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The complex numbers with integer part all zeroes in the base i – 1 system. Of particular interest are the quater-imaginary base (base 2i) and the base −1 ± i systems discussed below, both of which can be used to finitely represent the Gaussian integers without sign.
The definition of exponentiation can be extended in a natural way (preserving the multiplication rule) to define for any positive real base and any real number exponent . More involved definitions allow complex base and exponent, as well as certain types of matrices as base or exponent.
A strong base is a basic chemical compound that can remove a proton (H +) from (or deprotonate) a molecule of even a very weak acid (such as water) in an acid–base reaction. Common examples of strong bases include hydroxides of alkali metals and alkaline earth metals, like NaOH and Ca(OH) 2, respectively. Due to their low solubility, some ...
The same vector can be represented in two different bases (purple and red arrows). In mathematics, a set B of vectors in a vector space V is called a basis (pl.: bases) if every element of V may be written in a unique way as a finite linear combination of elements of B.
For example, this notion contains the split-complex numbers, which are elements of the ring [] / (as opposed to [] / (+) for complex numbers). In this ring, the equation a 2 = 1 has four solutions. The field R {\displaystyle \mathbb {R} } is the completion of Q , {\displaystyle \mathbb {Q} ,} the field of rational numbers , with respect to the ...
For example, the Euclidean topology on the plane admits as a base the set of all open rectangles with horizontal and vertical sides, and a nonempty intersection of two such basic open sets is also a basic open set. But another base for the same topology is the collection of all open disks; and here the full (B2) condition is necessary.
As usual, a weaker acid has a stronger conjugate base. Examples of Lewis bases based on the general definition of electron pair donor include: simple anions, such as H − and F −; other lone-pair-containing species, such as H 2 O, NH 3, HO −, and CH 3 −; complex anions, such as sulfate; electron-rich π-system Lewis bases, such as ethyne ...
The use of the term complex is often confused with the term complicated. In today's systems, this is the difference between myriad connecting "stovepipes" and effective "integrated" solutions. [ 17 ] This means that complex is the opposite of independent, while complicated is the opposite of simple.