.333~ is not 1/3. The two differ by an infinitesimal. An infinitesimal is a number so small that it may as well be 0. For practical purposes you can treat .333~ as the limit of the sequence (.3, .33, .333, …) which is equal to 1/3. Consequently you can say that .333~ is equal to 1/3 because it is defined as equal to it. There are situations where infinitesimals matter, such as probability theory and integration, but those situations aren’t common and aren’t something most people would ever care about.
But, 0.999… (repetend) does equal 1. Let’s see if I remember how to prove it…
Let 0.999… (repetend) = x.
Then multiply both sides by 10:
9.999… (repetend) = 10x.
Subtract the first equality from this one. You get:
9.000… (repetend) = 9x.
Divide by 9:
x = 1
It’s a nice proof, but it depends on the assumption that 9.999~ has one more digit than 0.999~. Otherwise the subtraction gives you 9.0 minus an infinitesimal. Not that this comment is meaningful, because you can’t talk about what happens at the end of something that never ends.
I’m not sure this assumption is correct, 1/3 = 0.333 …, and none other. Every real number can be expressed as a decimal or a fraction. So perhaps you could explain what the decimal notation of the number 1/3 is then we can go from there.
The only meaning any mathematical expression ever has is the meaning it’s defined as having. If 0.333~ is defined as 1/3, then it is 1/3. Your first statement is therefore incorrect.
Right. It’s not like one person thought of .333… and then tried to figure out what it represents. It was “born” representing 1/3.
Note that there are cases where two things that are seemingly the same are difficult to show are/are not different. For example the Continuum Hypothesis.
It seem really obvious at first glance that the power set of the integers equals Aleph-1. But that hasn’t been proven. In fact, we can’t prove it’s true (or false) without making additional assumptions (axioms) concerning standard Set Theory.
But .333… was worked out from creating the decimal expansion of 1/3.
Also, once you allow infinitesimals into a number system an incredible number of complications arise that make life way too difficult for no gain in everyday Arithmetic.
RED I don’t know what you mean here: do you mean that every real number can be expressed as a decimal AND as a fraction? Because that’s wrong – and if you mean every real number can be expressed EITHER as a decimal OR as a fraction (OR *perhaps *both), then GREEN does not follow from that.
I used the word “or” in my post, though I don’t think using “and” is wrong. I’m using the closure axiom here, any real number multiplied by another real number results in yet another real number, no exceptions.
What I mean here is that for any given real number, we can write it down in a several of different ways, all of which represents that real number: fractions, decimals, logarithms, determinants, the list goes on. So my question to the OP is how is the real number 1/3 expressed in decimal notation if not 0.333…?
How are you defining a fraction? Obviously any number can be 1/n or n/1. That is a valid fraction for any real number. What is not included in your definition?
I took it to be implicit from watchwolf’s comments that in that context *fractions *meant rational fractions. Maybe that’s not what they meant, but I can’t read it any other way.
I kinda figured … my apologies for being so confusing …
One definition of a rational number is any number than can be expressed as the fraction p/q where p and q are integers and q≠0.
However, if p and/or q (and q≠0) can be a real number, and any real number can be expressed this way. For example e/7 is an irrational but real number.
He said “real numbers” so that was the way I took it. That’s also what you quoted and highlighted in red and disagreed with. So I’m still wondering how you got rational fractions out of that.
While any number x can be expressed as x/1 or 2x/2 or whatever, that doesn’t give you a way to express real numbers, because before you can write that fraction, you have to already have a way of expressing the number. It only makes sense to say that you can “express a number as a fraction” if the numerator and denominator of the fraction are restricted to be some simpler sort of number that you already have some means of expressing (like the rational numbers being expressed as fractions of integers).
This “building up” of numbers is a good thing to understand.
First you start with the whole numbers, then the integers, then the rational numbers then the reals and then the complex numbers if so inclined.
All of these steps are fairly obvious except getting to the reals. My favorite way of defining them is via Dedekind cuts but Cauchy sequences might be more appropriate for people like the OP.
These types of constructions force .333… to represent 1/3. No wheedling around it.
There are a zillion ways to constructs sets of numbers. For example, surreal numbers. But doing arithmetic in those is worse than Roman numerals. And they have limited applications (and those can be done using other systems).
Notice that few people have heard of these other systems. Why? They’re not helpful for mundane calculations. Decimal forms are.
So you took, “Every real number can be expressed as a decimal or a fraction” to mean: “Every real number can be expressed as a decimal or a fraction of real numbers”.
I took it to mean: “Every real number can be expressed as a decimal or a fraction of whole numbers”.
After considering your arguments, I stand by my reading, because (with all due deference to you, as someone with considerable capacity in mathematics) your reading makes no sense.
Plus what Chronos said.
When the OP says “For practical purposes you can treat .333~ as the limit of the sequence (.3, .33, .333, …) which is equal to 1/3. Consequently you can say that .333~ is equal to 1/3 because it is defined as equal to it,” I have to wonder what other definition of .333` he’s thinking of where it’s not equal.
Not that this comment is meaningful, because you can’t talk about what happens at the end of something that never ends.