# Difference between revisions of "Talk:Square root"

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::: No, it cannot be said either way. It does not help to have two contradictory definitions of the square root sign. [[User:RSchlafly|RSchlafly]] 14:35, 24 April 2007 (EDT) | ::: No, it cannot be said either way. It does not help to have two contradictory definitions of the square root sign. [[User:RSchlafly|RSchlafly]] 14:35, 24 April 2007 (EDT) | ||

+ | ::::OK, so define it as the positive root, but point out that there are a pair of numbers that can be squared to produce any given number, and that is shown by our little formula, and is necessary to preserve multiple solutions of equations when taking the square root of each side? Using the Wolfram cite below to define it? [[User:Human|Human]] 14:52, 24 April 2007 (EDT) | ||

Wolfram says: | Wolfram says: | ||

Line 52: | Line 53: | ||

I want to make sure our article won't mislead some teen studying for the SAT. --[[User:Ed Poor|Ed Poor]] 13:03, 24 April 2007 (EDT) | I want to make sure our article won't mislead some teen studying for the SAT. --[[User:Ed Poor|Ed Poor]] 13:03, 24 April 2007 (EDT) | ||

+ | |||

+ | == redundant +/-? == | ||

+ | |||

+ | There is no need for a +/- on both sides. Having it on one side is sufficient, since the "other" verion of the equation can be obtained by multiplying by -1. | ||

+ | |||

+ | <math>\sqrt{9} = \pm 3</math> | ||

+ | |||

+ | and | ||

+ | |||

+ | <math>-\sqrt{9} = \pm 3</math> | ||

+ | |||

+ | are equivalent. | ||

+ | |||

+ | I don't know why people keep inserting the +/- on the left side? [[User:Human|Human]] 14:48, 24 April 2007 (EDT) | ||

+ | |||

+ | : Yes, they are equivalent, and both incorrect. [[User:RSchlafly|RSchlafly]] 15:06, 24 April 2007 (EDT) | ||

+ | |||

+ | ::Please solve for x: <math>x^2 = 9</math>, and show the steps (no calculators allowed). By the way, there is a hilarious science fiction story about a guy who calculates a massive slippage of the San Andreas fault, and predicts that Western CA (LA) will fall into the ocean. The "hook" at the end is he discovers he took the wrong root at some point, and it's the ''rest of the US'' that slides into the ocean. Presented for comic relief. [[User:Human|Human]] 15:12, 24 April 2007 (EDT) | ||

+ | |||

+ | :::It's a matter of notation. <math>\sqrt{x}</math> refers only to the positive square root, while <math>-\sqrt{x}</math> refers only to the negative square root. If we are describing both square roots, the +/- is required. [[User:Scriabin|Scriabin]] 15:15, 24 April 2007 (EDT) | ||

+ | |||

+ | ::::Correct, but we only need the +/- on the right hand side of the example equation. Right? I'd fix it, but that would amount to revert warring rather than consensus. [[User:Human|Human]] 16:53, 24 April 2007 (EDT) | ||

+ | |||

+ | == negative numbers == | ||

+ | Negative numbers don't have two roots that are differentiated by a factor of -1. You can look farther down the article to realize that the square roots of negative numbers are different than positive numbers. --<!-- What is the point to the hidden comment? -->[[User:Staple|Staple <small>-Sysop</small>]] | ||

+ | |||

+ | : Yes, negative numbers do have square roots, as explained later in the article. [[User:RSchlafly|RSchlafly]] 15:05, 24 April 2007 (EDT) | ||

+ | |||

+ | ::Thanks, I was adding this is you were typing - | ||

+ | ::Yes they do: 2i x 2i = -4, also -2i x -2i = -4 - just as solving a quadratic equation that has no real roots can result in a pair of imaginary numbers. | ||

+ | ::I think the hidden comment is out of date, from back when I didn't know the math markup at article creation. Anyone can delete it. [[User:Human|Human]] 15:08, 24 April 2007 (EDT) | ||

+ | |||

+ | Here's my solution to the challenge above: | ||

+ | |||

+ | 3 × 3 = 9 | ||

+ | -3 × -3 = 9 | ||

+ | |||

+ | So, x = {3, -3} | ||

+ | |||

+ | --[[User:Ed Poor|Ed Poor]] 15:18, 24 April 2007 (EDT) | ||

+ | |||

+ | :That's using "brute force" - you have to know the answers to solve it. | ||

+ | |||

+ | :Try: | ||

+ | |||

+ | ::<math>x^2 = 9</math> (given) | ||

+ | ::<math>x = \pm \sqrt{9}</math> (take square root of both sides) | ||

+ | ::<math>x = \pm 3</math> (looking up sqr(9) in table. | ||

+ | ::Giving the same result you got. | ||

+ | |||

+ | ::For clarity, apply the same technique to solve for ''x'' in <math>x^2 = y</math> - or, see [[Completing the square]]. [[User:Human|Human]] 15:23, 24 April 2007 (EDT) | ||

