Prove that irrational*non-zero rational number is equal to an irrational number

How can you prove that irrational*non-zero rational number is equal to an irrational number?

I suspect this is a university student asking me this.

One important step in how to make proofs is to understand clearly what you're asked to prove. It often involves statements that are true for ALL numbers in a certain set (such as all real numbers, all rational numbers etc.).

This one does not include the word "all" nor the word "any", but it still is of that type. I can reword it like this:

Prove that any irrational number multiplied by any non-zero rational number is equal to some irrational number.


Or, this way:

Prove that for all irrational numbers x and for all rational numbers y excluding 0 the following is true: xy is an irrational number.


This question basically ASKS for indirect proof.

In indirect proof, we assume the OPPOSITE is true, and show that would lead to a contradiction.

We are supposed to prove that if you take ANY irrational number times ANY non-zero rational number, then the result is an irrational number.

So assume the opposite: assume there is SOME irrational number X and SOME non-zero rational number R so that when you multiply those two, you get a RATIONAL one as an answer:

XR = S and S is rational


Well, from this equation X = S/R (remember R is not zero so this is legal). But then X would be a rational number (S) divided by another rational number (R).

You probably already know that if you divide a rational number by a rational number, the result is going to be a rational number - never an irrational number, so this is the contradiction needed for the proof. ALL DONE!

But just for a reminder: a rational number is like fraction: a/b where a and b are integers, b nonzero. So if you take a/b ÷ c/d, you know how that is done: multiply by d/c and it produces another fraction (rational number).

Tags: , ,

Comments

Maria Miller said…
First of all, if you're trying to prove something by contradiction, you need to assume the opposite of what you're trying to prove. If you're trying to rpove that something is irrational, then assume the opposite: that it is rational.

Then also:
r = p/x; if p and x are irrational, their quotient CAN be rational. For example:

2Pi / Pi = 2.
8:21 PM, April 20, 2006
Anonymous said…
Thank you for your post, it was very helpful to some Stanford student
10:53 PM, April 11, 2010
Anonymous said…
It's helpful, but I have a question. Do I need to prove that if X and M/N is rational that X/(M/N) is rational, or I can just use it?
10:49 PM, June 23, 2011
Maria Miller said…
If two numbers are rational, then their quotient is also rational. Most likely you can use that without proof--it would be assumed to have already been proven previously.
11:07 PM, June 23, 2011
Difference between Rational and Irrational Number said…
See if your trying to prove something by contradiction process then have to assume opposite of what you actual want means if you are trying to prove something rational then you have to assume it as a irrational number.
5:05 AM, February 15, 2012
Maria Miller said…
I am trying to prove something irrational, therefore for a proof by contradiction I need to assume it is rational.
8:09 AM, February 15, 2012
Anonymous said…
this link talking about irrational number

http://www.math-worksheets.co.uk/048-tmd-what-are-irrational-numbers/
2:03 PM, February 15, 2013
Post a Comment

Popular posts from this blog

Conversion chart for measuring units

Image
I was just sent a link to this site; all it is, is a handy one-page printable conversion chart for various US measures, metric measures, and US vs. metric measures. Includes even a comparative Fahrenheit vs. Celsius thermometer. http://metricconversioncharts.org/
Read more

Meaning of factors in multiplication: four groups of 2, or 4 taken two times?

Someone asked me about the interpretation or meaning of the two numbers in multiplication, such as 4 × 2. Do you view it as four groups of 2 , or as four, taken two times ? Here's their question: "...I noticed that your explanation about how to interpret a math phrase, such as 4 × 2, is different from how I have understood it and taught it to my older daughter who was using RightStart Math at this level. Let me explain:  I see that you are using the first factor to denote number of groups and the second factor tells how many are in each group (4 × 2 would be interpreted 4 groups of 2).  On the other hand, I have taught and thought of it reversed, with the first factor telling how many are in each group and the second factor telling you how many groups (4 × 2 would be 4 taken 2 times).  I know that the difference in explanations does not ultimately change the product, but I am wondering what you thoughts are.  When I conceptually picture what I'm doing I see i...
Read more

Geometric art project: seven-circle flower design

Image
I got inspired by the blogpost Art School | Geometric Design with Islamic Art where Deborah shows how to make a neat flower design with seven circles , using a compass, and then coloring it using 2, 3, or 4 colors (or however many of your own). I think it ties in neatly with mathematics, and lets students practice drawing circles with a compass.  My girls loved the art/math project. Here are pictures they made: Here are step-by-step instructions for the flower design: 1.  Draw a line and a circle so that the circle's center point is on the line. Then mark the points where this circle intersects the line. 2. Use those points as center points, and draw two more circles. The radius is the same all the time, so make sure you don't change it on your compass!We used 5 cm as the radius, and that made the whole design fit neatly on a regular letter size paper. 3. Now note the two points marked in the picture. They will be used as center points in the next...
Read more