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Decimal Square Roots

Date: 10/19/98 at 19:44:43
From: Ron Flitcraft
Subject: Decimal answers in square roots

Dr. Math,

Recently my Algebra 2 teacher stated in front of the class that
decimal points cannot be the answers to square roots. He said that any
decimal point that is the answer for a square root will be an
approximation. I doubted it and put a few numbers into my calculator.
When my friend and I found a couple of numbers that didn't fill the
number of the calculator space completely, our Algebra teacher admitted
he was wrong.

A couple of days later he put those same numbers in and multiplied them
by themselves. They came close to but were not the exact number -
omething like 46.999999999. I have devoted myself to finding a square
root that comes out to a decimal point for an answer and checks when it
is squared. Is there such a number? I have tried for quite some time

Date: 10/19/98 at 20:01:25
From: Doctor Sam
Subject: Re: Decimal answers in square roots

Ron,

A very interesting question, but I'm not exactly sure what you mean.
I am guessing that by "a decimal point" you mean a terminating decimal
like .6 rather than a decimal with infinitely many digits. (Do you see
my confusion? All decimals have "decimal points" but some decimals
have a fixed number of digits to the right of the decimal point while
others do not.)

If you mean "a fixed number of decimal digits" then there are many.
Here is an easy way to create such a number: pick a decimal and square
it. So for example, (1.5)^2 = 2.25 and the square root of 2.25 is
exactly 1.5

But maybe you mean that you want to take the square root of a whole
number, not a number with decimal digits to the right of its decimal
point. If that is what you mean, then your teacher is correct: the only
whole numbers that have a perfect and exact square root are numbers
that are perfect squares, like 4, 9, 16, 25, 49. Any whole number that
is not a perfect square will have infinitely many decimal digits in its
square root. Your calculator can only display 9 or 10 digits and so
rounds the answer. This gives a very good approximation of the square
root, but it is only an approximation.

For example, my calculator gives the square root of 2 as 1.414213562.
Now if I enter (square root of 2)^2 the calculator gives 2. But if I
take the displayed answer and square it I get:

(1.414213562)^2 = 1.99999999999

Whole numbers either have whole number square roots or their square
roots are irrational. An irrational number has infinitely many digits
in its decimal expansion, so no matter how accurate your calculator or
computer is it will never be able to compute all those digits and will
give you only an approximation.

I hope that helps.

- Doctor Sam, The Math Forum
http://mathforum.org/dr.math/

Associated Topics:
Middle School Square Roots

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 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

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The text is a question and answer exchange from the Dr. Math forum, where a student asks about the possibility of decimal numbers being exact square roots, and a math doctor responds with an explanation.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
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Decimal Square Roots

Date: 10/19/98 at 19:44:43
From: Ron Flitcraft
Subject: Decimal answers in square roots

Dr. Math,

Recently my Algebra 2 teacher stated in front of the class that
decimal points cannot be the answers to square roots. He said that any
decimal point that is the answer for a square root will be an
approximation. I doubted it and put a few numbers into my calculator.
When my friend and I found a couple of numbers that didn't fill the
number of the calculator space completely, our Algebra teacher admitted
he was wrong.

A couple of days later he put those same numbers in and multiplied them
by themselves. They came close to but were not the exact number -
omething like 46.999999999. I have devoted myself to finding a square
root that comes out to a decimal point for an answer and checks when it
is squared. Is there such a number? I have tried for quite some time

Date: 10/19/98 at 20:01:25
From: Doctor Sam
Subject: Re: Decimal answers in square roots

Ron,

A very interesting question, but I'm not exactly sure what you mean.
I am guessing that by "a decimal point" you mean a terminating decimal
like .6 rather than a decimal with infinitely many digits. (Do you see
my confusion? All decimals have "decimal points" but some decimals
have a fixed number of digits to the right of the decimal point while
others do not.)

If you mean "a fixed number of decimal digits" then there are many.
Here is an easy way to create such a number: pick a decimal and square
it. So for example, (1.5)^2 = 2.25 and the square root of 2.25 is
exactly 1.5

But maybe you mean that you want to take the square root of a whole
number, not a number with decimal digits to the right of its decimal
point. If that is what you mean, then your teacher is correct: the only
whole numbers that have a perfect and exact square root are numbers
that are perfect squares, like 4, 9, 16, 25, 49. Any whole number that
is not a perfect square will have infinitely many decimal digits in its
square root. Your calculator can only display 9 or 10 digits and so
rounds the answer. This gives a very good approximation of the square
root, but it is only an approximation.

For example, my calculator gives the square root of 2 as 1.414213562.
Now if I enter (square root of 2)^2 the calculator gives 2. But if I
take the displayed answer and square it I get:

(1.414213562)^2 = 1.99999999999

Whole numbers either have whole number square roots or their square
roots are irrational. An irrational number has infinitely many digits
in its decimal expansion, so no matter how accurate your calculator or
computer is it will never be able to compute all those digits and will
give you only an approximation.

I hope that helps.

- Doctor Sam, The Math Forum
http://mathforum.org/dr.math/

Associated Topics:
Middle School Square Roots

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search
user
The text is a question and answer exchange from the Dr. Math forum, where a student asks about the possibility of decimal numbers being exact square roots, and a math doctor responds with an explanation.

TYPE: SINGLE_QUESTION_WITH_ANSWER