Definition and Origin of the Word "Radical"

Date: 06/22/2005 at 18:35:27
From: Heather
Subject: Definition and origin of the word radical?

I was curious what the exact definition of "radical" was and also 
where and when the word originated?  I have found only a few general
definitions, such as square root symbol, which I already know.

Date: 06/22/2005 at 23:09:32
From: Doctor Peterson
Subject: Re: Definition and origin of the word radical?

Hi, Heather.

The word "radical" comes from the Latin word "radix", meaning "root",
from which we get not only the word "radical", but also "radish"!  The
other use of "radical" means "by the roots" or "from the roots", that
is, complete, such as a "radical" change in a person.  A political
"radical" is someone who wants radical change by radical means, that
is, an extremist.  Another word from the same source is "eradicate"
(originally, to "root out" or "pull out by the roots").

Now, why is a square root called a root in the first place?

Answer: a root is the source of something; if you square a number, the 
number it came from is the root, as if the square grew from it.

You can check a dictionary for the details of the history (when and
where); here are two sources that are listed in our FAQ:

  Math Words, and Some Other Words, of Interest
    http://www.pballew.net/etyindex.html 

  Earliest uses of mathematical words
    http://jeff560.tripod.com/mathword.html 

The latter says

  RADICAL. The word radical was used in English before 1668 by
  Recorde and others to refer to an irrational number.

  RADICAL SIGN appears in English in 1669 in An Introduction to
  Algebra edited in 1668 by John Pell (1611-1685):

  In the quotient subjoyn the surd part with its first radical Sign.

You may also find these pages helpful

  Etymology of the Word Radical
    http://mathforum.org/library/drmath/view/52670.html 

  History of Radicals
    http://mathforum.org/library/drmath/view/59082.html 

  History of the Radical Sign
    http://mathforum.org/library/drmath/view/52525.html 

As for an exact definition, you can of course check a dictionary; I
find that the term is used somewhat loosely, but in my usage it refers
to the symbol itself or the function represented by the symbol.  A
"radical expression" would then be an expression involving a radical,
that is, any kind of root.  Merriam-Webster says

  5 a : a mathematical expression indicating a root by means of a
  radical sign b : RADICAL SIGN

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/