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Topic: when is sqrt(a/b) not the same as sqrt(a)/sqrt(b) ?
Replies: 8   Last Post: Mar 5, 2014 3:34 PM

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Richard Fateman

Posts: 1,539
Registered: 12/7/04
Re: when is sqrt(a/b) not the same as sqrt(a)/sqrt(b) ?
Posted: Mar 1, 2014 7:54 PM
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There are simpler questions.

Like is sqrt(x) - sqrt(x) equal to 0 or 2*sqrt(x) or -2*sqrt(x).

Recall that there are two square roots. Which one did you mean
for the first sqrt? Which for the second? which for the third or fourth?

If you specify, as some systems seem to advise, that x is positive,
it still is ambiguous. Isn't it possible that sqrt(4) has 2 values,
sqrt(-4) has 2 values as well?

Now if you or your favorite CAS change the meaning of sqrt, then who knows?

As Lewis Caroll wrote,

"'When I use a word,' Humpty Dumpty said, in rather a scornful tone, 'it
means just what I choose it to mean ? neither more nor less.'

'The question is,' said Alice, 'whether you can make words mean so many
different things.'

'The question is,' said Humpty Dumpty, 'which is to be master ? that's
all.' "


On 3/1/2014 1:35 PM, Axel Vogt wrote:
> On 01.03.2014 22:19, Nasser M. Abbasi wrote:
> ...
> As Edgar already said:
> [sqrt(a/b) , sqrt(a)/sqrt(b)]:
> subs(a=1, %);
> 1/2 1
> [(1/b) , ----]
> 1/2
> b
> subs(b=-1, %);
> [I, -I]
> You need that for x * 1/x = 1
> The reason is the brunch cut: sqrt is 'only defined'
> except the negative real axis and continued to that
> by a special 'rule' in a non-continous way
> In Maple that is named "winding number", K:
> (z*w)^a = z^a*w^a * exp(-2*I*Pi*a*K(ln(z)+ln(w)))

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