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 :