why 2+2 and 2*2 = 4

[symoore22]

Hi,

I remember a maths class at school where the teacher spent an entire lesson prooving why 2+2 and 2*2 equals 4 (it was some long formula that took her an hour to write down and explain). Can anyone point me in the direction of where i can find this formula or perhaps its name.

many thanks

simon

[Logophobic]

I don't know, but I'm going to take a wild guess that it might have been something along these lines:

1 + 1 = 2
2 + 1 = 3
3 + 1 = 4

2 + 2
= (1 + 1) + (1 + 1)
= 1 + 1 + 1 + 1
= 2 + 1 + 1
= 3 + 1
= 4

2 * 2
= (1 + 1) * (1 + 1)
= 1 * (1 + 1) + 1 * (1 + 1)
= 1 + 1 + 1 + 1
= 2 + 1 + 1
= 3 + 1
= 4

or maybe just this:
2 * 2 = 2 * (1 + 1) = 2 + 2

[Code Doc]

Logophobic has the solution.

However, try proving that 2 is the only number other than 0 in the universe that has this characteristic. Any takers out there?

[zaza]

Sorry, is this homework?


a+b = ab

=> b = ab - a = a (b-1)
=> a = b / (b-1)

If a is an integer, then b / (b-1) is an integer.
The only way that this can be true is if b = 0 or if b divides exactly by b-1. The only non-zero value for which this is true is b=2.


If you don't mean for them to be integers, but actually numbers, then any pair such that a = b/(b-1) will do.


If you mean "where sum = product", then 1+2+3 = 1*2*3.




zaza



Edit: if you want to prove that b=2 is the only non-zero value, then let b=2+c. Then 2+c = a+ac and c = 2-a/(a-1). If a<0 then c<0 and b<2. If a=0, c=-2 and b=0. If a=1, then c is undefined. If a=2 then c=0 and b=2. If a>2 then c<0 and b<2.

[eranga262154]

2 + 2
= (1 + 1) + (1 + 1)
= 1 + 1 + 1 + 1
= 2 + 1 + 1
= 3 + 1
= 4

2 * 2
= (1 + 1) * (1 + 1)
= 1 * (1 + 1) + 1 * (1 + 1)
= 1 + 1 + 1 + 1
= 2 + 1 + 1
= 3 + 1
= 4

I think this make sense. Isn't it....

Beacse at the end you can found (1 + 1) + (1 + 1). That is same result.