4 1. Numbers and Logic
1.3. R
n
. W e will conside r th e plan e R
2
o f pair s (x,y) o f rea l number s an d
more generall y n-dimensiona l spac e R
n
o f n-tuple s (#i,X2,.. . ,x n) o f rea l
numbers. I n thes e spaces , b y rational s we'l l mea n n-tuple s o f rationals .
There ar e fancie r numbe r systems , suc h a s the comple x number s (whic h
will mak e a sol o appearanc e i n Chapte r 25 ) an d th e muc h fancie r "quater -
nions" (whic h mayb e you'l l se e i n som e futur e course) .
1.4. Distance . Th e (Euclidean ) distance betwee n rea l number s x an d y
is give n b y \y x\. Mor e generally , th e distanc e betwee n tw o vector s (xi),
(yi) i n R
n
i s give n b y
\y-x\ = ((yi - xi)
2
+ (y a - x 2f + + (y
n
- x
nf)1/2.
Distance satisfi.e s th e triangle inequality, whic h say s tha t th e lengt h o f on e
side o f a triangle wit h vertice s x, y, z i s less than o r equa l t o th e su m o f th e
lengths o f the othe r tw o sides :
\x z\ \x y\ + \y z\.
The Ancien t Greek s discovere d wit h disma y tha t no t ever y rea l i s
rational, wit h th e followin g example .
1.5. Proposition . \/2 is irrational
Proof. Suppos e tha t y/2 equal s p/q, i n lowes t terms , s o tha t p an d q hav e
no commo n factors . Sinc e 2 = p 2/q2,
2q2=p2.
Thus p mus t b e even : p = 2p'. Henc e
2q2
= 4p'
2,
q2
= 2p'
2
,
which mean s tha t q must b e even , a contradictio n o f lowes t terms , sinc e p
and q both hav e 2 as a factor .
There i s a n easie r example :
1.6. Proposition . log
1 0
5 is irrational.
Proof. Suppos e tha t log
1 0
5 equals p/q, whic h mean s tha t lCF/ " = 5 or
But 10p ends i n a 0 , while 5 q end s i n a 5 , contradiction .
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