it is at this juncture I feel morally obliged to point out

[silece]

that the real numbers R are not, in fact, one dimentional, and are not, in fact, scalars.

They are vectors, having two dimentions each, both a magnitude and direction.

|-------0-------|
-n +n
magnitude, n
direction, -/+

Yes, I understand the definition of vector applies to everything under consideration except the Real numbers,

Comments

Re: well... (of course this is the time I get busy at work)

[silece]

ordinary numbers are called "scalars" and have only one component to them, their size, or magnitude, for example the size of the number 5 is "5" (duh, yes, but carrying on...)

Now, there are types of 'numbers' called vectors, which have two components, a size and (usually) a direction. Velocity is a vector, and used properly, it will always be something like "5km/h east", as opposed to "speed" which is a scalar and will be something like "10km/h". It's useful if we're air traffic controllers or something and have to mess around with them in equations, so we give them names like V or P instead of always saying "900km/h due north".

That's why we need to know it has two components, so when we can work out how far away the bodies will land if two planes crash mid-air, we find the size of the resultant vector and start walking.

The important things there were the definitions. Scalars have one value, and vectors have two: magnitude and direction. This is how we are taught.

Now, my problem is that the examples of scalars we are given, ordinary numbers, ie. 2, 5, -47, are in fact comprised of two components themselves.

Take -47 for example. It has a magnitude, "47", and a direction, "-". It's not a direction in terms of compass points, but the sign tells us which side of 0 we're on. There are two "47"s on the number line, -47 and +47, both 47 units away from 0, but both in opposite directions.

I'm reasonably sure everyone would tell me I'm wrong, but I think that's only because it's convenient.