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# "sqrt" in UK....1 or +/-1? watch

1. In the U.S. the radical with implied index of 2 (i.e. the square root symbol) is usually taken to
mean only the principal root, e.g. sqrt(1) is 1, not -1, and -1 could be written as -sqrt(1).

What is the convention most widely used in the UK? The context is usual maths for 16-18 years old in
the UK. Is sqrt(1)=1 or both 1 and -1? This is not an invitation for explanations such as....there
are two numbers whos square is 1. This is simply a question of the meaning of the notation "sqrt" in

--
Darrell
2. Darrell,

It's been a long time but I think you are talking about 18 yr old study. At this point are we
talking about complex numbers ? I am an engineer so know it as 'j' (sqrt(-1)), in maths terms 'i',
that is the UK convention. sqrt(4)=2 in most other cases. I hope I am not oversimplifying. As an 18
yr old math student studying maths negative roots were considered as -1 and not -sqrt(1) but, it was
16 yrs ago. -sqrt(1) looks ambiguous to me ? Damn, I'm just rediscovering maths. In IT now.

Rich. "Darrell" <[email protected]> wrote in message
news:[email protected]...
[q1]> In the U.S. the radical with implied index of 2 (i.e. the square root symbol) is usually taken to[/q1]
[q1]> mean only the principal root, e.g. sqrt(1) is[/q1]
1,
[q1]> not -1, and -1 could be written as -sqrt(1).[/q1]
[q1]>[/q1]
[q1]> What is the convention most widely used in the UK? The context is usual maths for 16-18 years old[/q1]
[q1]> in the UK. Is sqrt(1)=1 or both 1 and -1? This is not an invitation for explanations such[/q1]
[q1]> as....there are two numbers[/q1]
whos
[q1]> square is 1. This is simply a question of the meaning of the notation "sqrt" in the UK. Thank you[/q1]
[q1]>[/q1]
[q1]> --[/q1]
[q1]> Darrell[/q1]
3. Thanks for the reply, but I am talking about reals only at this point. Perhaps my subject line could
have been formulated better. There are two square roots of a^2. The question is....does the
*notation* sqrt(a) (i.e. the radicand is a and the index is understood to be 2) in the UK evaluate
to both a and -a, or just a? In the U.S. it is just a and if you wish to denote -a, using this
notation, it would be -sqrt(a), i.e. sqrt(a), which is a, multiplied by -1 giving -a (hence
unambiguous.)

The purpose of the question is to confirm or refute a UK student's claim that in the UK the
convention is sqrt(a) evaluates to both a and -a. My feeling is that the UK in all probability
follows the same convention in normal practice, i.e. 16-18 year old algebra courses. If there are
any current UK teachers reading this, I would appreciate your responses as I am trying to find a
consensus one way or the other to confirm or refute my assumption.

--
Darrell

"Richard Battye" <[email protected]> wrote in message
news:[email protected]...
[q1]> Darrell,[/q1]
[q1]>[/q1]
[q1]> It's been a long time but I think you are talking about 18 yr old study.[/q1]
At
[q1]> this point are we talking about complex numbers ? I am an engineer so know it as 'j' (sqrt(-1)),[/q1]
[q1]> in maths terms 'i', that is the UK convention. sqrt(4)=2 in most other cases. I hope I am not[/q1]
[q1]> oversimplifying. As an 18[/q1]
yr
[q1]> old math student studying maths negative roots were considered as -1 and not -sqrt(1) but, it was[/q1]
[q1]> 16 yrs ago. -sqrt(1) looks ambiguous to me ?[/q1]
Damn,
[q1]> I'm just rediscovering maths. In IT now.[/q1]
[q1]>[/q1]
[q1]> Rich. "Darrell" <[email protected]> wrote in message[/q1]
[q1]> news:[email protected]...[/q1]
[q2]> > In the U.S. the radical with implied index of 2 (i.e. the square root symbol) is usually taken[/q2]
[q2]> > to mean only the principal root, e.g. sqrt(1)[/q2]
is
[q1]> 1,[/q1]
[q2]> > not -1, and -1 could be written as -sqrt(1).[/q2]
[q2]> >[/q2]
[q2]> > What is the convention most widely used in the UK? The context is usual maths for 16-18 years[/q2]
[q2]> > old in the UK. Is sqrt(1)=1 or both 1 and -1?[/q2]
This
[q2]> > is not an invitation for explanations such as....there are two numbers[/q2]
[q1]> whos[/q1]
[q2]> > square is 1. This is simply a question of the meaning of the notation "sqrt" in the UK. Thank[/q2]
[q2]> >[/q2]
[q2]> > --[/q2]
[q2]> > Darrell[/q2]
[q2]> >[/q2]
[q2]> >[/q2]
4. [q3]> > > What is the convention most widely used in the UK? The context is usual maths for 16-18 years[/q3]
[q3]> > > old in the UK. Is sqrt(1)=1 or both 1 and -1?[/q3]
[q1]> This[/q1]
[q3]> > > is not an invitation for explanations such as....there are two numbers[/q3]
[q2]> > whos[/q2]
[q3]> > > square is 1. This is simply a question of the meaning of the notation "sqrt" in the UK. Thank[/q3]
[q3]> > >[/q3]
[q3]> > > --[/q3]
[q3]> > > Darrell[/q3]

