complexity

Examples of complexity in the following topics:

• Roots of Complex Numbers

• Exponentials With Complex Arguments: Euler's Formula

• Trigonometry and Complex Numbers: De Moivre's Theorem

• Complex Logarithms

• Complex Conjugates and Division

  • The complex conjugate of x + yi is x - yi, and the division of two complex numbers can be defined using the complex conjugate.
  • The complex conjugate of the complex number z = x + yi is defined as x - yi.
  • The reciprocal of a nonzero complex number $z = x + yi$ is given by
  • The division of two complex numbers is defined in terms of complex multiplication (described above) and real division.
  • Practice dividing complex numbers by multiplying both the numerator and denominator by the complex conjugate of the denominator

• Introduction to Complex Numbers

  • Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part.
  • The complex number $a+bi$ can be identified with the point $(a,b)$.
  • Thus, for example, complex number $-2+3i$ would be associated with the point $(-2,3)$ and would be plotted in the complex plane as shown below.
  • A complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number.
  • The complex number $-2+3i$ is plotted in the complex plane, $2$ to the left on the real axis, and $3$ up on the imaginary axis.

• Reactions of Coordination Compounds

  • In chemistry, a coordination or metal complex consists of an atom or ion (usually metallic) and a surrounding array of bound molecules or anions known as ligands or complexing agents.
  • Many metal-containing compounds consist of coordination complexes.
  • These complexes are called chelate complexes, the formation of which is called chelation, complexation, and coordination.
  • Such labile complexes can be quite stable thermodynamically.
  • This complex, PtCl2(NH3)2, is an anti-tumor drug and an example of a coordination complex.

• More complex organic molecules and life

• What Happens When a Function Has a Complex Argument?

• Immune Complex Autoimmune Reactions

  • An immune complex is formed from the integral binding of an antibody to a soluble antigen and can function as an epitope.
  • An immune complex is formed from the integral binding of an antibody to a soluble antigen.
  • Immune complexes may cause disease when they are deposited in organs, e.g. in certain forms of vasculitis.
  • Type III hypersensitivity reactions are immune complex-mediated.
  • An immune complex is formed from the integral binding of an antibody to a soluble antigen.