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Origin of the Word Root

Date: 02/18/2002 at 22:54:59
From: Naomi
Subject: Origin of the word "root"

If we set an expression equal to zero, we call the solution the "root" 
of the equation. Why? Where does the expression come from? Did it use 
to only apply to square roots, or is the origin of the word completely 

Thanks so much!

Date: 02/19/2002 at 08:58:06
From: Doctor Peterson
Subject: Re: Origin of the word "root"

Hi, Naomi.

According to Pat Ballew's Math Words, and Some Other Words of 
Interest, which is listed in our FAQ under Math Reference   

    The Indo-European root werad was used for the branches or roots
    of plants. Later it was generalized to mean the origins, or
    beginnings of something whether it was physical or mental. In
    arithmetic the root of a number is the number that is used to
    build up another number by repeated multiplication. Since
    8 = 2*2*2 we say that 2 is the third root of eight. The word
    root is also used in the mathematics of functions to indicate
    the value that will produce a zero (a ground level number) for
    the function. If f(x)= x^2-9 then x=3 is one of the roots or
    zeros of the function. The word was used by al-Khowarizmi in
    his writings and was translated as radix in the Latin
    translations of his algebra. 

It's not entirely clear to me whether its first application in math 
was strictly to square or cube roots, or more generally to any 
"original number" that produces some result. But in any case, both 
uses in math come from that same "root" idea, that of the source from 
which something grows, whether the number that was squared to produce 
a given number, or the number that makes a given equation true. And 
the two ideas overlap when the equation under consideration is

    x^2 - a = 0

But I've never found any clear evidence whether the use of "root" for 
any equation arose from its use in this particular equation, extending 
the idea of a square root, or whether it comes directly from the idea 
of an origin applied to any equation.

Incidentally I disagree with both you and the quotation above as to 
the use of "root" to mean "a value that produces zero." The solutions 
of ANY equation are called roots, not just those of an equation that 
sets an expression or function equal to zero. The latter are called 
either the "roots" of the equation, or the "zeroes" of the function. 
It is common to confuse the two usages and talk about the "roots of 
the function," but that is sloppy. I prefer to keep the two terms 

- Doctor Peterson, The Math Forum   

Date: 05/22/2003 at 08:14:24
From: Evan Snyman
Subject: I am trying to find the etymology of the word "root"?

As a Latin scholar, I am interested in the origin of words.  I have 
tried to look this up in both real and online dictionaries, but 
cannot seem to find any joy.  What is the etymology of the 
word "root" - as in "x=2 is the root of the equation 2x=4"?

Date: 05/22/2003 at 08:26:34
From: Doctor Sarah
Subject: Re: I am trying to find the etymology of the word "root"?

Hi Evan - thanks for writing to Dr. Math.

Here's what Steven Schwartzman has to say, in his _The Words of 
Mathematics - An Etymological Dictionary of Mathematical Terms Used in 
English_ (Mathematical Association of America):

root (noun):
from Old Norse, from the Indo-European root wrad- "branch, root." A 
plant can't exist without its roots, so metaphorically speaking root 
means "basis" or "source." When you say that the cube root of 8 is 2, 
for example, 2 is the basic number that has to be raised up, like a 
plant from its roots, until it becomes 8. Also, when we solve an 
equation, we are looking for the solutions that we know or assume to 
be "rooted" in the equation; each solution is called a root of the 
equation. In graph theory, which abounds with botanical metaphors, a 
graph is said to be rooted if one of its points, known as the root, is 
distinguished from all the other points.

- Doctor Sarah, The Math Forum
Associated Topics:
High School Definitions
High School Square & Cube Roots
Middle School Definitions
Middle School Square Roots

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