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The only known powers of 2 with all digits even are 2 1 = 2, 2 2 = 4, 2 3 = 8, 2 6 = 64 and 2 11 = 2048. [12] 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.
This equivalence can be extended to the negative real numbers by noting () (+) = and taking the limit as n goes to infinity. Characterization 1 ⇔ characterization 3 [ edit ]
Powers of 2 appear in set theory, since a set with n members has a power set, the set of all of its subsets, which has 2 n members. Integer powers of 2 are important in computer science. The positive integer powers 2 n give the number of possible values for an n-bit integer binary number; for example, a byte may take 2 8 = 256 different values.
On the other hand, the function / cannot be continuously extended, because the function approaches as approaches 0 from below, and + as approaches 0 from above, i.e., the function not converging to the same value as its independent variable approaching to the same domain element from both the positive and negative value sides.
In mathematics and statistics, sums of powers occur in a number of contexts: . Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the analysis of variance involves summing the squares of quantities.
Some elementary teachers use raised minus signs before numbers to disambiguate them from the operation of subtraction. [21] The same convention is also used in some computer languages. For example, subtracting −5 from 3 might be read as "positive three take away negative 5", and be shown as 3 − − 5 becomes 3 + 5 = 8, which can be read as:
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Euler's identity therefore states that the limit, as n approaches infinity, of (+ /) is equal to −1. This limit is illustrated in the animation to the right. Euler's formula for a general angle. Euler's identity is a special case of Euler's formula, which states that for any real number x,