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surd

Type of Math Object: 
Definition
Major Section: 
Reference

Mathematics Subject Classification

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Hi,

I am sorry but I still think this term, surd, is quite ambiguous, and quite confusing. I don't really see any use for this term. For example, $\sqrt{2}$ can be written as $\exp(1/2 \log(2))$ as well, and in many other forms that may involve other sorts of symbols.

Alvaro

I read the entry as saying that given a number that is written using natural numbers, and only rational operations, and taking roots of integer degree, then it is a surd if there is no way to rewrite it so no roots are used (or equivalently if it is not a rational number). However, the entry needs to make it more clear.

However that does not seem to be the ``right'' definition of surd. In ``Mathematics Dictionary'' by Gleen James and Robert James published in 1959 by van Nostand Company it says

SURD, n. A sum of one of more irrational
indicated roots of numbers. Sometimes used
for irrational number. A surd of one term
is called quadratic, cubic, quartic,
quintic, etc., according to index of the
radical is two, three, four, five, etc.
It is an \emph{entire surd} if it does
not contain a rational factor or term
(e.g., \sqrt{3} or 5+\sqrt{2});
a \emph{mixed surd} if it contains a
rational factor or term (e.g., 2\sqrt{3}
or 5+\sqrt{2}); a \emph{pure surd} if
each term is a surd (e.g., 3\sqrt{2}+\sqrt{5}).
A binomial surd is a binomial, at least
one of whose terms is a surd, such as
2+\sqrt{3} or \sqrt[3]{2}-\sqrt{3}.
\emph{Conjugate binomial} surds are two
binomial surds of the form a\sqrt{b}+c\sqrt{d}
and a\sqrt{b}-c\sqrt{d} where a,b,c and
d are rational and \sqrt{b} and \sqrt{d}
are not both rational. The product of
two conjugate binomial surds is rational,
e.g., (a+\sqrt{b})(a-\sqrt{b})=a^2-b.
A \emph{trinomial surd} is a trinomial at
least two of whose terms are surds which
cannot be combined without evaluating them;
2+\sqrt{2}+\sqrt{3} and 3+\sqrt{5}+\sqrt[3]{2}
are trinomial surds.

Boris

>SURD, n. A sum of one of more irrational
> indicated roots of numbers. Sometimes used
> for irrational number.

Still a not very precise definition. Is $\sqrt{\pi}$ a surd then? What do they mean by "number" ? A natural number I assume. I think this is still a rather confusing definition that should be avoided.

I have updated the definition, it should now be correct and somewhat clearer.

--
"Do not meddle in the affairs of wizards for they are subtle and quick to anger."

Yes, it is very unclear. It seems to be a heritage of the epoch when people did not speak fields.

Boris

Maybe a definition like: a surd is an element of a field F over the rationals, generated by all real numbers r such that r^n is a (positive?) integer for some positive integer n.

I like that definition much better. Probably it would be enough to say that a "surd" is an algebraic integer of some radical extension of Q.

Alvaro

To me it appears that "surd" is used more in context of high-school education and it would seem unfit to use such a definition in the entry. Those who want to know can take this definition from the discussion.

--
"Do not meddle in the affairs of wizards for they are subtle and quick to anger."

> To me it appears that "surd" is used more in context of
> high-school education and it would seem unfit to use such a
> definition in the entry. Those who want to know can take
> this definition from the discussion.
>
>
> --
> "Do not meddle in the affairs of wizards for they are subtle
> and quick to anger."

Are you relly a mathwizard?

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