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The MathResource contains authoritative definitions for of all the terms you see on the left. These definitions were originally written by Jonathon Borwein and Ephraim Borowski for the Collins Dictionary of Mathematics - the world�s best selling math dictionary.

Borwein, one of the world�s most cited mathematicians over the past 20 years, has worked with MathResources to combine definitions from the original Collins Dictionary with the power of Maple, the world�s most respect math engine.

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The MathResource: Interactive Math Dictionary

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Sample Definitions from The MathResource:

dy/dx, f'(x), or D_x f(x),

while the ratio of differences of which this is the limit is written

x. The process of extracting the derivative is called differentiation. For example, the first derivative of axⁿ is anx^n-1. The second derivative is the first derivative of the first derivative, and is written

d²y/dx², f''(x), D² f(x), or D_xx f(x).

These definitions are readily extended to functions of several variables; see partial derivative.

2. the Gateaux derivative or Frèchet derivative of a vector space mapping.

eccentricity,

n. 1. also called linear eccentricity. a parameter, often denoted e, that identifies the shape of a conic section as a locus of points such that the ratio between the distance of any such point from a given fixed point (the focus) and its distance from a given fixed line (the directrix) equals this constant. Clearly this constant is independent of the position, orientation and size of the curve, and so identifies a family of similarly shaped curves.

2. also called numerical eccentricity. the ratio of the linear eccentricity to half the length of the major axis of a conic, often denoted e, that is constant for a family of similar curves. If the equation of the curve is given in vertex form as

y² = 2 p x - (1 - epsilon²) x²

with 2p the length of the latus rectum, then, as shown below, if e = 0, the curve is a circle; if e < 1, the curve is an ellipse; if e = 1, it is a parabola; and if e > 1, it is a hyperbola. Compare ellipticity.

of y² = 2x - (1 - e²) x² for e shown.

fundamental theorem of algebra,

n. the theorem that a complex polynomial of the nth degree has precisely n complex roots, counting multiplicity, and hence that the complex numbers are algebraically closed.

fundamental theorem of algebra,

or higher arithmetic n. the mathematical theory that studies the properties and relations of integers, and their extensions both algebraic and analytic. This includes the study of divisibility, primality and factorization properties, partition properties, irrational and transcendental numbers, algebraic independence, rates of approximation, the representation of numbers as, for example, sums of squares, and the integer solution of polynomials in several unknowns.

Gauss, Carl Friedrich

(1777-1855), German mathematician and astronomer who is generally regarded as one of the most prolific and influential mathematicians ever. In his doctoral thesis when he was only 22, he developed the concept of complex number and used it to establish the fundamental theorem of algebra. In 1801, he published Disquisitiones arithmeticae, which firmly established number theory as a well-defined branch of mathematics. He was Professor and Director of the observatory at Gottingen from 1807 onward, and was employed by the government to conduct a trigonometric survey of the kingdom of Hanover. He obtained a wide variety of essential results in geometry, algebra, analysis, astronomy, and statistics, as well as contributing to the mathematization of the physics of electricity, magnetism and gravitation.

half-plane,

n. 1a. any subset of the two-dimensional Cartesian space bounded by a line, possibly containing a half-line in the boundary line.

b. more often, the set of points (x, y) where

c <= ax + by or c >= ax + by (closed half-planes) or corresponding strict inequalities (open half-planes), especially such a subset when the line is parallel to an axis so that the set is defined by x >= a, x <= a, y >= b, or y <= b. The shaded area in the figure shows the half-plane x > b-(b/a) y.

The line is the boundary between two half-planes.

2. any subset of the complex numbers subject to such a condition on their real and imaginary parts.

hypocycloid,

n. a curve described by a point on the circumference of a circle as it rolls around the inside of another fixed coplanar circle. The form depends on the ratio of the radii of the circles. The astroid or star curve, as shown in the figures, is a common hypocycloid with equation

The second figure is an extended hypocycloid; in each case P₁ is the initial position of the point on the radius of the smaller circle, and P₂ is another position as it rolls around the inside of the larger circle. Compare epicycloid, cycloid.

Common and extended hypocycloid.

infinite series,

n. the sum of an infinite sequence of terms, usually indexed by, or expressed as a function on, the natural numbers, and often written in the form

or, more briefly,

_i a_i. If the sequence of partial sums tends to a limit as the number of terms increases, the series is said to converge; for example, the series

+...

converges to 2.

Klein bottle,

n. a closed surface with only one side and no interior; if it is cut in half lengthwise, two Möbius strips result. It cannot be realized in three-dimensional space, but a model can be formed by inserting the narrower end of an open tapered tube through the wider end and then sealing the ends to one another, as shown in the figure below.