I can't speak for now, but some 48 years ago at Oxford the usage was sqrt(1) = 1 and -sqrt(1) = -1.

The same ambiguities occurred then as now if the argument of a square root operation was allowed to
be negative or complex.
5. Virgil <[email protected]> wrote:
[q1]>I can't speak for now, but some 48 years ago at Oxford the usage was sqrt(1) = 1 and -sqrt(1) = -1.[/q1]

Slightly more recently, when I was at school about 25 years ago, sqrt (i.e. that thing shaped a bit
like a tick) meant the positive square root when applied to a positive real.

So the solutions of x^2 = 5 were x= +/- sqrt(5).

[q1]>The same ambiguities occurred then as now if the argument of a square root operation was allowed to[/q1]
[q1]>be negative or complex.[/q1]

Ditto.
--
Rob. http://www.mis.coventry.ac.uk/~mtx014/
6. In article <[email protected] prod.itd.earthlink.net>, Darrell
<[email protected]> writes
[q1]>If there are any current UK teachers reading this, I would appreciate your responses as I am trying[/q1]
[q1]>to find a consensus one way or the other to confirm or refute my assumption.[/q1]

You are right

sqrt(4)=2 (only) So the solutions to a^2=4 are +/-sqrt(4)

M.
--
Mark Houghton [email protected]
7. "Darrell" <[email protected]> a écrit dans le message de news:
[q1]> In the U.S. the radical with implied index of 2 (i.e. the square root symbol) is usually taken to[/q1]
[q1]> mean only the principal root, e.g. sqrt(1) is[/q1]
1,
[q1]> not -1, and -1 could be written as -sqrt(1).[/q1]
[q1]>[/q1]
[q1]> What is the convention most widely used in the UK? The context is usual maths for 16-18 years old[/q1]
[q1]> in the UK. Is sqrt(1)=1 or both >1 and -1?[/q1]

Then a number, sqrt(1), would be BOTH 1 and -1.... ;-) In the same subject, what is the argument of
a given complex number? In fact, there is several values y in C that verify: exp(y) = z, for a given
z in C. The question is to choose one... For the square root, it is evident that it is not a mater
of US or UK (I'm french so I know what I say..), but you must remember that the square root has a
physic sense: it is the lenght of the diagonal of a square. So, it is evident that it is a positive
number. Phytagore knew it! Then, at Descartes' time, we invented the negative numbers (even if he
didn't believe in it..)

Romain
8. Hi,

sqrt(x) is a function, so by the definition of a function can only return one value for a
certain x value. The value usually used is the positive root, the negative one is usually
referred to as -sqrt(x).

I do not believe that it is only the US and UK who teach this, as it is surely wrong to teach it
otherwise, or sqrt(x) could not be a function.

Regards,

Richard Hayden.

"Darrell" <[email protected]> wrote in message
news:[email protected]...
[q1]> In the U.S. the radical with implied index of 2 (i.e. the square root symbol) is usually taken to[/q1]
[q1]> mean only the principal root, e.g. sqrt(1) is[/q1]
1,
[q1]> not -1, and -1 could be written as -sqrt(1).[/q1]
[q1]>[/q1]
[q1]> What is the convention most widely used in the UK? The context is usual maths for 16-18 years old[/q1]
[q1]> in the UK. Is sqrt(1)=1 or both 1 and -1? This is not an invitation for explanations such[/q1]
[q1]> as....there are two numbers[/q1]
whos
[q1]> square is 1. This is simply a question of the meaning of the notation "sqrt" in the UK. Thank you[/q1]
[q1]>[/q1]
[q1]> --[/q1]
[q1]> Darrell[/q1]
[q1]>[/q1]
[q1]>[/q1]

---
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9. In article <[email protected]> , Mark Houghton <[email protected]> writes
[q1]>You are right[/q1]
[q1]>[/q1]
[q1]>sqrt(4)=2 (only) So the solutions to a^2=4 are +/-sqrt(4)[/q1]
[q1]>[/q1]
[q1]>M.[/q1]

... and I forgot to say, since it seems of some import, that I am a teacher of math(s) in the uk.

