differential 131

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 aand bis

the

ABSOLUTE VALUE

of the difference of the two quan-

tities: |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 dif-

ference: 8 ~ 13 = 5.

In

SET THEORY

, the difference of two sets Aand B

(also called the relative complement of Bin A) is the set

of elements that belong to Abut not to B. This differ-

ence is denoted A\\Bor 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 Aand B,

denoted either AB, A+B, or AΘB, is the set of all ele-

ments that belong to one, but not both, of the two sets

Aand B. It is the union of the differences A\\Band B\\A.

It is also the difference of the union of Aand Band

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 x3– a3=

(x–a)(x2+ ax + a2) 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 writ-

ten as a difference, x3+ a3= x3– (–a)3, we have a com-

panion equation x3+ a3= (x+ a)(x2– ax + a2).

The

DIFFERENCE OF TWO SQUARES

and the differ-

ence of two cubes formulae generalize for exponents

larger than 3. We have:

xn– an= (x– a)(xn–1 + axn–2 + a2xn–3 + … + an–2x+ an–1)

for n ≥2. This shows that the quantity x– ais always a

factor of xn– an. This observation is useful for factor-

ing numbers. For example, we see that 651 – 1 is divisi-

ble by 6 – 1 = 5. Since we can also write 651 – 1 =

(63)17 – 117, we have that 63– 1 = 215 is also a factor

of 651 – 1.

If nis odd, then there is a companion formula:

xn+ an= (x+ a)(xn–1 – axn–2 + a2xn–3 – …

– an–2x+ an–1)

This shows, for example, that 212 + 1 (which equals

(24)3+ 13) is divisible by 17.

See also M

ERSENNE PRIME

.

difference of two squares The equation x2– a2=

(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 ain one corner of a larger

square of side-length x. The area between the two

squares is x2– a2. But this L-shaped region can be

divided into two rectangles: one of length xand 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 x2– a2equals

(x– a)(x+ a).

The conjugate of a sum x+ ais the corresponding

difference x– a, and the conjugate of a difference x– a

is the corresponding sum x+ a. Multiplying an alge-

braic or numeric quantity by its conjugate and invok-

ing 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, x2+ a2, can be regarded as a

difference if one is willing to work with

COMPLEX NUM

-

BERS

. We have: x2+ a2= x2– (ia)2= (x– ia)(x+ ia).

See also

DIFFERENCE OF TWO CUBES

;

RATIONALIZ

-

ING THE DENOMINATOR

.

differential Close to any point x, the graph of a dif-

ferentiable function y = f(x) is well approximated by a