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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

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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