6 addition

a2+ b2> c2

b2+ c2> a2

c2+ a2> b2

See also

OBTUSE ANGLE

;

PERIGON

; P

YTHAGORAS

’

S

THEOREM

;

TRIANGLE

.

addition The process of finding the sum of two

numbers is called addition. In the elementary

ARITH

-

METIC

of whole numbers, addition can be regarded as

the process of accumulating sets of objects. For exam-

ple, if a set of three apples is combined with a set of

five apples, then the result is a set of eight apples. We

write: 3 + 5 = 8.

Two numbers that are added together are called

addends. For instance, in the equation 17 + 33 = 50,

the numbers 17 and 33 are the addends, and the num-

ber 50 is their sum. Addition can also be regarded as

the process of increasing one number (an addend) by

another (called, in this context, an augend). Thus when

17 is augmented by 33 units, the result is 50.

The

PLACE

-

VALUE SYSTEM

we use today for writing

numbers simplifies the process of adding large integers.

For instance, adding together 253 and 589 yields 2 + 5

= 7 units of 100, 5 + 8 = 13 units of 10, and 3 + 9 = 12

units of 1. So, in some sense, it is reasonable to write

the answer to this addition problem simply as 7 | 13 |

12 using a vertical bar to separate units of powers of

10. Since 13 units of 10 is equivalent to one unit of 100

and three units of 10, this is equivalent to 8 | 3 | 12.

Noting, also, that 12 units of one 12 is equivalent to

one unit of 10 and two single units, this can be rewrit-

ten as 8 | 4 | 2. Thus we have: 253 + 589 = 842.

The latter process of modifying the figures into sin-

gle-digit powers of 10 (that is, in our example, the pro-

cess of rewriting 7 | 13 | 12 as 8 | 4 | 2) is called

“carrying digits.” Students in schools are usually

taught an algorithm that has one carry digits early in

the process of completing an addition problem rather

than leaving this work as the final step. Either method

is valid. (The term “carry a digit” dates back to the

time of the

ABACUS

, where beads on rods represented

counts of powers of 10 and the person had to move—

“carry”—counters from one rod to another if any

count was greater than a single digit.)

The process of addition can be extended to

NEGA

-

TIVE NUMBERS

(yielding an operation called

SUBTRAC

-

TION

), the addition of

FRACTION

s (completed with the

aid of computing

COMMON DENOMINATOR

s),

REAL

NUMBERS

,

COMPLEX NUMBERS

,

VECTOR

s, and

MATRIX

addition. The number

ZERO

is an additive

IDENTITY

ELEMENT

in the theory of arithmetic. We have that a+ 0

= a= 0 + afor any number a.

The sum of two real-valued functions fand gis the

function f+ gwhose value at any input xis the sum of

the outputs of fand gat that input value: (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+ 7x+ 7.

A function with the property that f(x+ y) = f(x)+

f(y) for all inputs xand yis called “additive.” For

example, f(x) = 2xis additive.

The addition formulae in

TRIGONOMETRY

assert:

The symbol + used to denote addition is believed to

have derived from a popular shorthand for the Latin

word et meaning “and” and was widely used by math-

ematical scholars in the late 15th century. The symbol

first appeared in print in Johannes Widman’s 1489

book Behennde unnd hüpsche Rechnung auf fallen

Kauffmannschaften (Neat and handy calculations for

all tradesmen).

See also

ASSOCIATIVE

;

CASTING OUT NINES

;

COMMUTATIVE PROPERTY

;

DISTRIBUTIVE PROPERTY

;

MULTIPLICATION

;

SUMMATION

.

affine geometry The study of those properties of

geometric figures that remain unchanged by an

AFFINE

TRANSFORMATION

is called affine geometry. For exam-

ple, since an affine transformation preserves straight

lines and

RATIO

s of distances between

POINT

s, the

notions of

PARALLEL

lines,

MIDPOINT

s of

LINE

segments,

and tangency are valid concepts in affine geometry. The

notion of a

CIRCLE

, however, is not. (A circle can be

transformed into an

ELLIPSE

via an affine transforma-

tion. The equidistance of points on the circle from the

circle center need not be preserved.)

Affine geometry was first studied by Swiss mathe-

matician L

EONHARD

E

ULER

(1707–83). Only postulates