M.
--
Mark Houghton [email protected]
10. "Richard Hayden" <[email protected]> wrote:

[q1]>Hi,[/q1]
[q1]>[/q1]
[q1]>sqrt(x) is a function, so by the definition of a function can only return one value for a certain x[/q1]
[q1]>value. The value usually used is the positive root, the negative one is usually referred to as[/q1]
[q1]>-sqrt(x).[/q1]
[q1]>[/q1]
[q1]>I do not believe that it is only the US and UK who teach this, as it is surely wrong to teach it[/q1]
[q1]>otherwise, or sqrt(x) could not be a function.[/q1]

You are confusing things.......as I understand it, the notation sqrt(x) is being used in this thread
as a substitute for the regular root symbol (hard to represent in ascii). The root symbol is, of
course, not a function, but a symbol. The question then, is how do you interpret this symbol.
Certainly there is no a priori reason why it should be interpreted as a function - not all signs
inscribed on paper are interpreted as functions.

Gareth
11. "Darrell" <[email protected]> wrote:

[q1]>The purpose of the question is to confirm or refute a UK student's claim that in the UK the[/q1]
[q1]>convention is sqrt(a) evaluates to both a and -a. My feeling is that the UK in all probability[/q1]
[q1]>follows the same convention in normal practice, i.e. 16-18 year old algebra courses. If there are[/q1]
[q1]>any current UK teachers reading this, I would appreciate your responses as I am trying to find a[/q1]
[q1]>consensus one way or the other to confirm or refute my assumption.[/q1]

I'm not a teacher I'm afraid......

Certainly when I did A'Level maths, we assumed that the symbol implied both roots. e.g. in solving
an equation you might have:

x = sqrt(64)

therefore x = -8, x = 8.

However in cases when only a positive number made sense, we'd still just write sqrt(10) - and you'd
generally decide from context that the negative root wasn't relevant.

Gareth
12. Gareth Jones <[email protected]> wrote:
[q1]>You are confusing things.......as I understand it, the notation sqrt(x) is being used in this[/q1]
[q1]>thread as a substitute for the regular root symbol (hard to represent in ascii). The root symbol[/q1]
[q1]>is, of course, not a function, but a symbol. The question then, is how do you interpret this[/q1]
[q1]>symbol. Certainly there is no a priori reason why it should be interpreted as a function - not all[/q1]
[q1]>signs inscribed on paper are interpreted as functions.[/q1]

And the convention is that this symbol represents a function with domain and codomain both
consisting of the non-negative reals. sqrt(x) means, for non-negative x, that non-negative y such
that y^2=x.

Of course, one might take it to have domain the positive reals and codomain pairs of the form (x,-x)
(x a positive real). It would still be a function, but not a real-valued function.

--
Rob. http://www.mis.coventry.ac.uk/~mtx014/
13. In article <[email protected] c.uk>, Robert Low
<[email protected] ac.uk> wrote:
[q1]>Slightly more recently, when I was at school about 25 years ago, sqrt (i.e. that thing shaped a bit[/q1]
[q1]>like a tick) meant the positive square root when applied to a positive real.[/q1]

Back in the days [roughly the same as yours!] when we used to interview applicants
"seriously", one of our standard questions was to evaluate

sqrt (6 + sqrt (6 + sqrt (6 + sqrt (6 + sqrt (...))))).

Inasmuch as there is a proper answer, it is the positive zero of x^2 - x - 6, ie 3, which most
applicants got to with more-or-less help from the interviewer. But the follow-up question "what's
the significance of the negative zero, -2?" used to flummox all but the very best applicants. Then
we used to invite them to generalise, which gives the slightly surprising result that

sqrt (0 + sqrt (0 + sqrt (0 + sqrt (0 + sqrt (...)))))

is 1, not 0, and that

sqrt (-1/4 + sqrt (-1/4 + sqrt (-1/4 + sqrt (-1/4 + sqrt (...)))))

is 1/2, though it looks as though it must be complex.

