approach shows that any common factor of a given set
of integers is a divisor of the greatest common factor.)
The EUCLIDEAN 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 a and b as a linear combination of a
and b, that is, it is always possible to find integers x
and y so that:
gcd(a,b) = ax + by
Similarly, the greatest common factor of any finite set
of integers a 1 ,a 2 ,…,a n can be expressed as a linear combination of the form gcd(a 1 ,a 2 ,…,a n ) = a 1 x 1 + a 2 x 2 +
…+a n x n . 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 letters 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 GREEK 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
A
α
alpha
AL-fuh
α: often denotes
an angle
B
β
beta
BAY-tuh
β: often denotes
an angle
Γ
γ
gamma GAM-uh
γ: often denotes an
angle
Γ: the GAMMA
FUNCTION
∆
δ
delta
DEL-tuh
δ: a small quantity
(EPSILON-DELTA
DEFINITION)
∆: denotes change
Ε
ε
epsilon EP-sil-on
ε: a small quantity
(EPSILON-DELTA
DEFINITION)
Ζ
ζ
zeta
ZAY-tuh
ζ: the ZETA
FUNCTION
Η
η
eta
AY-tuh
Θ
θ
theta
THAY-tuh
θ: often denotes an
angle (POLAR
COORDINATES)
Ι
ι
iota
eye-OH-tuh
ι: a small quantity
Κ
κ
kappa KAP-uh
Λ
λ
lambda LAM-duh
λ: wavelength
(in physics)
Μ
µ
mu
MYOO
µ: denotes "microns”;
Möbius function
Ν
ν
nu
NYOO
ν: frequency
(physics)
Ξ
ξ
xi
kuh-SEYE
Ο
ο
omicron OM-ee-KRON
Π
π
pi
PIE
π: the ratio of a circle
to its diameter
Π: (infinite) product
Ρ
ρ
rho
ROH
ρ: radius of a sphere
(SPHERICAL
COORDINATES)
Σ
σ
sigma
SIG-ma
σ: STANDARD
DEVIATION
Σ: SUMMATION
Τ
τ
tau
TAU
Υ
υ
upsilon OOP-si-LON
Φ
ϕ
phi
FEE
ϕ: often denotes an
angle (SPHERICAL
COORDINATES)
Χ
χ
chi
K-EYE
χ: CHI-SQUARED TEST
Ψ
ψ
psi
SIGH
ψ: wave function
(physics)
Ω
ω
omega oh-MAY-guh ω: a complex
number; the first
transfinite
ORDINAL NUMBER.
of integers is a divisor of the greatest common factor.)
The EUCLIDEAN 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 a and b as a linear combination of a
and b, that is, it is always possible to find integers x
and y so that:
gcd(a,b) = ax + by
Similarly, the greatest common factor of any finite set
of integers a 1 ,a 2 ,…,a n can be expressed as a linear combination of the form gcd(a 1 ,a 2 ,…,a n ) = a 1 x 1 + a 2 x 2 +
…+a n x n . 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 letters 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 GREEK 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
A
α
alpha
AL-fuh
α: often denotes
an angle
B
β
beta
BAY-tuh
β: often denotes
an angle
Γ
γ
gamma GAM-uh
γ: often denotes an
angle
Γ: the GAMMA
FUNCTION
∆
δ
delta
DEL-tuh
δ: a small quantity
(EPSILON-DELTA
DEFINITION)
∆: denotes change
Ε
ε
epsilon EP-sil-on
ε: a small quantity
(EPSILON-DELTA
DEFINITION)
Ζ
ζ
zeta
ZAY-tuh
ζ: the ZETA
FUNCTION
Η
η
eta
AY-tuh
Θ
θ
theta
THAY-tuh
θ: often denotes an
angle (POLAR
COORDINATES)
Ι
ι
iota
eye-OH-tuh
ι: a small quantity
Κ
κ
kappa KAP-uh
Λ
λ
lambda LAM-duh
λ: wavelength
(in physics)
Μ
µ
mu
MYOO
µ: denotes "microns”;
Möbius function
Ν
ν
nu
NYOO
ν: frequency
(physics)
Ξ
ξ
xi
kuh-SEYE
Ο
ο
omicron OM-ee-KRON
Π
π
pi
PIE
π: the ratio of a circle
to its diameter
Π: (infinite) product
Ρ
ρ
rho
ROH
ρ: radius of a sphere
(SPHERICAL
COORDINATES)
Σ
σ
sigma
SIG-ma
σ: STANDARD
DEVIATION
Σ: SUMMATION
Τ
τ
tau
TAU
Υ
υ
upsilon OOP-si-LON
Φ
ϕ
phi
FEE
ϕ: often denotes an
angle (SPHERICAL
COORDINATES)
Χ
χ
chi
K-EYE
χ: CHI-SQUARED TEST
Ψ
ψ
psi
SIGH
ψ: wave function
(physics)
Ω
ω
omega oh-MAY-guh ω: a complex
number; the first
transfinite
ORDINAL NUMBER.