+ | |||

+ | == This is really a usage question == | ||

+ | |||

+ | I think we all understand the point that there are two solutions to the equation | ||

+ | |||

+ | :x<sup>2</sup> = 9 | ||

+ | |||

+ | so what's at stake here is a question of usage: in mathematics and elsewhere, what is the customary meaning of the phrase ''square root'' (does it refer to both solutions to that equation, or only to the positive solution); what is the customary meaning of the radical sign? Usage questions can't be answered by logical reasoning; they always require finding an authority for the usage. | ||

+ | |||

+ | One ordinary dictionary says: | ||

+ | |||

+ | square root NOUN: A divisor of a quantity that when squared gives the quantity. For example, the square roots of 25 are 5 and -5 because 5 × 5 = 25 and (-5) × (-5) = 25 [http://www.bartleby.com/61/26/S0682600.html AHD online]. | ||

+ | |||

+ | Another says: | ||

+ | |||

+ | Main Entry: square root | ||

+ | Function: noun | ||

+ | : a factor of a number that when squared gives the number <the square root of 9 is ±3> [http://www.m-w.com/dictionary/square%20root Merriam-Webster online] | ||

+ | |||

+ | Wolfram's ''Mathworld'' site, which I've found to be very good, has a long article on [http://mathworld.wolfram.com/SquareRoot.html square root], which opens by saying | ||

+ | |||

+ | :A square root, also called a radical or surd, of x is a number such that r<sup>2</sup> = x. The function is therefore the inverse function of f(x) = x<sup>2</sup> for x ≥ 0. | ||

+ | |||

+ | It goes on to say that | ||

+ | |||

+ | :Note that any positive real number has two square roots, one positive and one negative. For example, the square roots of 9 are -3 and +3. | ||

+ | |||

+ | It then introduces the phrase "principal square root:" | ||

+ | |||

+ | :Any nonnegative real number x has a unique nonnegative square root r; this is called the ''principal square root'' and is written r = x<sup>1/2</sup> or r = √x For example, the principal square root of 9 is +3, while the other square root of 9 is -3. In common usage, unless otherwise specified, "the" square root is generally taken to mean the principal square root. | ||

+ | |||

+ | So this source seems to be saying that the ''phrase'' square root includes both values, i.e. there are two "square roots," but that the mathematical notation using the symbols x<sup>1/2</sup> and r = √x refer ''only'' to the principal or positive-valued square root. Thus, according to this source, one could write: | ||

+ | :The two square roots of 9 are ±3. | ||

+ | They don't actually say this but they do use parallel language later, when they say: | ||

+ | :The two square roots of -9 are ±√-9 = ±3''i'' | ||

+ | |||

+ | P. S. No time to look further into this now, but two other relevant sources would be some reasonably popular current algebra textbook, and... I really think this is worth looking into... the State of Massachusetts releases the test questions and answers used on the standardized tests required for high-school graduation, and there's various curriculum stuff available, and I know I saw that New Jersey and probably numerous other states do something similar. I'll betcha a nickel that there's a test question about this somewhere, and it is probably possible to say "If the State of Massachusetts asks you about this, the answer that will be scored correct is thus-and-such..." [[User:Dpbsmith|Dpbsmith]] 16:26, 24 April 2007 (EDT) | ||

+ | |||

+ | :After struggling through some of the math coding, I think I have clarified the article by defining the radical symbol as the principal square root only, while making it more clear that there are two square roots. | ||

+ | |||

+ | Another usage data point, although biased by the fact that I searched for the phrase "principal square root." And, darn it all, it points out that not every number has two square roots: the exception is 0. | ||

+ | :All positive numbers have two square roots, one positive, one negative. The two square roots of 9 are 3 and -3, and the two square roots of 144 are 12 and -12. The number 0 is the only number that has one square root, which is zero. The '''principal square root''' of a positive number is its positive square root.... The symbol √, which is called a '''radical symbol,''' is used to represent the the principal square root of a number, and -√ is used to represent the negative square root of a number. | ||

