of two quantities could be negative. We have, for
example, 5 – 7 = –2. The minus sign was first used in a
printed text in 1489 by German mathematician
Johannes Widman (1462–98).
The absolute difference of two quantities a and b is
the ABSOLUTE VALUE of the difference of the two quantities: |a – b|. The absolute difference of 13 and 8, for
example, is 5, as is the absolute difference of 8 and 13.
Some authors use the symbol ~ to denote absolute difference: 8 ~ 13 = 5.
In SET THEORY, the difference of two sets A and B
(also called the relative complement of B in A) is the set
of elements that belong to A but not to B. This difference is denoted A\B or A – B. For example, A =
{1,2,3,6,8,} and B = {2,4,5,6}, the A\B = {1,3,8}. Also,
B\A = {4,5}.
The symmetric difference of two sets A and B,
denoted either A᭞B, A+B, or AΘB, is the set of all elements that belong to one, but not both, of the two sets
A and B. It is the union of the differences A\B and B\A.
It is also the difference of the union of A and B and
their intersection:
A᭞B = (A\B)∪(B\A)
=(A∪B) – (A∩B)
For the example above, we have: A᭞B = {1,3,4,5,8}.
See also FINITE DIFFERENCES.
difference of two cubes The equation x
3 – a
3 =
(x – a)(x
2 + ax + a
2
) is called the difference of two cubes
formula. One can check that it is valid by EXPANDING
BRACKETS. Since the sum of two cubes can also be written as a difference, x
3 + a
3 = x
3 – (–a)
3
, we have a companion equation x
3 + a
3 = (x + a)(x
2 – ax + a
2
).
The DIFFERENCE OF TWO SQUARES and the difference of two cubes formulae generalize for exponents
larger than 3. We have:
x
n
– a
n = (x – a)(x
n–1 + ax
n–2 + a
2
x
n–3 + … + a
n–2
x + a
n–1
)
for n ≥ 2. This shows that the quantity x – a is always a
factor of x
n – a
n
. This observation is useful for factoring numbers. For example, we see that 6
51 – 1 is divisible by 6 – 1 = 5. Since we can also write 6
51 – 1 =
(6
3
)
17 – 1
17 , we have that 6
3 – 1 = 215 is also a factor
of 6
51 – 1.
If n is odd, then there is a companion formula:
x n + a
n = (x + a)(x
n–1 – ax
n–2 + a
2 x
n–3 – …
– a
n–2
x + a
n–1
)
This shows, for example, that 2
12 + 1 (which equals
(2
4
)
3 + 1
3
) is divisible by 17.
See also MERSENNE PRIME.
difference of two squares The equation x
2 – a
2 =
(x – a)(x + a) is called the difference-of-two-squares
formula. One can check that it is valid by EXPANDING
BRACKETS. It can also be verified geometrically: place a
small square of side-length a in one corner of a larger
square of side-length x. The area between the two
squares is x
2 – a
2
. But this L-shaped region can be
divided into two rectangles: one of length x and width
(x – a) and a second of length a and width (x – a).
These stack together to form a single (x – a) × (x + a)
rectangle. Thus it must be the case that x
2 – a
2 equals
(x – a)(x + a).
The conjugate of a sum x + a is the corresponding
difference x – a, and the conjugate of a difference x – a
is the corresponding sum x + a. Multiplying an algebraic or numeric quantity by its conjugate and invoking the difference-of-two-squares formula can often
simplify an expression. For example, if we multiply the
quantity
by “one,” we obtain:
(We have “rationalized” the denominator.)
A sum of two squares, x
2 + a
2
, can be regarded as a
difference if one is willing to work with COMPLEX NUMBERS. We have: x
2 + a
2 = x
2 – (ia)
2 = (x – ia)(x + ia).
See also DIFFERENCE OF TWO CUBES; RATIONALIZING THE DENOMINATOR.
differential Close to any point x, the graph of a differentiable function y = f(x) is well approximated by a
1
2
3
1
2
3
2
3
2
3
2
3
2
3
2
3
1
2
3
2
2
−
= −
⋅
+
+
=
+
− ( )
=
+
= +
1
2
3
−
differential 131