1/3 cannot be expressed in decimal form?

[Holocene]

Somome told me this. So, this is wrong:

\displaystyle{\frac{1}{3} = 0.\overline{3}}

?

 

[Dragonfall]

Sure it can. What you wrote is the decimal form of 1/3.

 

[Holocene]

Sure it can. What you wrote is the decimal form of 1/3.

It seems so obvious that it can be. But someone claimed "it's not allowed in college algebra"...

 

[tony873004]

I did that once on a homework assignment: wrote 0.3 with a bar over the 3 instead of writing 1/3. My teacher, who had a great sense of humor, pointed it out to the class, explaining that the bar prevented me from using an infinite amount of paper to express my answer.

 

[epkid08]

Doesn't that go against the definition of a rational number?

 

[Defennder]

This appears to be yet another question on 0.999...=1 in disguise.

 

[Holocene]

Doesn't that go against the definition of a rational number?

pi isn't rational, but that doesn't stop us from using it in algebra?

 

[epkid08]

I wasn't talking about using irrational numbers in algebra...

I was saying that a number that can be expressed as a simplified fraction, can also be expressed as a repeating decimal. That's one of the definitions of a rational number.

 

[Holocene]

I wasn't talking about using irrational numbers in algebra...

I was saying that a number that can be expressed as a simplified fraction, can also be expressed as a repeating decimal. That's one of the definitions of a rational number.

My mistake. I now read your original post as advocating that, yes, you can in fact represent 1/3 in decimal form.

 

[arildno]

You cannot write 1/3 in a FINITE series of decimals.

That does not prohibit, you, however, from inventing a symbol to signify an INFINITE series of decimals, by which 1/3 might certainly be written down.

 

[HallsofIvy]

Doesn't that go against the definition of a rational number?

No, it doesn't. A rational number is one that can be represented as a fraction. It doesn't mean that it cannot also be represented in other forms also.

 

[Ignea_unda]

Was the person who told you this trying to tell you that 1/3 could not be expressed in a finite decimal? That would turn out to be a true statement. Otherwise I agree with all the above posts.

 

[epkid08]

No, it doesn't. A rational number is one that can be represented as a fraction. It doesn't mean that it cannot also be represented in other forms also.

I don't understand your post. You're saying I'm wrong, but you're also agreeing with me?

 

[arildno]

I don't understand your post. You're saying I'm wrong, but you're also agreeing with me?

HallsofIvy responded to this:

Doesn't that go against the definition of a rational number?

His comment is entirely explicable due to the complete paucity of content in that post!

By an oversight, he didn't identify this with you:

I wasn't talking about using irrational numbers in algebra...

I was saying that a number that can be expressed as a simplified fraction, can also be expressed as a repeating decimal. That's one of the definitions of a rational number.

This is NOT what you said in your first post, it is impossible to deduce the contents of this post from the first one, so given HoI's oversight, how should he have understood what your first post meant?

 

[epkid08]

My first comment was directed at your initial post, "1/3 cannot be expressed in decimal form." I responded, "[If 1/3 is not equal to .3~] Wouldn't that go against the definition of a rational number?" Which it does. How are people confused by this?

 

[uart]

Don't worry epkid08. I realized that your first reply was a direct response to the original poster and took it the way that I'm sure you meant it to be interpreted.

Some people (quite understandably) thought that you were responding to the immediately previous reply and that made it look a little like you were saying the opposite of what I (and I'm sure others) understood you to be saying.

 

[yenchin]

Might be a good idea to quote which ever post you want to reply to...

 

[epkid08]

Might be a good idea to quote which ever post you want to reply to...

Who are you talking to? how ironic

 

[uman]

This is off-topic, but I've always wondered why many people who refuse to believe that 1 = 0.9... readily believe that 1/3 = 0.3...

 

[sketchtrack]

Reminds me of 1/2+1/4+1/8+1/16+1/32...=1

 

[yenchin]

Who are you talking to? how ironic

No one in particular. A general comment.

 

[Ignea_unda]

This is off-topic, but I've always wondered why many people who refuse to believe that 1 = 0.9... readily believe that 1/3 = 0.3...

Please clarify that 1 = 0.9...

I could see if you are rounding but I do not see those to be equal by definition.

 

[LukeD]

Please clarify that 1 = 0.9...

I could see if you are rounding but I do not see those to be equal by definition.

That's "1 = 0.9999..." and they are equal by definition
but most people have a very fuzzy notion of what a real number is, so few people have no idea how to interpret 0.9999... correctly

 

[rock.freak667]

Reminds me of 1/2+1/4+1/8+1/16+1/32...=1

Reminds me of a thread I once posted where I had to describe the series of 0.3,0.33,0.333,0.333, ... and all I needed to do was to write it out as a geometric progression

Please clarify that 1 = 0.9...

I could see if you are rounding but I do not see those to be equal by definition.

There's also a thread on why 0.9999...=1 as well.
Fun way (for me) to show it was to say that there is no number,a, such that 0.999999999999...<a<1. Those were some strange times.

 

[uman]

Idnea_unda: 0.9... is just a shorthand way of writing \lim_{n\to\infty}\sum_{k=1}^n \frac{9}{10^k}, which clearly equals one.