Numbers

This boo k deal s with number s o f the simples t kind , th e one s we lear n

as children whe n w e learn t o count, th e number s 0 , 1,2, .. . . (Zer o is

included becaus e the outcome of a count ca n be "none." ) The y ar e of -

ten calle d "natura l numbers " o r "countin g numbers " o r "nonnegativ e

integers." Her e they—an d onl y they—wil l b e calle d numbers .

Numbers ar e ordered in th e sens e tha t tw o give n number s a an d

b satisfy eithe r a b or a = b or a 6 , meanin g tha t i f tw o count s

are don e simultaneously , on e t o a an d on e t o 6 , eithe r th e coun t t o

a wil l finish first, o r the y wil l finish a t th e sam e time , o r th e coun t

to b will finish first. Normall y number s ar e visualize d a s a sequence

written fro m lef t t o right , startin g wit h 0 an d listin g th e number s

in order , continuin g (i n th e imagination ) forever . I n term s o f thi s

image, th e orde r relatio n become s th e relatio n o f a lyin g t o th e lef t

of, o r coincidin g with , o r lyin g t o th e righ t of , b.

Addition o f numbers i s very clos e conceptually t o the basi c mean -

ing o f number s a s th e outcome s o f counts . I f a se t containin g a ob -

jects i s united wit h a set containin g b objects, th e new set wil l contai n

a + b objects. Th e basi c propertie s o f additio n ar e commutativity —

the statemen t tha t a + b = b + a —and associativity—th e statemen t

that (a + b) + c = a + (b + c) . Bot h o f the m follo w fro m th e ver y

meaning o f th e operatio n o f counting .

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