+ | ::Gustafson, R. David and Alan S. Tussy (2006) ''Introductory Algebra,'' Thomson Brooks/Coles, [http://books.google.com/books?id=8iHLN8eXuUwC&pg=PA565&ots=gCTZiHg9yo&dq=%22principal+square+root%22&sig=90fhwp5PYuHtepVC-e59D0G97Zk p. 565]. [[User:Dpbsmith|Dpbsmith]] 18:10, 24 April 2007 (EDT) | ||

+ | |||

+ | ==Generalisation== | ||

+ | |||

+ | Should this page not mention that <math>x^2 -2 = 0</math> is a special case of the Fundamental theorem of algebra which states that every non-zero single-variable polynomial, (possibly with complex coefficients), has exactly as many complex roots as its degree, if repeated roots are counted up to their multiplicity?--[[User:CatWatcher|CatWatcher]] 16:59, 24 April 2007 (EDT) | ||

+ | |||

+ | :Perhaps, but even better why not create the [[Fundamental theorem of algebra|article]]? And then it can be under "see also". [[User:Human|Human]] 17:30, 24 April 2007 (EDT) | ||

+ | |||

+ | :No, that's way too complicated; stuff like that can go into [[Algebra]]. When the dust settles I'm going to cut 80% of the page so that it fits into 16 square inches of screen space. How long will each side be? Why sqrt(16) of course! ;-) |

## Latest revision as of 16:10, 24 April 2007

I have commented out part of the text, until I figure out how to generate the radical symbol for roots. Also, it looks like a few basic math files need to be created... Human 16:27, 12 April 2007 (EDT)

- like this Jaques 17:46, 12 April 2007 (EDT)

## Contents

## Order?

The current article jumps back and forth with concepts. We go from the intro to irrational numbers, then to imaginary numbers, and only then do we get the whole "two square roots per number" and "notation" stuff laid down. Anybody mind if I try to re-arrange some stuff? --Sid 3050 17:43, 12 April 2007 (EDT)

- I'll take that as a "No". Going in... --Sid 3050 18:04, 12 April 2007 (EDT)
- Done. In my eyes, this is a bit more structured than before. Opinions, edits and expansions welcome, of course. Just my suggestion. --Sid 3050 18:30, 12 April 2007 (EDT)
- Thanks, you're right.
~~I think it might need some more copyediting in order to truly "introduce" topics in order of complexity. But at least now the article exists - and is~~already better than it started. You didn't get a response earlier because you were fixing it about eleven seconds after I saved it ;) Human 19:13, 12 April 2007 (EDT) Oops, reason for strikethrough, it's chaged again for the even better! Human 19:18, 12 April 2007 (EDT)

- Thanks, you're right.

- Done. In my eyes, this is a bit more structured than before. Opinions, edits and expansions welcome, of course. Just my suggestion. --Sid 3050 18:30, 12 April 2007 (EDT)

## Zero

Wouldn't only all nonzero numbers have two square roots? MountainDew 17:53, 12 April 2007 (EDT)

- Cross-checking with Wikipedia (my Calculus books are out of reach at the moment) says that the square roots do not have to be distinct, so . But there are indeed two. --Sid 3050 17:57, 12 April 2007 (EDT)

- Whoa, thanks for fixing this thing up so fast, folks! Human 18:06, 12 April 2007 (EDT)

## +/- notation deleted?

Why is this:

- "As such, the technically correct way of writing is
- "

Being deleted? It is not only correct, it is very important. For instance, the quadratic formula page would need it if the details of completing the square were laid out. Earlier today, Rschlafly removed it, I reverted it back in, and now Jacques has removed it. So let us discuss it here? Human 13:50, 22 April 2007 (EDT)

- Hearing no objection or reason, I am putting it back in. Human 01:12, 24 April 2007 (EDT)

- I removed it, because it is not technically correct. It is just not true that the quadratic formula page needs it. That page uses a plus-or-minus symbol in front of the square root. It would just need a plus sign, if the plus-or-minus were really technically correct for the square root. RSchlafly 01:37, 24 April 2007 (EDT)
- How is it not technically correct? Please explain this new math you're using. ColinR
^{talk}01:38, 24 April 2007 (EDT)

- How is it not technically correct? Please explain this new math you're using. ColinR

- I removed it, because it is not technically correct. It is just not true that the quadratic formula page needs it. That page uses a plus-or-minus symbol in front of the square root. It would just need a plus sign, if the plus-or-minus were really technically correct for the square root. RSchlafly 01:37, 24 April 2007 (EDT)

