Why Does (1/3)*3 = 1 and Not .9 Repeating?

[Rogerio]

...
To answer this question from the beginning:
The reason why 1/3 *3 = 1 where 0.333 (repeating) *3 is not:
is because 1/3 is perfectly rational (or simply as an elementary school kid would call it a fraction) while 0.33(repeating) is not.
...

First: there is no 'perfect' rational number - a number is rational or irrational.

And second: 0.33... is rational, since it can be represented as 2/6 , for instance.

:-)

 

[ohwilleke]

Why the hell have you introduced this spurious analogy?

I know how to make the decimals a model of the real numbers, thank you.

Why so hostile? Nothing spurious here. The best way to explain something, when one approach isn't working, it to explain it in another correct manner in that hope that useing a different approach makes the matter more clear.

Your statement to which I replied says that a matter is definitional. But, that really understates the situation. Why did somebody use a particular definition? Because no other definition makes sense to give what we expect the thing we call "real numbers" to do.

The point of my discussion was to illustrate explicitly, as okidream doesn't seem to follow (as indicated his response which seems to erroneously state that the value of a number should be independent of the base of the number system in which it is represented), why the relationship between the base ten number system and what it means to represent numbers in more than one number system, makes the definitions the imply that 0.333 repeating=1/3 the uniquely sensible choice, rather than being an arbitrary definition.

 

[matt grime]

Your analogy was completely bizarre (frames of reference in GR? In what sense can this be 'equivalent'?) and I always get hostile when people use patronizing language to explain to *me* something that is bleeding obvious, and that *I* don't need to be told, especially if they don't explain what it means to produce a model of a complete ordered field.

 

[Hurkyl]

Ok, nobody seems to be convincing anyone of anything (per usual, *sigh*), and temperatures are rising, so I'm going to close this.