The graph of a function, drawn in black, and a tangent line to that function, drawn in red. The slope of the tangent line is equal to the derivative of the function at the marked point. The derivative of a function at a chosen input value describes the best linear approximation of the function near that input value. The process of finding a derivative is called differentiation. Differentiation and the derivative The simplest case, apart from the trivial case of a constant function, is when y is a linear function of x, meaning that the graph of y divided by x is a line. y + Δy = f(x + Δx) = m (x + Δx) + b = m x + m Δx + b = y + m Δx. It follows that Δy = m Δx. This gives an exact value for the slope of a line. Rate of change as a limit value Figure 1. Figure 2. Figure 3. Figure 4. The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the limit value of the ratio of the differences Δy / Δx as Δx becomes infinitely small. Notation Rigorous definition
Tetration, for n = 1, 2, 3 ..., showing convergence to the infinite power tower between the two dots of infinite power tower converges for the bases In mathematics, tetration (or hyper-4) is the next hyper operator after exponentiation, and is defined as iterated exponentiation. The word was coined by Reuben Louis Goodstein, from tetra- (four) and iteration. Tetration is used for the notation of very large numbers. Shown here are examples of the first four hyper operators, with tetration as the fourth (and succession, the unary operation denoted taking and yielding the number after , as the 0th): Additiona succeeded n times.Multiplicationa added to itself, n times.Exponentiationa multiplied by itself, n times.Tetration a exponentiated by itself, n times. where each operation is defined by iterating the previous one (the next operation in the sequence is pentation). Here, succession ( ) is the most basic operation; addition ( ) can be thought of as a chained multiplication involving n numbers a. by: . .
e (mathematical constant)Functions f(x) = ax are shown for several values of a. e is the unique value of a, such that the derivative of f(x) = ax at the point x = 0 is equal to 1. The blue curve illustrates this case, ex. For comparison, functions 2x (dotted curve) and 4x (dashed curve) are shown; they are not tangent to the line of slope 1 and y-intercept 1 (red). 2.71828182845904523536028747135266249775724709369995... (sequence A001113 in OEIS). The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant. The first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. The effect of earning 20% annual interest on an initial $1,000 investment at various compounding frequencies An account starts with $1.00 and pays 100 percent interest per year. 1.
Imaginary uniti in the complex or cartesian plane. Real numbers lie on the horizontal axis, and imaginary numbers lie on the vertical axis There are in fact two complex square roots of −1, namely i and −i, just as there are two complex square roots of every other real number, except zero, which has one double square root. In contexts where i is ambiguous or problematic, j or the Greek ι (see alternative notations) is sometimes used. In the disciplines of electrical engineering and control systems engineering, the imaginary unit is often denoted by j instead of i, because i is commonly used to denote electric current in these disciplines. For the history of the imaginary unit, see Complex number: History. Definition With i defined this way, it follows directly from algebra that i and −i are both square roots of −1. Similarly, as with any non-zero real number: i and −i and are solutions to the matrix equation Proper use (incorrect). (ambiguous). Similarly: The calculation rules Properties
Euclidean algorithmEuclid's method for finding the greatest common divisor (GCD) of two starting lengths BA and DC, both defined to be multiples of a common "unit" length. The length DC being shorter, it is used to "measure" BA, but only once because remainder EA is less than CD. EA now measures (twice) the shorter length DC, with remainder FC shorter than EA. Then FC measures (three times) length EA. Because there is no remainder, the process ends with FC being the GCD. In mathematics, the Euclidean algorithm[a], or Euclid's algorithm, is a method for computing the greatest common divisor (GCD) of two (usually positive) integers, also known as the greatest common factor (GCF) or highest common factor (HCF). The GCD of two positive integers is the largest integer that divides both of them without leaving a remainder (the GCD of two integers in general is defined in a more subtle way). The main principle is that the GCD does not change if the smaller number is subtracted from the larger number.
Quantum nonlocalityQuantum nonlocality is the phenomenon by which the measurements made at a microscopic level necessarily refute one or more notions (often referred to as local realism) that are regarded as intuitively true in classical mechanics. Rigorously, quantum nonlocality refers to quantum mechanical predictions of many-system measurement correlations that cannot be simulated by any local hidden variable theory. Many entangled quantum states produce such correlations when measured, as demonstrated by Bell's theorem. Experiments have generally favoured quantum mechanics as a description of nature, over local hidden variable theories. Any physical theory that supersedes or replaces quantum theory must make similar experimental predictions and must therefore also be nonlocal in this sense; quantum nonlocality is a property of the universe that is independent of our description of nature. Example Imagine two experimentalists, Alice and Bob, situated in separate laboratories. and P(b0|A1) = or
ParabolaThe parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas. Strictly, the adjective parabolic should be applied only to things that are shaped as a parabola, which is a two-dimensional shape. However, as shown in the last paragraph, the same adjective is commonly used for three-dimensional objects, such as parabolic reflectors, which are really paraboloids. Sometimes, the noun parabola is also used to refer to these objects. Part of a parabola (blue), with various features (other colours). Introductory images Click on any image to enlarge it. Description of final image History The earliest known work on conic sections was by Menaechmus in the fourth century BC. Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity. Thus: Substituting: is
Flower of LifeOccurrences of the ornament The Flower of Life symbol drawn in red ochre Temple of Osiris at Abydos, Egypt Abydos (Egypt) Possibly five patterns resembling the Flower of Life can be seen on one of the granite columns of the Temple of Osiris in Abydos, Egypt, and a further five on a column opposite of the building. They are drawn in red ochre and some are very faint and hard to distinguish. As a New Age symbol The New York Times quoted a New Age artist as saying, "The Flower of Life has been found in sacred sites throughout the world. References ^ Jump up to: a b Melchizedek, Drunvalo.
CardioidA cardioid generated by a rolling circle around another circle and tracing one point on the edge of it. A cardioid given as the envelope of circles whose centers lie on a given circle and which pass through a fixed point on the given circle. The name was coined by de Castillon in 1741 but had been the subject of study decades beforehand. Named for its heart-like form, it is shaped more like the outline of the cross section of a round apple without the stalk. A cardioid microphone exhibits an acoustic pickup pattern that, when graphed in two dimensions, resembles a cardioid, (any 2d plane containing the 3d straight line of the microphone body.) In three dimensions, the cardioid is shaped like an apple centred on the microphone which is the "stalk" of the apple. Equations Based on the rolling circle description, with the fixed circle having the origin as its center, and both circles having radius a, the cardioid is given by the following parametric equations: or Inverse curve