“stack” pieces of the cube during the slicing process so
as to cut through several pieces at once. (To see this,
consider the innermost cube among the 27 little cubes.
It has six planar faces, each of which need to be cut. As
no single planar cut will slice two of those faces, a minimum of six cuts is indeed required to form the 27 little
cubes.) It is possible, however, to dice a large cube into
4 × 4 × 4 = 64 smaller cubes in fewer than nine slices if
“stacking” is permitted (six will suffice). Nine slices
suffice to dice a cube into 5 × 5 × 5 = 125 smaller
cubes. In general, the minimum number of slices
required to subdivide a large cube into n × n × n = n
3
smaller cubes is given by the formula:
3 log 2 n
making use of the CEILING FUNCTION.
The four-dimensional analog of a cube is a
HYPERCUBE.
See also FLOOR/CEILING/FRACTIONAL PART FUNCTIONS; DUPLICATING THE CUBE; PARALLELEPIPED; PRISM.
cube root/nth root The cube root of a number a is a
value x such that x
3 = a. We write
3
√
–
a for the cube root
of a.
Every real number a has exactly one real cube root.
For instance, the cube root of 27 is 3 (since 3 × 3 × 3
equals 27), and the cube root of –8 is –2 (since –2 × –2
× –2 equals –8). If one works within the realm of complex numbers, then the FUNDAMENTAL THEOREM OF
ALGEBRA shows that every number has exactly three
complex cube roots. For instance, the three cube roots
of 27 are
, and 3. A study of
the nth roots of unity shows that the three complex
cube roots of a number lie on the vertices of an equilateral triangle in the complex plane.
In general, the nth root of a number a is a value x
such that x
n = a. Again, the fundamental theorem of
algebra shows that every number has exactly n complex nth roots. If a is a real number, then a has a real
nth root if a is positive. (For example, a fourth root of
16 is 2.) If a is a negative real number, then a real nth
root exists only if n is odd. (For example, –243 has a
fifth root, namely –3, but no real fourth root.) The real
nth root of a number a, if it exists, is denoted
n
√
–
a .
See also CUBIC EQUATION; ROOT; SQUARE ROOT.
cubic equation Any degree-three POLYNOMIAL equation of the form ax
3 + bx
2 + cx + d = 0 with a ≠ 0 is
called a cubic equation.
During the Renaissance, scholars sought for a general arithmetic formula in terms of the coefficients a, b,
c, and d that would solve all cubic equations (one akin
to the famous QUADRATIC formula for solving degreetwo equations). At the time, however, scholars were not
comfortable working with NEGATIVE NUMBERS, or with
ZERO as a number, and wrote equations in a form that
avoided their appearance. (For instance, the cubic x
3 –
2x + 5 = 0 was cast as x
3 + 5 = 2x.) Mathematicians
consequently thought that there were eight different
types of cubic equations to solve.
Italian scholar SCIPIONE DEL FERRO (1465–1526)
was the first to make progress in solving certain cubics,
but never published his results. Later, the scholar NICCOLÒ TARTAGLIA (ca. 1499–1557) succeeded in solving
some additional classes of cubics. GIROLAMO CARDANO (1501–76) published Tartaglia’s work (without
Tartaglia’s consent) in his epic 1545 piece Ars magna
(The great art) and developed a general approach that
solves all cubic equations. The formula he devised is
today called “Cardano’s formula” or the “CardanoTartaglia formula.” We describe it here using modern
notation:
By dividing the cubic equation through by the leading coefficient a, we can assume that we are working
with a cubic of the form:
x
3 + Bx
2 + Cx + D = 0
for numbers
,
, and
. Substituting
simplifies the equation further to one without
a square term:
y
3 + py + q = 0
(Here
and
.) This form
of the cubic is called the reduced cubic, and any solution y to this equation corresponds to a solution
of the original equation.
Assume that the equation has a solution that can
be written as the sum of two quantities: y = u + v. Substituting these variables yields:
b
x = y – —
3a
2B
3
BC
q = — – — + D
27
3
B
2
p = C – —
3
B
x = y – —
3
d
D = — a
c
C = — a
b
B = — a
− +
− −
3
2
3 3
2
3
2
3 3
2
i
i
,
112 cube root/nth root
as to cut through several pieces at once. (To see this,
consider the innermost cube among the 27 little cubes.
