A function f:X → Y is said to be “one-to-one” (or
“injective”) if no two different elements of X yield the
same output. For example, the “is the mother of” function is not one-to-one: two different people could have
the same mother. The squaring function, thought of as
a map from the set {1,2,3,4} to the set{1,4,9,16}, is oneto-one. It is not one-to-one, however, when thought of
as a number from the set of all real numbers to the set
of all real numbers: the numbers 2 and –2, for example, yield the same output.
A function that is both one-to-one and onto is
called a “bijection” (or sometimes a PERMUTATION). A
bijection f:X → Y has the property that each element of
Y “comes from” one, and only one, element of X. Thus
it is possible to define the inverse function, denoted f
–1
:
Y → X, which associates to each element y of Y the element of X from whence it came. Algebraically, if the
dependent variable y is given as an explicit formula in
terms of x, then the inverse function determines the
independent variable x as a formula in terms of y. For
example, the inverse of the function y = 3x + 2 is given
by x = (y – 2)/3. (That is, the inverse operation of
“tripling a number and adding two” is to “subtract
two and then divide by three.”) The roles of the variables x and y have switched, and thus the graph of the
inverse of a function can be found by switching the xand y-axes on the graph of the original function.
In the theory of CARDINALITY, bijections play a key
role in determining whether or not two sets X and Y
have the same “size.”
See also ALGEBRAIC NUMBER; HISTORY OF FUNCTIONS (essay).
fundamental theorem of algebra (d’Alembert’s theorem) The following important theorem in mathematics is deemed fundamental to the theory of algebra:
Every polynomial p(z) = a n z
n + a n–1 z
n–1 +…+
a 1 z + a 0 with coefficients a i either real or complex numbers, a n ≠ 0, has at least one root.
That is, there is at least one complex number α
such that p(α) = 0.
By the FACTOR THEOREM we must then have p(z) =
(z – α)q(z) for some polynomial q(z) of degree n – 1.
Applying the fundamental theorem of algebra to the
polynomial q(z), and again to each polynomial of
degree greater than one that appears, shows that the
polynomial p(z) factors completely into n (not necessarily distinct) linear factors. We have as a consequence:
Every polynomial p(z) = a n z
n + a n–1 z
n–1 +…+
a 1 z + a 0 with coefficients a i either real or complex numbers, a n ≠ 0, factors completely as
p(z) = a n (z – α 1 )(z – α 2 )…(z – α n ) for some
complex numbers α 1 , α 2 , …, α n .
Consequently, in the FIELD of complex numbers, every
polynomial of degree n has precisely n roots (when
counted with multiplicity). For instance, the polynomial z
4 – 2z
3 + 2z
2 – 2z + 1 factors as (z – 1)(z – 1)
(z – i)(z + i) with the root 1 appearing twice. Mathematicians call a field algebraically closed if every
degree-n polynomial with coefficients from that field
has precisely n roots in that field. The set of complex
numbers is thus algebraically closed. (The field of real
numbers, however, is not. The polynomial p(x) = x
2 + 1,
for instance, does not factor within the reals.)
The fundamental theorem of algebra was first conjectured by Dutch mathematician Albert Girard in 1629
in his investigation of imaginary roots. CARL FRIEDRICH
GAUSS (1777–1855) was the first to prove the result in
his 1799 doctoral thesis. He later re-proved the result
several times throughout his life using a variety of different mathematical approaches, and he gave it the
name the “fundamental theorem of algebra.” In France,
the result is known as d’Alembert’s theorem to honor
the work of JEAN LE ROND D’ALEMBERT (1717–83) and
his many (unsuccessful) attempts to prove it.
To prove the theorem, it suffices to consider a complex polynomial with leading coefficient equal to one:
p(z) = z
n + a n –1 z
n– 1 +…+ a 1 z + a 0 . (Divide through by
a n if necessary.) Notice that if a 0 = 0, then the polynomial has one root, namely z = 0, and there is nothing
more to establish. Suppose then that a 0 is a complex
number different from zero.
Using EULER’S FORMULA, regard the variable z as a
complex number of the form z = Re
iθ = R(cosθ + isinθ),
where R is a nonzero real number and θ is an angle.
Notice that as θ varies from zero to 360°, z = R(cosθ +
isinθ) traces a circle of radius R and z
n = R
n e
inθ =
R n (cos(nθ) + isin(nθ)) wraps around a circle of radius R n
n times. Notice, too, that if R is large, then p(z) = z
n +
a n –1 z
n– 1 +…+ a 1 z + a 0 = z
n
is well approximated as z
n (1 + 0 +…+ 0 + 0) = z
n . Thus
1
1
1
1
0
+
+ +
+
−
−
a
z
a
z
a
z
n
n
n
L
fundamental theorem of algebra 209
“injective”) if no two different elements of X yield the
same output. For example, the “is the mother of” function is not one-to-one: two different people could have
the same mother. The squaring function, thought of as
a map from the set {1,2,3,4} to the set{1,4,9,16}, is oneto-one. It is not one-to-one, however, when thought of
as a number from the set of all real numbers to the set
of all real numbers: the numbers 2 and –2, for example, yield the same output.
