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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, 
which is neither irrational nor an integer. Please help me help my 

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 

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   
Associated Topics:
High School Exponents
High School Number Theory

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