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

[jgens]

That is interesting. I have never seen the term 'whole numbers' in any real math textbook. Also, I think the convention for taking N to include 0 is more recent, so that might account for some of the difference.

On an unrelated note, the logicians I have met start counting from zero. That is, textbooks start with chapter 0. When they list sequences they being at x₀. When they number properties, they start with 'property 0'

 

[Mark44]

feel free to disagree.

Natural numbers - counting numbers - are neither positive nor negative;

This is nonsense. This would imply that, for example, 5 cannot be placed anywhere on the real number line.

there is not even a zero, because if there are none, you can't count them.

We only needed positive and negative when we invented the concept of integers, in order to give closure under subtraction.

As soon as you find a negative length to measure, let me know and I will then see we may need integers for measuring, and not just natural numbers.

I already touched on this in post 23.

If we are talking about lengths of things, they are nonnegative numbers.

 

[PeterO]

This is nonsense. This would imply that, for example, 5 cannot be placed anywhere on the real number line.

I always thought Real numbers - another conceptual imagination of man to give closure under involution - had a positive and negative sub-set, so of course will included the integer +5, not the Natural Number 5.

 

[jgens]

I always thought Real numbers - another conceptual imagination of man to give closure under involution - had a positive and negative sub-set, so of course will included the integer +5, not the Natural Number 5.

1) What does involution have to do with R? The special thing about R is that it is the unique ordered field with the least upper bound property.

2) The notion of signless natural numbers is not standard. There are several reasons for this. First, we like to view the natural numbers as a subset of Z, Q, R, etc. Second, one way of defining an ordered field involves specifying a collection of positive numbers that satisfy certain properties. The standard order on N gives the natural numbers all of these properties, and so it makes sense to call them positive.

 

[Mark44]

I always thought Real numbers - another conceptual imagination of man to give closure under involution - had a positive and negative sub-set, so of course will included the integer +5, not the Natural Number 5.

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.

 

[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.