A function that is both one-to-one and onto is
called a “bijection” (or sometimes a PERMUTATION). A
bijection f:X → Y has the property that each element of
Y “comes from” one, and only one, element of X. Thus
it is possible to define the inverse function, denoted f
–1
:
Y → X, which associates to each element y of Y the element of X from whence it came. Algebraically, if the
dependent variable y is given as an explicit formula in
terms of x, then the inverse function determines the
independent variable x as a formula in terms of y. For
example, the inverse of the function y = 3x + 2 is given
by x = (y – 2)/3. (That is, the inverse operation of
“tripling a number and adding two” is to “subtract
two and then divide by three.”) The roles of the variables x and y have switched, and thus the graph of the
inverse of a function can be found by switching the xand y-axes on the graph of the original function.
In the theory of CARDINALITY, bijections play a key
role in determining whether or not two sets X and Y
have the same “size.”
See also ALGEBRAIC NUMBER; HISTORY OF FUNCTIONS (essay).
fundamental theorem of algebra (d’Alembert’s theorem) The following important theorem in mathematics is deemed fundamental to the theory of algebra:
Every polynomial p(z) = a n z
n + a n–1 z
n–1 +…+
a 1 z + a 0 with coefficients a i either real or complex numbers, a n ≠ 0, has at least one root.
That is, there is at least one complex number α
such that p(α) = 0.
By the FACTOR THEOREM we must then have p(z) =
(z – α)q(z) for some polynomial q(z) of degree n – 1.
Applying the fundamental theorem of algebra to the
polynomial q(z), and again to each polynomial of
degree greater than one that appears, shows that the
polynomial p(z) factors completely into n (not necessarily distinct) linear factors. We have as a consequence:
Every polynomial p(z) = a n z
n + a n–1 z
n–1 +…+
a 1 z + a 0 with coefficients a i either real or complex numbers, a n ≠ 0, factors completely as
p(z) = a n (z – α 1 )(z – α 2 )…(z – α n ) for some
complex numbers α 1 , α 2 , …, α n .
Consequently, in the FIELD of complex numbers, every
polynomial of degree n has precisely n roots (when
counted with multiplicity). For instance, the polynomial z
4 – 2z
3 + 2z
2 – 2z + 1 factors as (z – 1)(z – 1)
(z – i)(z + i) with the root 1 appearing twice. Mathematicians call a field algebraically closed if every
degree-n polynomial with coefficients from that field
has precisely n roots in that field. The set of complex
numbers is thus algebraically closed. (The field of real
numbers, however, is not. The polynomial p(x) = x
2 + 1,
for instance, does not factor within the reals.)
The fundamental theorem of algebra was first conjectured by Dutch mathematician Albert Girard in 1629
in his investigation of imaginary roots. CARL FRIEDRICH
GAUSS (1777–1855) was the first to prove the result in
his 1799 doctoral thesis. He later re-proved the result
several times throughout his life using a variety of different mathematical approaches, and he gave it the
name the “fundamental theorem of algebra.” In France,
the result is known as d’Alembert’s theorem to honor
the work of JEAN LE ROND D’ALEMBERT (1717–83) and
his many (unsuccessful) attempts to prove it.
To prove the theorem, it suffices to consider a complex polynomial with leading coefficient equal to one:
p(z) = z
n + a n –1 z
n– 1 +…+ a 1 z + a 0 . (Divide through by
a n if necessary.) Notice that if a 0 = 0, then the polynomial has one root, namely z = 0, and there is nothing
more to establish. Suppose then that a 0 is a complex
number different from zero.
Using EULER’S FORMULA, regard the variable z as a
complex number of the form z = Re
iθ = R(cosθ + isinθ),
where R is a nonzero real number and θ is an angle.
Notice that as θ varies from zero to 360°, z = R(cosθ +
isinθ) traces a circle of radius R and z
n = R
n e
inθ =
R n (cos(nθ) + isin(nθ)) wraps around a circle of radius R n
n times. Notice, too, that if R is large, then p(z) = z
n +
a n –1 z
n– 1 +…+ a 1 z + a 0 = z
n
is well approximated as z
n (1 + 0 +…+ 0 + 0) = z
n . Thus
1
1
1
1
0
+
+ +
+
−
−
a
z
a
z
a
z
n
n
n
L
fundamental theorem of algebra 209