Does Solving x - √2 = 0 Imply x = ±√2?

[PeterO]

The integer +5 (the plus sign is unnecessary) and the natural number 5 can both be found at exactly the same place on the real number line.

I have no idea what "closure under involution" means. This didn't come up in any of the numerous math classes I took.

Sorry, I meant evolution, not involution - neither term in common usage.

Natural numbers are closed under addition [and of course multiplication which is just repeated addition].
ie, select any two natural numbers, add them, and the answer is a natural number.

Subtraction - even difference - is not closed, because if you happen to choose the same number twice, or insist on subtracting your second choice from your first choice, the answer is not always a Natural number. 8 - 3 = 5 ; 6 - 6 = ? ; 4 - 7 = ?

If we expand our thinking to include integers, we again have a closed set of numbers under subtraction as well s addition and multpilication.

But what of division? For closure, starting with Natural numbers, we need fractions, but if we use our newly developed number system we need Rationals - and the condition that the denominator is not 0 [not a problem with Natural numbers]

And so we have a number system closed under the four basic operations addition, subtraction, multiplication and division.

The set is also closed under involution since we can raise any rational to a power, and get another [or the same in the case of +1] rational.

However evolution [taking the root] does not work out for all numbers: √3 for example.

If we expand our system to real numbers, we have solved half the problem - but need to go further to complex numbers for taking even the second root of a negative number.

We then have a system closed under addition, subtraction, multiplication, division, involution and evolution.

No doubt someone will find [or already has] some operation where even that set of numbers is not closed.

 

[jgens]

I am fairly certain that Mark understands what closure means. He was confused by your use of the term involution, since you are using it in a completely non-standard way. This wikipedia page has a more common usage of the term involution: http://en.wikipedia.org/wiki/Involution_(mathematics )

Also, if you want closure under nth roots, then the full complex number system is actually a much stronger number system than you actually need.