It has six planar faces, each of which need to be cut. As
no single planar cut will slice two of those faces, a minimum of six cuts is indeed required to form the 27 little
cubes.) It is possible, however, to dice a large cube into
4 × 4 × 4 = 64 smaller cubes in fewer than nine slices if
“stacking” is permitted (six will suffice). Nine slices
suffice to dice a cube into 5 × 5 × 5 = 125 smaller
cubes. In general, the minimum number of slices
required to subdivide a large cube into n × n × n = n
3
smaller cubes is given by the formula:
3 log 2 n
making use of the CEILING FUNCTION.
The four-dimensional analog of a cube is a
HYPERCUBE.
See also FLOOR/CEILING/FRACTIONAL PART FUNCTIONS; DUPLICATING THE CUBE; PARALLELEPIPED; PRISM.
cube root/nth root The cube root of a number a is a
value x such that x
3 = a. We write
3
√
–
a for the cube root
of a.
Every real number a has exactly one real cube root.
For instance, the cube root of 27 is 3 (since 3 × 3 × 3
equals 27), and the cube root of –8 is –2 (since –2 × –2
× –2 equals –8). If one works within the realm of complex numbers, then the FUNDAMENTAL THEOREM OF
ALGEBRA shows that every number has exactly three
complex cube roots. For instance, the three cube roots
of 27 are
, and 3. A study of
the nth roots of unity shows that the three complex
cube roots of a number lie on the vertices of an equilateral triangle in the complex plane.
In general, the nth root of a number a is a value x
such that x
n = a. Again, the fundamental theorem of
algebra shows that every number has exactly n complex nth roots. If a is a real number, then a has a real
nth root if a is positive. (For example, a fourth root of
16 is 2.) If a is a negative real number, then a real nth
root exists only if n is odd. (For example, –243 has a
fifth root, namely –3, but no real fourth root.) The real
nth root of a number a, if it exists, is denoted
n
√
–
a .
See also CUBIC EQUATION; ROOT; SQUARE ROOT.
cubic equation Any degree-three POLYNOMIAL equation of the form ax
3 + bx
2 + cx + d = 0 with a ≠ 0 is
called a cubic equation.
During the Renaissance, scholars sought for a general arithmetic formula in terms of the coefficients a, b,
c, and d that would solve all cubic equations (one akin
to the famous QUADRATIC formula for solving degreetwo equations). At the time, however, scholars were not
comfortable working with NEGATIVE NUMBERS, or with
ZERO as a number, and wrote equations in a form that
avoided their appearance. (For instance, the cubic x
3 –
2x + 5 = 0 was cast as x
3 + 5 = 2x.) Mathematicians
consequently thought that there were eight different
types of cubic equations to solve.
Italian scholar SCIPIONE DEL FERRO (1465–1526)
was the first to make progress in solving certain cubics,
but never published his results. Later, the scholar NICCOLÒ TARTAGLIA (ca. 1499–1557) succeeded in solving
some additional classes of cubics. GIROLAMO CARDANO (1501–76) published Tartaglia’s work (without
Tartaglia’s consent) in his epic 1545 piece Ars magna
(The great art) and developed a general approach that
solves all cubic equations. The formula he devised is
today called “Cardano’s formula” or the “CardanoTartaglia formula.” We describe it here using modern
notation:
By dividing the cubic equation through by the leading coefficient a, we can assume that we are working
with a cubic of the form:
x
3 + Bx
2 + Cx + D = 0
for numbers
,
, and
. Substituting
simplifies the equation further to one without
a square term:
y
3 + py + q = 0
(Here
and
.) This form
of the cubic is called the reduced cubic, and any solution y to this equation corresponds to a solution
of the original equation.
Assume that the equation has a solution that can
be written as the sum of two quantities: y = u + v. Substituting these variables yields:
b
x = y – —
3a
2B
3
BC
q = — – — + D
27
3
B
2
p = C – —
3
B
x = y – —
3
d
D = — a
c
C = — a
b
B = — a
− +
− −
3
2
3 3
2
3
2
3 3
2
i
i
,
112 cube root/nth root