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Also unlike addition and multiplication, exponentiation is not associative: for example, (2 3) 2 = 8 2 = 64, whereas 2 (3 2) = 2 9 = 512. Without parentheses, the conventional order of operations for serial exponentiation in superscript notation is top-down (or right -associative), not bottom-up [ 23 ] [ 24 ] [ 25 ] (or left -associative).
The first 3 powers of 2 with all but last digit odd is 2 4 = 16, 2 5 = 32 and 2 9 = 512. The next such power of 2 of form 2 n should have n of at least 6 digits. The only powers of 2 with all digits distinct are 2 0 = 1 to 2 15 = 32 768 , 2 20 = 1 048 576 and 2 29 = 536 870 912 .
Characterization 2 ⇒ characterization 5 [ edit ] In the sense of definition 2, the equation exp ( x + y ) = exp ( x ) exp ( y ) {\displaystyle \exp(x+y)=\exp(x)\exp(y)} follows from the term-by-term manipulation of power series justified by uniform convergence , and the resulting equality of coefficients is just the Binomial theorem .
The sequence starts with a unary operation (the successor function with n = 0), and continues with the binary operations of addition (n = 1), multiplication (n = 2), exponentiation (n = 3), tetration (n = 4), pentation (n = 5), etc. Various notations have been used to represent hyperoperations.
The zeta function values listed below include function values at the negative even numbers (s = −2, −4, etc.), for which ζ(s) = 0 and which make up the so-called trivial zeros. The Riemann zeta function article includes a colour plot illustrating how the function varies over a continuous rectangular region of the complex plane.
Geometrically, when moving increasingly farther to the right along the -axis, the value of / approaches 0. This limiting behavior is similar to the limit of a function lim x → x 0 f ( x ) {\textstyle \lim _{x\to x_{0}}f(x)} in which the real number x {\displaystyle x} approaches x 0 , {\displaystyle x_{0},} except that there is no real number ...
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{1, 3, 5, 7, 9, ...; 2, 4, 6, 8, 10, ...} is an ordering of the set (with cardinality ℵ 0) of positive integers. If the axiom of countable choice (a weaker version of the axiom of choice) holds, then ℵ 0 is smaller than any other infinite cardinal, and is therefore the (unique) least infinite ordinal.