complex fraction

Examples of complex fraction in the following topics:

• Complex Fractions

  • A complex fraction is one in which the numerator, denominator, or both are fractions, which can contain variables, constants, or both.
  • A complex fraction, also called a complex rational expression, is one in which the numerator, denominator, or both are fractions.
  • When dealing with equations that involve complex fractions, it is useful to simplify the complex fraction before solving the equation.
  • The process of simplifying complex fractions, known as the "combine-divide method," is as follows:
  • Let's apply this method to the first complex fraction presented above:

• Diprotic and Polyprotic Acids

  • Given the pH and the values of Ka for each dissociation step, we can calculate each species' fractional concentration, α (alpha).
  • The fractional concentration is defined as the concentration of a particular conjugate base of interest, divided by the sum of all species' concentrations.
  • For example, a generic diprotic acid will generate three species in solution: H2A, HA-, and A2-, and the fractional concentration of HA-, which is given by:
  • The following formula shows how to find this fractional concentration of HA-, in which pH and the acid dissociation constants for each dissociation step are known:
  • The above complex equations can determine the fractional concentration of various ions from polyprotic acids.

• Partial Fractions

  • In algebra, partial fraction decomposition (sometimes called partial fraction expansion) is a procedure used to reduce the degree of either the numerator or the denominator of a rational function.
  • Reducing complex mathematical problems via partial fraction decomposition allows us to focus on computing each single element of the decomposition rather than the more complex rational function.
  • We can then write $R(x)$ as the sum of partial fractions:
  • We have rewritten the initial rational function in terms of partial fractions.
  • We have solved for each constant and have our partial fraction expansion:

• MHC Polymorphism and Antigen Binding

  • MHC molecules display a molecular fraction called an epitope and mediate interactions of leukocytes with other leukocytes or body cells.
  • Major histocompatibility complex (MHC) is a cell-surface molecule encoded by a large gene family in all vertebrates.
  • MHC molecules display a molecular fraction called an epitope and mediate interactions of leukocytes with other leukocytes or body cells.
  • The human leukocyte antigen (HLA) system is the name of the major histocompatibility complex (MHC) in humans.

• Fractions

• Mole Fraction and Mole Percent

  • Mole fractions are dimensionless, and the sum of all mole fractions in a given mixture is always equal to 1.
  • What is the mole fraction of nitrogen in the mixture?
  • What is the mole fraction of NaCl?
  • We can now find the mole fraction of the sugar:
  • Mole fraction increases proportionally to mass fraction in a solution of sodium chloride.

• Fractions

  • A fraction represents a part of a whole.
  • Find a common denominator, and change each fraction to an equivalent fraction using that common denominator.
  • To subtract a fraction from a whole number or to subtract a whole number from a fraction, rewrite the whole number as a fraction and then follow the above process for subtracting fractions.
  • To multiply a fraction by a whole number, simply multiply that number by the numerator of the fraction:
  • The process for dividing a number by a fraction entails multiplying the number by the fraction's reciprocal.

• DNA Mobility Shifts

  • Under the correct experimental conditions, the interaction between the DNA and protein is stabilized and the ratio of bound to unbound nucleic acid on the gel reflects the fraction of free and bound probe molecules as the binding reaction enters the gel.
  • An antibody that recognizes the protein can be added to this mixture to create an even larger complex with a greater shift.
  • This method is referred to as a supershift assay, and is used to unambiguously identify a protein present in the protein-nucleic acid complex.
  • Lane 3 contains protein and a DNA fragment that does react; the resulting complex is larger, heavier, and slower-moving.
  • The pattern shown in lane 3 is the one that would result if all the DNA were bound and no dissociation of complex occurred during electrophoresis.

• Fractions Involving Radicals

  • Root rationalization is a process by which any roots in the denominator of an irrational fraction are eliminated.
  • In mathematics, we are often given terms in the form of fractions with radicals in the numerator and/or denominator.
  • This same principal can be applied to fractions: whatever we do to the numerator, we must also do to the denominator, and vice versa.
  • You are given the fraction $\frac{10}{\sqrt{3}}$, and you want to simplify it by eliminating the radical from the denominator.
  • Therefore, multiply the top and bottom of the fraction by $\frac{\sqrt{3}}{\sqrt{3}}$, and watch how the radical expression disappears from the denominator:$\displaystyle \frac{10}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = {\frac{10\cdot\sqrt{3}}{{\sqrt{3}}^2}} = {\frac{10\sqrt{3}}{3}}$

• Biomolecules

  • Coordination complexes are found in many biomolecules, especially as essential ingredients for the active site of enzymes.
  • Coordination complexes (also called coordination compounds) and transition metals are widespread in nature.
  • While there are other biologically relevant molecules that also contain metals, coordination complexes contain a central metal ion and are important in many biological processes.
  • A large fraction of all proteins are metalloenzymes, and they have many diverse functions including transport, storage, and signal transduction.