# surd

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Definition
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Reference

## Mathematics Subject Classification

### surds

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

### Re: surds

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
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

### Re: surds

>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.

### Re: surds

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."

### Re: surds

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

Boris

### Re: surds

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.

### Re: surds

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

### Re: surds

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."

### Re: surds

> 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?