three to get

所以你只會認為負3就只是負3.

zero, and on Hank’s adding machine if you add 3 plus 9,997.00, you get zero, so 9,997

但是如果你不介意含糊一點，你就可以在加法運算機版本的負3（既9997）的前面

is literally negative three!

加上一堆9，這麼這個數就幾乎就等於減3

Unfortunately, there are an infinite number of numbers when you’re doing regular arithmetic,

只要你的視線不要看得太遠。

so you might think that negative three is just negative three.

因為顯然999...9997加3，就等於0，幾乎

But if you’re willing to fudge a little, you can just take the adding machine’s version

順便說一下，這種相減來相加的方法也正是計算機做減法的方式

of negative three, add a bunch of 9s out in front, and when you add that to stuff it’s

它們只是在簡單得多的二進制中進行運算

BASICALLY the same as subtracting three, as long as you don’t look too far.

這叫 "subtracting using the two's complement"，如果你想額外了解一下的話。

Because, you know, nine gadgillion nine hundred and ninety bajillion nine hundred and ninety

seven plus three, is zero.

Almost.

By the way, this method of subtracting by adding is how computers subtract as well,

they just do it in binary which makes it a lot easier.

It's called "subtracting using the two's complement," if

you want to

look it up.