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In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers, then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.}
One use of conjugate acids and bases lies in buffering systems, which include a buffer solution. In a buffer, a weak acid and its conjugate base (in the form of a salt), or a weak base and its conjugate acid, are used in order to limit the pH change during a titration process. Buffers have both organic and non-organic chemical applications.
Conjugate gradient, assuming exact arithmetic, converges in at most n steps, where n is the size of the matrix of the system (here n = 2). In mathematics , the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations , namely those whose matrix is positive-semidefinite .
The conjugate transpose of a matrix with real entries reduces to the transpose of , as the conjugate of a real number is the number itself. The conjugate transpose can be motivated by noting that complex numbers can be usefully represented by 2 × 2 {\displaystyle 2\times 2} real matrices, obeying matrix addition and multiplication: a + i b ≡ ...
In mathematics, the complex conjugate root theorem states that if P is a polynomial in one variable with real coefficients, and a + bi is a root of P with a and b being real numbers, then its complex conjugate a − bi is also a root of P. [1]
Conjugate addition of organocuprates is widely used in organic synthesis. Vinyl ether cuprates serve as convenient acyl anion equivalents in conjugate addition reactions to enones. The resulting enol ethers can be hydrolyzed to 1,4-diketones, which are difficult to access using conventional carbonyl chemistry.
H 2 O is a base because it accepts a proton from CH 3 COOH and becomes its conjugate acid, the hydronium ion, (H 3 O +). [9] The reverse of an acid–base reaction is also an acid–base reaction, between the conjugate acid of the base in the first reaction and the conjugate base of the acid.
The subgroups can thus be divided into conjugacy classes, with two subgroups belonging to the same class if and only if they are conjugate. Conjugate subgroups are isomorphic, but isomorphic subgroups need not be conjugate. For example, an abelian group may have two different subgroups which are isomorphic, but they are never conjugate.
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