Excuse me. I'm going to vent. RSchlafly, you are completely wrong. As in, no ideology needed. The square root of 9 *is* plus or minus 3. The equation shown expresses that clearly. When a square root is taken, the plus or minus becomes required in order to keep track of multiple possible solutions. Unfortunately, the last math text I kept was my calculus book, so I don't have a ready cite from a high school algebra text. I can't think of any "polite" ways to say that you don't understand what you are talking about. perhaps when I struggle to remember the steps in completing the square and post it on one of these math files, you'll see why it matters. Human 02:05, 24 April 2007 (EDT)

- Technically the plus/minus should go in front of the radical if you are referring to both square roots. Usually if there is no sign in front of radical it means just the positive square root. Scriabin 02:08, 24 April 2007 (EDT)

- Exactly. I took a few minutes to think about what Rschlafly is saying again, and I apologise for the outburst. We make the point in the article that a number has two square roots, but fail to say that
*by convention*we use the plus/minus symbol when we are making sure we haven't thrown out a solution. In other words, simply using the radical implies both answers, based on what the article says. What has to be explained is how this works in solving equations. Say we take the square root of "both sides" - often what has to happen then is that the two different roots force us into simplifying two separate possible equations in two separate columns, at least until the two possibilities resolve. It's a hard core math thing, in that in a lot of "real world" situations, there are no negative numbers. Think carpentry... So, to conclude, I think the expression should be included - but explained better. Human 02:18, 24 April 2007 (EDT)

- Exactly. I took a few minutes to think about what Rschlafly is saying again, and I apologise for the outburst. We make the point in the article that a number has two square roots, but fail to say that

OK, I tried to make it clearer. In the interest of a harmonious community, RSchlafly, do you think this makes more sense now? Human 02:34, 24 April 2007 (EDT)

- I am still confused by "Since in many situations only the positive square root has any meaning". Don't you mean "Since in many situations both square roots are needed"? The square root symbol is for the nonnegative root, but often both roots are needed. RSchlafly 02:50, 24 April 2007 (EDT)
- It could be said either way. The way I worded it, I hope, the positive root takes precendence, and both are included when the situation requires it. It could just as easily be worded the other way. Whichever consensus thinks is clearest and most useful to the target audience? Human 14:12, 24 April 2007 (EDT)

- No, it cannot be said either way. It does not help to have two contradictory definitions of the square root sign. RSchlafly 14:35, 24 April 2007 (EDT)
- OK, so define it as the positive root, but point out that there are a pair of numbers that can be squared to produce any given number, and that is shown by our little formula, and is necessary to preserve multiple solutions of equations when taking the square root of each side? Using the Wolfram cite below to define it? Human 14:52, 24 April 2007 (EDT)

- No, it cannot be said either way. It does not help to have two contradictory definitions of the square root sign. RSchlafly 14:35, 24 April 2007 (EDT)

Wolfram says:

- The unique nonnegative square root of a nonnegative real number. For example, the principal square root of 9 is 3, although both -3 and 3 are square roots of 9. [1]

I want to make sure our article won't mislead some teen studying for the SAT. --Ed Poor 13:03, 24 April 2007 (EDT)

## redundant +/-?

There is no need for a +/- on both sides. Having it on one side is sufficient, since the "other" verion of the equation can be obtained by multiplying by -1.

and

are equivalent.

I don't know why people keep inserting the +/- on the left side? Human 14:48, 24 April 2007 (EDT)

- Yes, they are equivalent, and both incorrect. RSchlafly 15:06, 24 April 2007 (EDT)

- Please solve for x: , and show the steps (no calculators allowed). By the way, there is a hilarious science fiction story about a guy who calculates a massive slippage of the San Andreas fault, and predicts that Western CA (LA) will fall into the ocean. The "hook" at the end is he discovers he took the wrong root at some point, and it's the
*rest of the US*that slides into the ocean. Presented for comic relief. Human 15:12, 24 April 2007 (EDT)

- Please solve for x: , and show the steps (no calculators allowed). By the way, there is a hilarious science fiction story about a guy who calculates a massive slippage of the San Andreas fault, and predicts that Western CA (LA) will fall into the ocean. The "hook" at the end is he discovers he took the wrong root at some point, and it's the

- It's a matter of notation. refers only to the positive square root, while refers only to the negative square root. If we are describing both square roots, the +/- is required. Scriabin 15:15, 24 April 2007 (EDT)

- Correct, but we only need the +/- on the right hand side of the example equation. Right? I'd fix it, but that would amount to revert warring rather than consensus. Human 16:53, 24 April 2007 (EDT)

## negative numbers

Negative numbers don't have two roots that are differentiated by a factor of -1. You can look farther down the article to realize that the square roots of negative numbers are different than positive numbers. --Staple -Sysop

- Yes, negative numbers do have square roots, as explained later in the article. RSchlafly 15:05, 24 April 2007 (EDT)

- Thanks, I was adding this is you were typing -
- Yes they do: 2i x 2i = -4, also -2i x -2i = -4 - just as solving a quadratic equation that has no real roots can result in a pair of imaginary numbers.
- I think the hidden comment is out of date, from back when I didn't know the math markup at article creation. Anyone can delete it. Human 15:08, 24 April 2007 (EDT)

Here's my solution to the challenge above:

3 × 3 = 9 -3 × -3 = 9 So, x = {3, -3}

--Ed Poor 15:18, 24 April 2007 (EDT)

- That's using "brute force" - you have to know the answers to solve it.

