Opening Thoughts:
Welcome t o th e Jungl e
One o f the mos t fundamenta l notion s i n mathematic s i s that o f
ber. Althoug h th e ide a o f numbe r i s basic , th e number s them
possess bot h nuanc e an d complexit y tha t spar k th e imaginatio
Mathematics is the queen of the sciences a
number theory is the queen of mathematics
Carl Friedrich
Welcome t o diophantin e analysis—a n are a o f numbe r theo
which w e attemp t t o discove r hidde n treasure s an d truth s w
the jungl e o f number s b y explorin g th e rationa l numbers . Dio
tine analysi s comprise s tw o differen t bu t interconnecte d doma
diophantine approximatio n an d diophantin e equations .
The rationa l number s ar e creature s tha t appea r bot h fa
and understandable . I n comparison , th e irrationa l numbers , i n
sense, see m completel y enigmatic . I f w c thin k abou t th e ra
numbers a s sittin g i n th e rea l line , the n diophantin e approxim
can b e viewe d a s a n exploratio n o f th e numbe r lin e unde r a
scope. I f w e vie w a piec e o f th e rea l numbe r lin e unde r highe
higher magnificatio n w e would se e the following :
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