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For example, 0.24999... equals 0.25, exactly as in the special case considered. These numbers are exactly the decimal fractions, and they are dense. Second, a comparable theorem applies in each radix or base. For example, in base 2 (the binary numeral system) 0.111... equals 1, and in base 3 (the ternary numeral system) 0.222
The fundamental theorem of algebra, also called d'Alembert's theorem [1] or the d'Alembert–Gauss theorem, [2] states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomials with real coefficients, since every real number is a complex number with its imaginary part ...
p is an integer factor of the constant term a 0, and; q is an integer factor of the leading coefficient a n. The rational root theorem is a special case (for a single linear factor) of Gauss's lemma on the factorization of polynomials. The integral root theorem is the special case of the rational root theorem when the leading coefficient is a n ...
Division by zero. The reciprocal function y = 1 x. As x approaches zero from the right, y tends to positive infinity. As x approaches zero from the left, y tends to negative infinity. In mathematics, division by zero, division where the divisor (denominator) is zero, is a unique and problematic special case. Using fraction notation, the general ...
Cantor's diagonal argument (among various similar names [note 1]) is a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural numbers – informally, that there are sets which in some sense contain more elements than there are positive integers.
The fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of Euclid 's Elements . If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers. — Euclid, Elements Book VII, Proposition 30.
The topmost point in the middle shows f (1/2) = 1/2. Thomae's function is a real -valued function of a real variable that can be defined as: [1] : 531. It is named after Carl Johannes Thomae, but has many other names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function ...
In mathematics, a constructive proof is a method of proof that demonstrates the existence of a mathematical object by creating or providing a method for creating the object. This is in contrast to a non-constructive proof (also known as an existence proof or pure existence theorem ), which proves the existence of a particular kind of object ...