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# surd

In mathematics, numbers often have to be written using roots or fractional exponents. For example, the number $\sqrt{2}$ cannot be written without using a root or writing $2^{{1/2}}$. A *surd* is an irrational number which can be written as the sum of rational powers of rational numbers.
Another example would be $3\sqrt{2}-\sqrt[3]{5}$.

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## Comments

## 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

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

## 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?