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A result is called "deep" if its proof requires concepts and methods that are advanced beyond the concepts needed to formulate the result. For example, the prime number theorem — originally proved using techniques of complex analysis — was once thought to be a deep result until elementary proofs were found. [1]
is a sentence. This sentence means that for every y, there is an x such that =. This sentence is true for positive real numbers, false for real numbers, and true for complex numbers. However, the formula (=) is not a sentence because of the presence of the free variable y.
Once trained, such a model can detect synonymous words or suggest additional words for a partial sentence. Word2vec was developed by Tomáš Mikolov and colleagues at Google and published in 2013. Word2vec represents a word as a high-dimension vector of numbers which capture relationships between words.
A complex number can be visually represented as a pair of numbers (a, b) forming a vector on a diagram called an Argand diagram, representing the complex plane. Re is the real axis, Im is the imaginary axis, and i is the "imaginary unit", that satisfies i 2 = −1.
Binary coding systems of complex numbers, i.e. systems with the digits = {,}, are of practical interest. [9] Listed below are some coding systems , (all are special cases of the systems above) and resp. codes for the (decimal) numbers −1, 2, −2, i. The standard binary (which requires a sign, first line) and the "negabinary" systems (second ...
A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula.
A form of unary notation called Church encoding is used to represent numbers within lambda calculus. Some email spam filters tag messages with a number of asterisks in an e-mail header such as X-Spam-Bar or X-SPAM-LEVEL. The larger the number, the more likely the email is considered spam. 10: Bijective base-10: To avoid zero: 26: Bijective base-26
Algebraic numbers on the complex plane colored by degree (red=1, green=2, blue=3, yellow=4) A real number is called a real algebraic number if there is a polynomial (), with only integer coefficients, so that is a root of , that is, () =. Each real algebraic number can be defined individually using the order relation on the reals.