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### Perfect Squares and Irrational Numbers

```
Date: 02/13/2002 at 21:11:18
From: Mimi O'Shea
Subject: Perfect square - definition

My class is confused about the definition of a "perfect square."
Generally, I see it defined as a number whose square root is an
integer. On the other hand, isn't any non-perfect square an
irrational number? What is the number 0.49? Its square root is 0.7,
students.
```

```
Date: 02/13/2002 at 23:23:27
From: Doctor Peterson
Subject: Re: Perfect square - definition

Hi, Mimi.

The word "number" in this context means a whole number (or more
generally, an integer). So a perfect square is the square of an
integer.

It is the square roots of INTEGERS that are not perfect squares that
are always irrational. As you point out, the square of any rational
number is a rational number that has a rational square root, so it is
not true that any NUMBER that is not a perfect square has an
irrational square root.

You might want to point out to your students that this illustrates why
mathematicians are so picky about definitions (more so than many
textbook authors, unfortunately). It is so easy to give a quick
statement that is true in the context in which we are currently
thinking (say, whole numbers) and leave out part of the definition
that we just assume you understand. I'm sure I give off-the-cuff
definitions or theorems in my e-mail answers that are lacking in
detail; but a textbook owes you the complete version. So don't
complain when the next definition you read seems too complicated;
if it is written right, every word counts! (I suppose math is like
poetry in that sense ...)

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

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