如何證明1+1=2

Proof that 1 + 1 = 2

Date: 06/10/99 at 00:03:59
From: mike
Subject: Need the math proof for 1 + 1 = 2

My wife the math teacher wants the mathematical proof for

1 + 1 = 2

I have spent quite a bit of time looking for it on the Web - help!



Date: 06/10/99 at 10:15:58
From: Doctor Rob
Subject: Re: Need the math proof for 1 + 1 = 2

The proof starts from the Peano Postulates, which define the natural
numbers N. N is the smallest set satisfying these postulates:

P1. 1 is in N.
P2. If x is in N, then its "successor" x' is in N.
P3. There is no x such that x' = 1.
P4. If x isn't 1, then there is a y in N such that y' = x.
P5. If S is a subset of N, 1 is in S, and the implication
(x in S => x' in S) holds, then S = N.

Then you have to define addition recursively:
Def: Let a and b be in N. If b = 1, then define a + b = a'
(using P1 and P2). If b isn't 1, then let c' = b, with c in N
(using P4), and define a + b = (a + c)'.

Then you have to define 2:
Def: 2 = 1'

2 is in N by P1, P2, and the definition of 2.

Theorem: 1 + 1 = 2

Proof: Use the first part of the definition of + with a = b = 1.
Then 1 + 1 = 1' = 2 Q.E.D.

Note: There is an alternate formulation of the Peano Postulates which
replaces 1 with 0 in P1, P3, P4, and P5. Then you have to change the
definition of addition to this:
Def: Let a and b be in N. If b = 0, then define a + b = a.
If b isn't 0, then let c' = b, with c in N, and define
a + b = (a + c)'.

You also have to define 1 = 0', and 2 = 1'. Then the proof of the
Theorem above is a little different:

Proof: Use the second part of the definition of + first:
1 + 1 = (1 + 0)'
Now use the first part of the definition of + on the sum in
parentheses: 1 + 1 = (1)' = 1' = 2 Q.E.D.

- Doctor Rob, The Math Forum

2014-08-18 07:47:31 補充：
畫圈圈來表示:

o+o=oo

對! 单是出題 1+1 = 2 係未完全的.祇係一般當佢十進制啫.

其實在二進制係=10.....(電腦, 計算機, 太空科學, 導弹,衛星等等, 都係二進制)

尚有三進制,四進進制....等等.


就如在某一地方, 問你一蚊買到幾多野一樣, 美金同港纸就差一大截!

其實1+1唔一定=2，你聰明你知我講乜。

I know that everyone of you are interesting in proving 1+1=2, but, as a professional and an experienced teacher in mathematics, I must tell you the proving is quiet challenging that it does not work by yourself as you are a student or do not very get familarise in all those mathematic area. In fact, the proving of 1+1=2 combines of several topics like trigonometry, calculus, and so on, simply say you cannot attain this proving at the moment.

呢條問題唔係第一次有人問。

每次都可以反問問的人，你如何定義「２」？

2014-08-17 23:18:49 補充：
如果問的人無法定義２，即是題目也不成立，那任憑你用什麼方法去所謂「證明」都是徒勞無功。

都是等發問者告訴清楚以下的定義：
✿ 1 這個數字的意思
✿ 2 這個數字的意思
✿ + 這個符號的意思
✿ = 這個符號的意思

然後才回答吧。

http://mathforum.org/library/drmath/view/51551.html

2014-08-17 23:09:25 補充：
http://www.tngs.tn.edu.tw/teaching/math/research/1%2B1=2.pdf

2014-08-17 23:27:24 補充：
某程度上，

1+1=2可以話係定義黎