of a general ellipsoid. This has proved to be a very difficult problem, and no closed-form formula exists. (The
surface area of an ellipse can only be expressed in
terms of a difficult “elliptic integral.”)
empty set (null set) Any set that contains no elements is called an empty set. For example, the set of all
real numbers greater than three and less than two is
empty, as is the set of all people with gills.
A set A is said to be a subset of a set B, written
A
B, if all elements of A belong to B. Consequently
any empty set is a subset of any other set. In particular,
if A and B are both empty, then A B and B A, and
the two empty sets are equal. This shows that there is
only one empty set. It is usually denoted as Ø, but it
can also be written as { }.
The set with the empty set as its one member is
written {Ø}, and the set with the set containing the
empty set as its lone member is written {{Ø}}. In this
way we construct a chain of sets:
Ø, {Ø}, {{Ø}}, {{{Ø}}},…
which naturally corresponds to the sequence of counting numbers 0, 1, 2, 3, … In this context one could
argue that all of mathematics arises from the empty set.
It is an interesting exercise then to give a numerical
interpretation to a two-member set of the form
{Ø,{Ø}}, for instance.
A set that is not empty is called nonempty.
See also SET THEORY.
endpoint See INTERVAL.
epicycle See CYCLOID.
epsilon-delta definition See LIMIT.
equality Two quantities are said to be equal if, in
some meaningful sense, they are equivalent. For example, the quantities 2 + 3 and 5 have the same value and
so are equal. The two sets {a,b,c} and {c,a,b} are equal
since they contain the same elements. The symbol = is
used to denote the equivalence of two quantities, and
so we write 2+3 = 5 and {a,b,c} = {c,a,b}.
Two algebraic expressions are said to be equal if
one can be transformed into the other by the standard
rules of algebra. For instance, (x + 1) 2 + 3 = x
2 + 2x +
4. Two functions are said to be equal if they have the
same domains and produce the same output value for
each input. For example, the functions f(x) = 9x and
, defined on positive values of x, are
equal.
The symbol = (a pair of parallel line segments to
denote equality) was introduced in 1557 by Welsh mathematician ROBERT RECORDE (ca. 1510–58) “because
noe 2 thynges can be more equalle.”
See also EQUATION.
equating coefficients Two polynomials f(x) = a n x
n +
a n–1 x
n–1 + … + a 1 x + a 0 and g(x) = b n x
n + b n–1 x
n–1 + …
+ b 1 x + b 0 are identical as functions, that is, give the
same output values for each input value of x, only if
the coefficients of the polynomials match: a n = b n , a n–1
= b n–1 ,…,a 0 = b 0 . (The general study of POLYNOMIALs
establishes this.) The process of matching coefficients if
two polynomials are known to be the same is called
“equating coefficients.”
For example, if x
2 equals a polynomial of the form
A + B(x – 1) + C(x – 1)(x – 2), then, after EXPANDING
BRACKETS, we have x
2 = Cx
2 + (B – 3C)x + (A – B +
2C). Equating coefficients yields: C = 1, B – 3C = 0
(and so B = 3), and A – B + 2C = 0 (and so A = 1).
Thus x
2 = 1 + 3(x – 1) + (x – 1)(x – 2). (This technique
is often used in the method of PARTIAL FRACTIONS.)
As another example, if α and β are the roots of a
quadratic equation of the form x
2 – mx + n, then: x
2 –
mx + n = (x – α) (x – β) = x
2 – (α + β)x + αβ. We conclude then that m is the sum of the roots, and n their
product.
equating real and imaginary parts Two COMPLEX
NUMBERS a + ib and c + id are equal only if a = c and b
= d. Using this fact is called “equating real and imaginary parts.” For example, if (x + iy)(2 + 3i) = 4 + 5i,
then we must have 2x – 3y = 4 and 3x + 2y = 5.
