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Connecting the numbers /,1,3,"ß ! The five numbers in the title are arguably the most important constants in mathematics. Without proof (just manipulating series mechanically) we can find a connection between them. You recall the “imaginary” or “complex number” symbol 3 from the days of studying the quadratic formula. The number 3 is introduced as a new “number” with the property that 3# œ ", that is 3 œ È " . We can also consider other complex numbers like " $3ß ( $) 3, *3, etc. In general, a complex number looks like D œ + ,3 where + and , are real numbers. Note that 3& 3' 3( 3) 3 3# œ " 3$ œ 3 † 3# œ 3Ð "Ñ œ 3 3% œ 3 # † 3 # œ " etc. œ 3% † 3 œ 3 œ 3% † 3# œ " œ 3% † 3$ œ 3 œ 3% † 3% œ " It turns out that all the facts of calculus can be developed using complex numbers instead of real numbers. In particular, Taylor series can be developed and it turns out that exactly the same formulas for the sin, cos, and exponential functions hold true: D# #x /D œ " D D$ $x sin D œ D cos D œ " D& &x # D #x D$ $x D %x D( (x % D% %x ' D 'x D& &x œ ! D8x for all D _ ÞÞÞÞ 8œ! ÞÞÞ 8 D œ ! Ð "Ñ8 Ð#8"Ñx for all D _ #8" D œ ! Ð "Ñ8 Ð#8Ñx for all D 8œ! _ ÞÞÞ #8 8œ! For any real number B, we can substitute the complex number D œ 3B into these series and simplify. Watch what happens: /3B œ " Ð3BÑ œ " 3B œ Ð" B# #x B# #x Ð3BÑ# #x $ Ð3BÑ$ $x 3 B$x B% %x B' 'x B% %x Ð3BÑ% %x & 3 B&x Ð3BÑ& &x B' 'x ÞÞÞ Ñ 3ÐB Ð3BÑ' 'x Ð3BÑ( (x ÞÞÞ œ ( 3 B(x ÞÞÞ œ B$ $x B& &x B( (x Ñ œ œ cos B 3 sin B (for all real B) This shows that the sin, cos and exponential functions are all intimately tied together through the complex numbers (see Math 416, Complex Variables). In particular, if we let B œ 1ß we get /31 œ cos 1 3 sin 1 œ " ! œ "Þ Therefore /31 " œ !, an equation that relates what are (arguably) the 5 most important constants in mathematics!