Rational + Irrational

Status
Not open for further replies.
mathdad

mathdad

Full Member
Joined
Apr 24, 2015
Messages
737
Let r = rational number

Let i = irrational number

Why does r + i = i?
 
mathdad said:
Let r = rational number
Let i = irrational number. Why does r + i = i?
Click to expand...
First, it is common practice to use Q (\(\displaystyle \mathbb{Q}\)) for the set of rational numbers.
If \(\displaystyle x\in\mathbb{R}\) i.e. \(\displaystyle x\) is a real number & \(\displaystyle x\notin\mathbb{Q}\) the we say that \(\displaystyle x\) is irrational, [less common is \(\displaystyle x\in\mathscr{I}].\).
Say \(\displaystyle a\in\mathbb{Q}~\&~s\in\mathscr{I}\), one rational the other irrational.
If \(\displaystyle a+s\) were rational, lets say \(\displaystyle a+s=z\in\mathbb{Q}\) then \(\displaystyle s=z-a\).
However the difference of two rational numbers is rational. BUT \(\displaystyle s\) not rational.
 
pka said:
First, it is common practice to use Q (\(\displaystyle \mathbb{Q}\)) for the set of rational numbers.
If \(\displaystyle x\in\mathbb{R}\) i.e. \(\displaystyle x\) is a real number & \(\displaystyle x\notin\mathbb{Q}\) the we say that \(\displaystyle x\) is irrational, [less common is \(\displaystyle x\in\mathscr{I}].\).
Say \(\displaystyle a\in\mathbb{Q}~\&~s\in\mathscr{I}\), one rational the other irrational.
If \(\displaystyle a+s\) were rational, lets say \(\displaystyle a+s=z\in\mathbb{Q}\) then \(\displaystyle s=z-a\).
However the difference of two rational numbers is rational. BUT \(\displaystyle s\) not rational.
Click to expand...

Can you explain this at my level? This is what good teachers and tutors do. You know that I am reviewing math learned long ago (in my college days). The symbols used here are cool but slightly confusing to me.
 
mathdad said:
Can you explain this at my level? This is what good teachers and tutors do.
Click to expand...
No that is a false statement. In my division, the best professors were the ones who set out the first day of class what level of background was expected in the class. Samples are given out and students in the class are asked to sign that they understand the prerequisite material.
If you lack basic understanding of operation with rational numbers why are you asking questions about them?
 
pka said:
No that is a false statement. In my division, the best professors were the ones who set out the first day of class what level of background was expected in the class. Samples are given out and students in the class are asked to sign that they understand the prerequisite material.
If you lack basic understanding of operation with rational numbers why are you asking questions about them?
Click to expand...

I cannot win with you, right? It was just a simple comment.
 
mathdad said:
Let r = rational number
Let i = irrational number
Why does r + i = i?
Click to expand...
Here's an answer which is more of a visual demonstration than a rigorous explanation. But first, we need to remember the following patterns about decimal forms of rational and irrational numbers.

The decimal form of a rational number always ends with a repeating pattern of digits. Here are some examples:

12/5 = 2.4000... (repeats zeros forever)

4/9 = 0.4444... (repeats fours)

45/7 = 6.428571428571... (repeats group of digits)

All rational numbers show a repeating form, when written as a decimal number. On the other hand, irrational numbers written in decimal form never show a repeating pattern:

Pi = 3.141592653589... (digits continue without pattern)

e = 2.718281828459045... (no pattern)

Now, let's add a rational and an irrational: 12/5 + Pi

2.400000000000...
3.141592653589...
-------------------------------
5.541592653589...

Can you agree that adding a random string of digits to a repeating pattern of digits will never result in a repeating pattern? Such a sum will never represent a rational number.

?
 
tkhunny said:
Is it more important to win or more important to learn. Philosophy matters. Attitude matters.
Click to expand...

I am just tired of getting criticized for posting math questions, easy and not so easy, in a MATH SITE.
 
Otis said:
Here's an answer which is more of a visual demonstration than a rigorous explanation. But first, we need to remember the following patterns about decimal forms of rational and irrational numbers.

The decimal form of a rational number always ends with a repeating pattern of digits. Here are some examples:

12/5 = 2.4000... (repeats zeros forever)

4/9 = 0.4444... (repeats fours)

45/7 = 6.428571428571... (repeats group of digits)

All rational numbers show a repeating form, when written as a decimal number. On the other hand, irrational numbers written in decimal form never show a repeating pattern:

Pi = 3.141592653589... (digits continue without pattern)

e = 2.718281828459045... (no pattern)

Now, let's add a rational and an irrational: 12/5 + Pi

2.400000000000...
3.141592653589...
-------------------------------
5.541592653589...

Can you agree that adding a random string of digits to a repeating pattern of digits will never result in a repeating pattern? Such a sum will never represent a rational number.

?
Click to expand...

This is exactly what I was looking for. Thank you.
 
mathdad said:
... exactly what I was looking for ...
Click to expand...
Is your textbook talking about closure properties for sets of numbers? (That's what pka was getting at.) I'm not sure what prompted the question.

\(\;\)
 
Otis said:
Is your textbook talking about closure properties for sets of numbers? (That's what pka was getting at.) I'm not sure what prompted the question.

\(\;\)
Click to expand...

Boy, am I ever sorry for posting this thread.
 
mathdad said:
I am just tired of getting criticized for posting math questions, easy and not so easy, in a MATH SITE.
Click to expand...
"criticize" doesn't have to be pejorative. If you seek help from volunteer mathematics professionals, why not accept the help you receive - no matter how you choose to frame it at first glance? Take it at face value and learn from it. Simple.
 
tkhunny said:
"criticize" doesn't have to be pejorative. If you seek help from volunteer mathematics professionals, why not accept the help you receive - no matter how you choose to frame it at first glance? Take it at face value and learn from it. Simple.
Click to expand...

I am beyond this post. Moved on to other more exciting questions.
 
This is the thing with volunteers. If you wish to control the engagement, you should consider paying for it. If you wish simply to learn, then one would do well simply to choose to listen. It is a pleasure top help the honest student who has a true desire to learn. It is the nature of volunteers.
 
If the question comes from your algebra review, then my visual demonstration is probably not adequate. That is, there's a specific lesson to be learned about the properties of the set of rational numbers, if that's what motivates the question.

?
 
Otis said:
If the question comes from your algebra review, then my visual demonstration is probably not adequate. That is, there's a specific lesson to be learned about the properties of the set of rational numbers, if that's what motivates the question.

?
Click to expand...

The author discusses the world of real numbers in the early pages of the textbook. He simply made a comment about adding a rational number with an irrational. My post stems from that ONE COMMENT by Michael Sullivan.
 
mathdad said:
Can you explain this at my level? This is what good teachers and tutors do.
Click to expand...

This is not a teaching site. Good forum users will realize that. Tutors are also
coaches. And good math tutors as coaches will advise, recommend, or tell you that
you need to study more before posting in a particularmy advanced topic, that you already should
have known a prior relatively basic topic, and/or that your topic is non-math related and belongs elsewhere, etc.
 
Last edited:
Status
Not open for further replies.
Top