- Try:

- (given)
- (take square root of both sides)
- (looking up sqr(9) in table.
- Giving the same result you got.

- For clarity, apply the same technique to solve for
*x*in - or, see Completing the square. Human 15:23, 24 April 2007 (EDT)

- For clarity, apply the same technique to solve for

## This is really a usage question

I think we all understand the point that there are two solutions to the equation

- x
^{2}= 9

so what's at stake here is a question of usage: in mathematics and elsewhere, what is the customary meaning of the phrase *square root* (does it refer to both solutions to that equation, or only to the positive solution); what is the customary meaning of the radical sign? Usage questions can't be answered by logical reasoning; they always require finding an authority for the usage.

One ordinary dictionary says:

square root NOUN: A divisor of a quantity that when squared gives the quantity. For example, the square roots of 25 are 5 and -5 because 5 × 5 = 25 and (-5) × (-5) = 25 AHD online.

Another says:

Main Entry: square root Function: noun

- a factor of a number that when squared gives the number <the square root of 9 is ±3> Merriam-Webster online

Wolfram's *Mathworld* site, which I've found to be very good, has a long article on square root, which opens by saying

- A square root, also called a radical or surd, of x is a number such that r
^{2}= x. The function is therefore the inverse function of f(x) = x^{2}for x ≥ 0.

It goes on to say that

- Note that any positive real number has two square roots, one positive and one negative. For example, the square roots of 9 are -3 and +3.

It then introduces the phrase "principal square root:"

- Any nonnegative real number x has a unique nonnegative square root r; this is called the
*principal square root*and is written r = x^{1/2}or r = √x For example, the principal square root of 9 is +3, while the other square root of 9 is -3. In common usage, unless otherwise specified, "the" square root is generally taken to mean the principal square root.

So this source seems to be saying that the *phrase* square root includes both values, i.e. there are two "square roots," but that the mathematical notation using the symbols x^{1/2} and r = √x refer *only* to the principal or positive-valued square root. Thus, according to this source, one could write:

- The two square roots of 9 are ±3.

They don't actually say this but they do use parallel language later, when they say:

- The two square roots of -9 are ±√-9 = ±3
*i*

P. S. No time to look further into this now, but two other relevant sources would be some reasonably popular current algebra textbook, and... I really think this is worth looking into... the State of Massachusetts releases the test questions and answers used on the standardized tests required for high-school graduation, and there's various curriculum stuff available, and I know I saw that New Jersey and probably numerous other states do something similar. I'll betcha a nickel that there's a test question about this somewhere, and it is probably possible to say "If the State of Massachusetts asks you about this, the answer that will be scored correct is thus-and-such..." Dpbsmith 16:26, 24 April 2007 (EDT)

- After struggling through some of the math coding, I think I have clarified the article by defining the radical symbol as the principal square root only, while making it more clear that there are two square roots.

Another usage data point, although biased by the fact that I searched for the phrase "principal square root." And, darn it all, it points out that not every number has two square roots: the exception is 0.

- All positive numbers have two square roots, one positive, one negative. The two square roots of 9 are 3 and -3, and the two square roots of 144 are 12 and -12. The number 0 is the only number that has one square root, which is zero. The
**principal square root**of a positive number is its positive square root.... The symbol √, which is called a**radical symbol,**is used to represent the the principal square root of a number, and -√ is used to represent the negative square root of a number.

## Generalisation

Should this page not mention that is a special case of the Fundamental theorem of algebra which states that every non-zero single-variable polynomial, (possibly with complex coefficients), has exactly as many complex roots as its degree, if repeated roots are counted up to their multiplicity?--CatWatcher 16:59, 24 April 2007 (EDT)

- Perhaps, but even better why not create the article? And then it can be under "see also". Human 17:30, 24 April 2007 (EDT)

- No, that's way too complicated; stuff like that can go into Algebra. When the dust settles I'm going to cut 80% of the page so that it fits into 16 square inches of screen space. How long will each side be? Why sqrt(16) of course! ;-)