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 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/
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