LEONHARD EULER (1707–83) made clever use of
this technique to find formulae for PYTHAGOREAN
g x x
x
x
( )
log
log
=
+
2
3
3
⊃
⊃
⊃
162 empty set
surface area of an ellipse can only be expressed in
terms of a difficult “elliptic integral.”)
empty set (null set) Any set that contains no elements is called an empty set. For example, the set of all
real numbers greater than three and less than two is
empty, as is the set of all people with gills.
A set A is said to be a subset of a set B, written
A
B, if all elements of A belong to B. Consequently
any empty set is a subset of any other set. In particular,
if A and B are both empty, then A B and B A, and
the two empty sets are equal. This shows that there is
only one empty set. It is usually denoted as Ø, but it
can also be written as { }.
The set with the empty set as its one member is
written {Ø}, and the set with the set containing the
empty set as its lone member is written {{Ø}}. In this
way we construct a chain of sets:
Ø, {Ø}, {{Ø}}, {{{Ø}}},…
which naturally corresponds to the sequence of counting numbers 0, 1, 2, 3, … In this context one could
argue that all of mathematics arises from the empty set.
It is an interesting exercise then to give a numerical
interpretation to a two-member set of the form
{Ø,{Ø}}, for instance.
A set that is not empty is called nonempty.
See also SET THEORY.
endpoint See INTERVAL.
epicycle See CYCLOID.
epsilon-delta definition See LIMIT.
equality Two quantities are said to be equal if, in
some meaningful sense, they are equivalent. For example, the quantities 2 + 3 and 5 have the same value and
so are equal. The two sets {a,b,c} and {c,a,b} are equal
since they contain the same elements. The symbol = is
used to denote the equivalence of two quantities, and
so we write 2+3 = 5 and {a,b,c} = {c,a,b}.
Two algebraic expressions are said to be equal if
one can be transformed into the other by the standard
rules of algebra. For instance, (x + 1) 2 + 3 = x
2 + 2x +
4. Two functions are said to be equal if they have the
same domains and produce the same output value for
each input. For example, the functions f(x) = 9x and
, defined on positive values of x, are
equal.
The symbol = (a pair of parallel line segments to
denote equality) was introduced in 1557 by Welsh mathematician ROBERT RECORDE (ca. 1510–58) “because
noe 2 thynges can be more equalle.”
See also EQUATION.
equating coefficients Two polynomials f(x) = a n x
n +
a n–1 x
n–1 + … + a 1 x + a 0 and g(x) = b n x
n + b n–1 x
n–1 + …
+ b 1 x + b 0 are identical as functions, that is, give the
same output values for each input value of x, only if
the coefficients of the polynomials match: a n = b n , a n–1
= b n–1 ,…,a 0 = b 0 . (The general study of POLYNOMIALs
establishes this.) The process of matching coefficients if
two polynomials are known to be the same is called
“equating coefficients.”
For example, if x
2 equals a polynomial of the form
A + B(x – 1) + C(x – 1)(x – 2), then, after EXPANDING
BRACKETS, we have x
2 = Cx
2 + (B – 3C)x + (A – B +
2C). Equating coefficients yields: C = 1, B – 3C = 0
(and so B = 3), and A – B + 2C = 0 (and so A = 1).
Thus x
2 = 1 + 3(x – 1) + (x – 1)(x – 2). (This technique
is often used in the method of PARTIAL FRACTIONS.)
As another example, if α and β are the roots of a
quadratic equation of the form x
2 – mx + n, then: x
2 –
mx + n = (x – α) (x – β) = x
2 – (α + β)x + αβ. We conclude then that m is the sum of the roots, and n their
product.
equating real and imaginary parts Two COMPLEX
NUMBERS a + ib and c + id are equal only if a = c and b
= d. Using this fact is called “equating real and imaginary parts.” For example, if (x + iy)(2 + 3i) = 4 + 5i,
then we must have 2x – 3y = 4 and 3x + 2y = 5.
LEONHARD EULER (1707–83) made clever use of
this technique to find formulae for PYTHAGOREAN
g x x
x
x
( )
log
log
=
+
2
3
3
⊃
⊃
⊃
162 empty set