summation 487

Substituting this formula into itself multiple times yields

the following

CONTINUED FRACTION

expansion for √

–

2:

See also

INTEGRATION BY SUBSTITUTION

;

TRANS

-

FORMATION OF COORDINATES

.

substitution rule for integration See

INTEGRATION

BY SUBSTITUTION

.

subtraction The process of finding the

DIFFERENCE

of two numbers is called subtraction. In the elementary

ARITHMETIC

of

WHOLE NUMBERS

, subtraction can be

thought of as the process of removing a subset from a

set. For example, if three apples are removed from a set

of eight apples, five apples remain. We write 8 – 3 = 5.

In a more general context, subtraction is best

defined as the inverse operation of

ADDITION

: the dif-

ference a– bis defined to be a quantity that, when

added to b, gives the answer a. Here ais called the

minuend, bthe subtrahend, and the result a– bthe dif-

ference. (Thus the subtrahend plus the difference gives

the minuend.) In some contexts it is convenient to

regard subtraction simply as the addition of

NEGATIVE

NUMBERS

. For instance, the difference 8 – 3 may be

viewed as a sum 8 + (–3).

The process of subtraction does not satisfy the

COMMUTATIVE PROPERTY

. For example, 8 – 3 does not

yield the same result as 3 – 8. (This is clear if one

rewrites these differences in terms of sums of negative

quantities: 8 + (–3) and 3 + (–8) are sums of two differ-

ent pairs of numbers.)

The

PLACE

-

VALUE SYSTEM

we use today for writing

numbers simplifies the process of subtracting large inte-

gers. For instance, subtracting 216 from 589 yields 5

– 2 = 3 units of 100, 8 – 1 = 7 units of 10, and 9 – 6 = 3

units of 1. Thus, 589 – 216 = 373. This process is still

valid even if one encounters negative quantities of units.

For example, 463 – 198 may be computed as 3 |–3| –5,

where vertical bars are used to separate powers of 10.

“Borrowing” 1 unit of 100 from the first column, which

is equivalent to 10 units of 10 in the second column,

allows us to rewrite this as 2 |7| –5. Borrowing 1 unit of

10 from the second column, which is equivalent to 10

single units in the third column, now permits us to

rewrite this as 2 |6| 5. Thus we have: 463 – 198 = 265.

Students in schools are usually taught an algorithm that

has one borrow digits early in the process of completing

a subtraction problem rather than leave this work as the

final step. Either method is valid.

The process of subtraction can be extended to

FRACTION

s (completed with the aid of computing

COM

-

MON DENOMINATOR

s),

REAL NUMBERS

,

COMPLEX NUM

-

BERS

,

VECTOR

s, and

MATRICES

.

The difference of two real-valued functions fand g

is the function f– g, whose value at any input xis the

difference of the outputs of fand gat that input: (f–g)

(x) = f(x) – g(x).For example, if f(x) = x2+ 2xand g(x)

= 5x+ 7, then (f– g)(x) = x2+ 2x– 5x– 7 = x2– 3x– 7.

The subtraction formulae in

TRIGONOMETRY

assert:

The mathematician F

IBONACCI

(ca. 1175–1250)

introduced the term minus for subtraction in his 1202

text Liber abaci (The book of the abacus), which schol-

ars later abbreviated to –

min their own work. It has

been suggested that perhaps the letter “m” was later

dropped to leave the bar “–” as the symbol of choice

for subtraction. This symbol first appeared in print in

Johannes Widman’s 1489 book Behennde und hüpsche

Rechnung auf fallen Kauffmannschaften (Neat and

handy calculations for all tradesman).

See also

ARITHMETIC

.

sufficient condition See

CONDITION

—

NECESSARY

AND SUFFICIENT

.

summation The process of finding the sum of a col-

lection of numbers is called summation. Mathematicians