--
Andy Walker, School of MathSci., Univ. of Nott'm, UK. [email protected]
14. Dr A. N. Walker <[email protected]> wrote:
[q1]> Back in the days [roughly the same as yours!] when we used to interview applicants[/q1]
[q1]> "seriously", one of our standard questions was to evaluate[/q1]
[q1]>[/q1]
[q1]> sqrt (6 + sqrt (6 + sqrt (6 + sqrt (6 + sqrt (...))))).[/q1]
[q1]>[/q1]
[q1]>Inasmuch as there is a proper answer, it is the positive zero of[/q1]

Did any of the candidates ever ask 'what does the ... mean?'

(It's a question I try to persuade my students they *ought* to ask when they see an ellipsis. They
tend to think it's obvious.)

A little thought shows that there's actually quite a bit of interesting elementary analysis
hiding in there.

Do you look for a fixed point of the function x -> sqrt(6+x)?

Or do you define a sequence by x_0 = a, x_{n+1}=sqrt(6+x_n), and find the limit of that? If the
latter, how do you show convergence, and does the answer depend on a?

[q1]>the very best applicants. Then we used to invite them to generalise, which gives the slightly[/q1]
[q1]>surprising result that[/q1]
[q1]>[/q1]
[q1]> sqrt (0 + sqrt (0 + sqrt (0 + sqrt (0 + sqrt (...)))))[/q1]
[q1]>[/q1]
[q1]>is 1, not 0, and that[/q1]

Well, kind of. If you take a>0 (and 6=0, so to speak) in the above sequence, then this always
converges to 1. But it you take a=0, then the sequence is constant and every term is 0. Or if you
take 6 very small, and take the limit, you get 1.

It's a nice case of a pair of limits not commuting, anyway.

And life gets considerably more interesting with the negative square roots, especially when you
actually get have to take the square root of a negative number.

Formally, you still get the two solutions---one for each square root: but now it's not so clear at
all just whether the sequence converges.

[q1]> sqrt (-1/4 + sqrt (-1/4 + sqrt (-1/4 + sqrt (-1/4 + sqrt (...)))))[/q1]
[q1]>[/q1]
[q1]>is 1/2, though it looks as though it must be complex.[/q1]

A cutey, since the roots coincide at 1/2. (Haven't figured out the just how to define the sequence,
since there's no canonical positive square root to take. But it looks at first glance as if either
choice of square root at each stage will give a sequence converging on 1/2. I'm prepared to be

Alas, this stuff looks more like a final year project than an interview question these days...

--
Rob. http://www.mis.coventry.ac.uk/~mtx014/
15. In article <[email protected] .com>, Jo L <[email protected]> writes
[q1]>I was always of the understanding that the square root of say 4 was always +2 or -2, indeed a[/q1]
[q1]>similar question has appeared on a KS3 SATS paper, where you were told something like one answer to[/q1]
[q1]>square root of 9 was +3, what was the other answer.[/q1]
[q1]>[/q1]
[q1]>Jo[/q1]

Now - I haven't seen that paper, and I need to, really...

but....

solve ax^2+bx+c=0

x=(-b+/-sqrt(b^2-4ac))/2a

Now why is it that we take care to write in the +/-?

If sqrt can take both positive *and* negative values, surleythat would be uneccessary?

OTOH One solution to X^2=9 is 3. The other is -3. I must look up the paper...

M.
--
Mark Houghton [email protected]
16. In article <[email protected] .com>,
"Jo L" <[email protected]> wrote:

[q1]> I was always of the understanding that the square root of say 4 was always +2 or -2, indeed a[/q1]
[q1]> similar question has appeared on a KS3 SATS paper, where you were told something like one answer[/q1]
[q1]> to square root of 9 was +3, what was the other answer.[/q1]

"A square root" of 4, could be either +2 or -2.

"The square roots" of 4 are +2 and -2.

"The square root" of 4 is +2, by the convention that, for positive real radicands, "the square root"
means the principal (or positive) square root.

