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Rational and Irrational Numbers: Multiplication, Division


Date: 10/15/2001 at 07:42:51
From: jess
Subject: Rational and irrational numbers

I really need help with irrational and rational numbers.

The main thing I would like explained is the rules for:

   irrational multiplied by irrational
   rational multipiled by rational
   irrational divided by rational

Please please can you help?
Thank you so much for your time.


Date: 10/15/2001 at 11:31:35
From: Doctor Ian
Subject: Re: Rational and irrational numbers

Hi Jess, 

Any rational number can be written as one integer divided by another, 
right?  That's the definition of a rational number. Now, suppose you 
have two of them, and you multiply them together. What happens? 

  a   c   a * c
  - * - = -----
  b   d   b * d

Now, if a and c are integers, then a*c is also an integer, right?  
Similarly for b and d. So the product of two rational numbers must be 
a rational number. 

Similar reasoning can show that dividing one rational number by 
another must also produce a rational result. (Recall that to divide by 
a fraction, you invert and multiply.)

There are no corresponding rules for two irrational numbers. For 
example, 

    __     __
  \| 2 * \| 3

is irrational, but 

    __     __
  \| 2 * \| 2

is clearly rational.

Also, 
    __
  \| 2 
 ------
    __
  \| 3

is irrational; but 

  \| 2 
 ------
    __
  \| 2

is clearly rational.


So far, here's what we know:

     *,/        rational     irrational

  rational      rational         ?
  irrational       ?           either


So what about a rational and an irrational?   

Well, suppose we multiply a rational number (a/b) by some unknown 
number (?) and end up with another rational number (p/q).  

  a       p
  - * ? = -  
  b       q

What can we say about the mystery number?  Well, from the definition 
of division, we know that 

            p
            -
            q      p   b   p * b
      ? = -----  = - * - = -----
            a      q   a   q * a
            -
            b

Now, we know that a, b, p, and q are all integers. So this says that 
if we multiply a rational number by something, and get a rational 
number, the something _must_ be a rational number.  

What this means is that if we multiply a rational by an irrational, we 
can't end up with a rational product, because to get a rational 
product, both factors have to be rational. Therefore, the product of a 
rational and an irrational must be irrational. 

If you've never seen this kind of reasoning - called 'proof by
contradiction' - before, it can seem a little dizzying.  But you may 
as well get used to it now, since you'll be seeing a lot of it in the 
years ahead. The basic idea is this: You assume that something is 
true, and then show that this assumption must lead you to conclude 
something that you know for sure is _not_ true, which means the thing 
you assumed must be false instead of true. 

This may be easier to follow if you think about something other than 
numbers. For example, suppose you're accused of a crime, but there are 
witnesses who say that you were on the other side of town when it 
happened.  

The court would reason this way: Let's assume that you committed the 
crime. Then you must have been at the scene of the crime when it was 
committed. But we know that you were somewhere else! Therefore, we 
know that you didn't commit the crime.

Similar reasoning can show that the quotient of a rational and an 
irrational - or the quotient of an irrational and a rational - must be 
irrational.  

So here is our final table:

     *,/        rational     irrational

  rational      rational     irrational
  irrational    irrational     either

Does this help?  Write back if you'd like to talk about this some 
more, or if you have any other questions. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Logic
High School Number Theory

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