approach shows that any common factor of a given set
of integers is a divisor of the greatest common factor.)
The E
UCLIDEAN ALGORITHM
can be used to find
the greatest common divisor of two integers if either of
these methods is infeasible. Repeated use of the
Euclidean algorithm will find the greatest common
divisor of more than two integers.
The Euclidean algorithm also shows that it is
always possible to write the greatest common divisor
of two integers aand bas a linear combination of a
and b, that is, it is always possible to find integers x
and yso that:
gcd(a,b) = ax + by
Similarly, the greatest common factor of any finite set
of integers a1,a2,…,ancan be expressed as a linear com-
bination of the form gcd(a1,a2,…,an) = a1x1+ a2x2+
…+anxn. In our example:
12 = gcd(72, 120, 180)
= 72 ×(1) + 120 ×(1) + 180 ×(–1)
See also
FUNDAMENTAL THEOREM OF ARITHMETIC
;
JUG
-
FILLING PROBLEM
;
RELATIVELY PRIME
.
Greek alphabet To honor the mathematical scholars
of Greek antiquity, mathematicians today often use let-
ters of the Greek alphabet to represent variables and
symbols in equations. Typically, lowercase letters are
used to represent variables (such as angles,
COMPLEX
NUMBERS
, and quantities studied in
STATISTICS
), and
uppercase letters are used for standard arithmetical and
statistical operations. The uses can vary from author to
author, however.
The following table lists the letters of the Greek
alphabet along with the common uses of some
characters.
See also G
REEK MATHEMATICS
.
Greek mathematics The ancient Greeks of ca. 600
B
.
C
.
E
. to ca. 480
C
.
E
. set the current standards of logical
rigor in mathematics. Although many ancient cultures
practiced and developed mathematics, it was the Greeks
who developed the explicit art of “proof” and explored
the power of pure deductive reasoning to its fullest.
Greek mathematics 237
Upper Lower
Case Case Name Pronunciation Use
Aαalpha AL-fuh α: often denotes
an angle
Bβbeta BAY-tuh β: often denotes
an angle
Γγgamma GAM-uh γ: often denotes an
angle
Γ: the
GAMMA
FUNCTION
∆δdelta DEL-tuh δ: a small quantity
(
EPSILON
-
DELTA
DEFINITION
)
∆: denotes change
Εεepsilon EP-sil-on ε: a small quantity
(
EPSILON
-
DELTA
DEFINITION
)
Ζζzeta ZAY-tuh ζ: the
ZETA
FUNCTION
Ηηeta AY-tuh
Θθtheta THAY-tuh θ: often denotes an
angle (
POLAR
COORDINATES
)
Ιιiota eye-OH-tuh ι: a small quantity
Κκkappa KAP-uh
Λλlambda LAM-duh λ: wavelength
(in physics)
Μµmu MYOO µ: denotes "microns”;
Möbius function
Ννnu NYOO ν: frequency
(physics)
Ξξxi kuh-SEYE
Οοomicron OM-ee-KRON
Ππpi PIE π: the ratio of a circle
to its diameter
Π: (infinite) product
Ρρrho ROH ρ: radius of a sphere
(
SPHERICAL
COORDINATES
)
Σσsigma SIG-ma σ:
STANDARD
DEVIATION
Σ:
SUMMATION
Ττtau TAU
Υυupsilon OOP-si-LON
Φϕphi FEE ϕ: often denotes an
angle (
SPHERICAL
COORDINATES
)
Χχchi K-EYE χ:
CHI
-
SQUARED TEST
Ψψpsi SIGH ψ: wave function
(physics)
Ωωomega oh-MAY-guh ω: a complex
number; the first
transfinite
ORDINAL NUMBER
.