References to square roots of negative reals or complexes should be made very carefully.
17. "Mark Houghton" <[email protected]> wrote in message news:[email protected]...
[q1]> In article <[email protected] .com>, Jo L <[email protected]> writes[/q1]
[q2]> >I was always of the understanding that the square root of say 4 was[/q2]
always
[q2]> >+2 or -2, indeed a similar question has appeared on a KS3 SATS paper,[/q2]
where
[q2]> >you were told something like one answer to square root of 9 was +3, what[/q2]
was
[q2]> >[/q2]
[q2]> >Jo[/q2]
[q1]>[/q1]
[q1]> Now - I haven't seen that paper, and I need to, really...[/q1]
[q1]>[/q1]
[q1]> but....[/q1]
[q1]>[/q1]
[q1]> solve ax^2+bx+c=0[/q1]
[q1]>[/q1]
[q1]> x=(-b+/-sqrt(b^2-4ac))/2a[/q1]
[q1]>[/q1]
[q1]> Now why is it that we take care to write in the +/-?[/q1]
[q1]>[/q1]
[q1]> If sqrt can take both positive *and* negative values, surleythat would be uneccessary?[/q1]
[q1]>[/q1]
[q1]> OTOH One solution to X^2=9 is 3. The other is -3. I must look up the paper...[/q1]

...This is the precise reason I made the point of stating in my inquiry that I am not asking whether
or not a positive real number has one, or two, "square roots." I am specifically asking if it is
true or not that in the UK the normal convention is that the mathematical *symbol* sqrt (the radical
sign with index of 2" represents only the nonnegative root. We all know a positive integer has two
square roots.

Based on the consensus formed from previous responses, I'll wager one million bits that the paper in
question is literally asking something along the lines of "One square root of 9 is 3, what is the
other?" as opposed to something along the lines of sqrt(9)=3, and sqrt(9)=??? ...Although I would be
interested in knowing if this assumption proves to be incorrect, in the interest of my survey.

--
Darrell
18. Dr A. N. Walker <[email protected]> wrote:
[q1]>Robert Low <[email protected] ac.uk> wrote:[/q1]
[q2]>>Did any of the candidates ever ask 'what does the ... mean?' (It's a question I try to persuade my[/q2]
[q2]>>students they *ought* to ask when they see an ellipsis. They tend to think it's obvious.)[/q2]
[q1]>[/q1]
[q1]> It *ought* to be obvious![/q1]

I agree absolutely, in the sense that one should never use an ellipsis if there's any ambiguity.
Unfortunately, that doesn't guarantee that I will agree with the studnet about the obvious sense.
(Does the string 0.999... ring any warning bells? :-))

[q2]>>A little thought shows that there's actually quite a bit of interesting elementary analysis hiding[/q2]
[q2]>>in there.[/q2]
[q1]>[/q1]
[q1]> Absolutely, though [of course] that's not the point of an interview question. But some of[/q1]
[q1]> the applicants got hooked,[/q1]

Sure...I was just running with a rather interesting ball that I hadn't thought about before.

[q1]> Right. Now I've got you hooked, .... Consider the set S(p) [p real] defined by the rules:[/q1]
[q1]>[/q1]
[q1]> 1. p is in S(p)[/q1]
[q1]> 2. if x > -6 is in S(p), then so are sqrt(6+x) and -sqrt(6+x) [equivalently, y is in S(p) if[/q1]
[q1]> y^2-6 is][/q1]
[q1]> 3. no other numbers are in S(p)[/q1]
[q1]>[/q1]
[q1]>What can you say about S(p)? Generalise [6 -> q; p,q -> complex; +-sqrt -> other functions; links[/q1]
[q1]>with surreals; others?]. Yeah, I think that'd be a decent 3rd/4th-yr project -- I may give it a[/q1]
[q1]>whirl next year![/q1]

Let us know how it went!
--
Rob. http://www.mis.coventry.ac.uk/~mtx014/
19. On Wed, 15 May 2002 1951 GMT, Gareth Jones <[email protected]> wrote:

[q1]>[/q1]
[q1]>x = sqrt(64)[/q1]
[q1]>[/q1]
[q1]>therefore x = -8, x = 8.[/q1]

I think he said "sqrt" meant the square root tick symbol so:

if x^2=2 then x= + or - sqrt(2)

--
Dave
20. In article <[email protected] .prod.itd.earthlink.net>, Darrell
<[email protected]> writes
[q1]>Based on the consensus formed from previous responses, I'll wager one million bits that the[/q1]
[q1]>paper in question is literally asking something along the lines of "One square root of 9 is 3,[/q1]
[q1]>what is the other?" as opposed to something along the lines of sqrt(9)=3, and sqrt(9)=???[/q1]
[q1]>...Although I would be interested in knowing if this assumption proves to be incorrect, in the[/q1]
[q1]>interest of my survey.[/q1]

O.k. - I found the "official" line in the National Numeracy Strategy document. I would've looked
there before, but I like to leave all these documents at work :-)

Students should:

"Know that a positive integer has two square roots, one positive and one negative; by convention the
square root sign denotes the postive square root."

P.
--
Mark Houghton [email protected]

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