By Luṭf Allāh Yār Muḥammadī
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Since 1 aM M M = aM = a we see that a1/M is the M th root of a. This is sometimes written 1 ∆ aM = √ M a The M th root of a real (or complex) number is not unique. , 4 = ±2). In the general case of M th roots, there are M distinct values, in general. How do we come up with M diﬀerent numbers which when raised to the M th power will yield a? The answer is to consider complex numbers in polar form. By Euler’s Identity, the real number a > 0 can be expressed, for any integer k, as a·ej2πk = a·cos(2πk)+j ·a·sin(2πk) = a+j ·a·0 = a.
1415926535897932384626433832795028841971693993751058209749 . . 7182818284590452353602874713526624977572470936999595749669 . . Let x ˆn denote the n-digit decimal expansion of an arbitrary real number x. Then x ˆn is a rational number (some integer over 10n ). 7 A First Look at Taylor Series Any “smooth” function f (x) can be expanded in the form of a Taylor series: f (x) = f (x0 ) + f (x0 ) f (x0 ) f (x0 ) (x − x0 ) + (x − x0 )2 + (x − x0 )3 + · · · . 1 1·2 1·2·3 This can be written more compactly as ∞ f (x) = f (n) (x0 ) (x − x0 )n .
3 The Exponent Zero How should we deﬁne a0 in a manner that is consistent with the properties of integer exponents? Multiplying it by a gives a0 · a = a0 a1 = a0+1 = a1 = a by property (1) of exponents. Solving a0 · a = a for a0 then gives a0 = 1 . 4 Negative Exponents What should a−1 be? , ∀M ∈ Z. O. Smith, CCRMA, Stanford, Winter 2002. edu/~jos/mdft/. CHAPTER 3. 5 Page 29 Rational Exponents A rational number is a real number that can be expressed as a ratio of two integers: L , L ∈ Z, M ∈ Z x= M Applying property (2) of exponents, we have 1 ax = aL/M = a M L Thus, the only thing new is a1/M .