John Gabriel <firstname.lastname@example.org> wrote in news:email@example.com:
> Construction of rational numbers: > > 1. A magnitude is the idea of size of extent.
So it is just an idea
> We can either tell that > two magnitudes are equal or not.
Lets say I have a tree in my back yard. You have a tree in your back yard.
How do we tell if your tree has the same idea of size of extent of height as my tree does
> If we can tell they are not equal, > then we know which is smaller or bigger, but we can't tell how much > bigger or smaller. This is called qualitative measurement (without > numbers).
There is a ladder attached the the ground. There is a wall near it. How do you tell if the length of the ladder is the same magnitutde of length as the height of the wall?
> From this concept, we only know about size. There are no numbers yet. > We have a notion of smaller, bigger and equal, but all these notions > are vague at this stage.
> 2. We can form ratios of magnitudes. AB : CD where AB and CD are line > segments. The expression AB : CD means the comparison of magnitudes AB > and CD.
What is AB:CD .. what does it mean? You haven't defined it. What is a ratio accord to you? Is it the same as a number? If so, where did numbers come from. If not, how do you create numbers?
> At this stage we begin to compare magnitudes. All we can do is tell > that magnitudes are equal or not. If not equal, we can only tell if > smaller or greater. Still no numbers.
So we're in the same position as at point 1. Point 2 hasn't gotten us anywhere
> 3. A ratio of equal magnitudes, say AB : AB or CD : CD allows us to > use either as the standard of measurement, that is, the unit.
So a ratio of two equal magnitudes allows us to use either (which must mean on of the two magnitudes, as there is only one ratio) as a unit. So we can use AB as a unit. Or CD as a unit.
> The unit > is a ratio of equal magnitudes.
Hang on .. the preceding sentence says that the magnitude is a unit
> The first number is born. It is the result of comparing equal > magnitudes. A unit is the ratio k:k where k is any magnitude.
OK .. so your first sentence in point 3 was really just very sloppy and poorly expressed.
You stil haven't defined what a ratio is.
> 4. The unit enables us now to compare AB and CD
The unit is just a ratio (which is still undefined by you) of equal magnitudes. That has nothing to do with unequal magnitudes .. other than we can say that their ratio (whatever that is) is not a unit
> if both are exact > multiples of the unit that measures both.
To have multuiples you need to already have numbers. You were supposed to be constructing them, yet now you assume they exists so that you can use them as multiples
> We can now perform > quantitative measurement,
Because we already had numbers. You didn't construct them
> because we can tell how much greater AB is > than CD or how much less AB is than CD.
So magically numbers appear from nowhere to let you express magnitudes as a number and a unit. And no .. all yiou can show is different ratios. There is nothing there yet that says how MUCH less AB is than CD (say). Only that they have a certain ratio
> 2:3 or 3:2 tells us that there is a difference of 1 unit.
is 2:3 the same as 20:30, or is it a different number
> 5. Finally, if a magnitude is only part of a unit,
Hang on .. a unit is a ratio. Now you are saying a unit is made up of magnitudes?
> then we arrive at a > ratio of numbers, say AB : CD where AB and CD are multiples of the > unit.
That makes no sense .. a unit is a ratio. How can a magnitude (like AB) be a multiple of a ratio?
> AB : CD now means the comparison of numbers AB and CD.
Which means what? You need to define this. And since when are AB and CD numbers? .. they were magnitutes everywhere else previously.
And how does that relate to the use of the same name and symbol for the ratio of magnitudes (which are not numbers).
And where did these number come from? You were supposed to construct them. But you just assumed they were pre-existing.
> When we > write AB/CD, it is called a fraction.
Which means what? You need to define this. How does it relate to this idea of ratios that you talk about and do not define.
> 2:1 : 3:1 = 2/3 which is a rational number known as a proper > fraction.
So a ratio of ratios is a rational number. What about other ratios of ratios .. what method do you use to convert them to rational numbers. You only showed one example.
> 3:1 : 2:1 = 3/2 which is a rational number known as an improper > fraction.
So you say some rational numbers are are proper fractions and some are improper .. though you don't say what that means. Where is your defintion?
> So, in five steps I have derived the concept of number for you.
No .. you suddenly used numbers as pre-existing in the middle of your supposed construction.
An analogy would be "How to construct a bridge: Go to a river, you will find a bridge there that lets you get to the other side. This is how you construct a bridge"
> There > is one thing left - what happens when you can't measure a magnitude > that is not a multiple of a unit and can't be expressed exactly using > any part of a unit?
You've not even shown that yet
> This is called an incommensurable magnitude and > the best you can do is provide an approximation such as 3.14159... or > 1.414..., etc. > > > So, from these axioms, one can construct ALL the rational numbers - in > only 5 steps. :-)
Shame that that doesn't work and it is full of holes and things left undefined and ambiguous and that it doesn't construct numbers at all, rather is pre-supposes that numbers already exist.
3/10 for your homework. Please try it again and resubmit it what you when fix up the mistakes. Keep trying, you might make it one day.