Exponential function

In mathematics, an exponential function is a function of the form where b is a positive real number, in which the argument x occurs as an exponent. For real numbers c and d, a function of the form f = a b c x + d is an exponential function, as it can be rewritten as a b c x + d = x; as functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function is directly proportional to the value of the function. The constant of proportionality of this relationship is the natural logarithm of the base b: For b = 1 the real exponential function is a constant and the derivative is zero because log e b = 0, for positive a and b > 1 the real exponential functions are monotonically increasing, because the derivative is greater than zero for all arguments, for b < 1 they are monotonically decreasing, because the derivative is less than zero for all arguments. The constant e = 2.71828... is the unique base for which the constant of proportionality is 1, so that the function's derivative is itself: Since changing the base of the exponential function results in the appearance of an additional constant factor, it is computationally convenient to reduce the study of exponential functions in mathematical analysis to the study of this particular function, conventionally called the "natural exponential function", or "the exponential function" and denoted by While both notations are common, the former notation is used for simpler exponents, while the latter tends to be used when the exponent is a complicated expression.

The exponential function satisfies the fundamental multiplicative identity This identity extends to complex-valued exponents. It can be shown that every continuous, nonzero solution of the functional equation f = f f is an exponential function, f: R → R, x ↦ b x, with b > 0. The fundamental multiplicative identity, along with the definition of the number e as e1, shows that e n = e × ⋯ × e ⏟ n terms for positive integers n and relates the exponential function to the elementary notion of exponentiation; the argument of the exponential function can be any real or complex number or an different kind of mathematical object. Its ubiquitous occurrence in pure and applied mathematics has led mathematician W. Rudin to opine that the exponential function is "the most important function in mathematics". In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same proportional change in the dependent variable; this occurs in the natural and social sciences.

The graph of y = e x is upward-sloping, increases faster as x increases. The graph always lies above the x-axis but can be arbitrarily close to it for negative x; the slope of the tangent to the graph at each point is equal to its y-coordinate at that point, as implied by its derivative function. Its inverse function is the natural logarithm, denoted log, ln, or log e; the real exponential function exp: R → R can be characterized in a variety of equivalent ways. Most it is defined by the following power series: exp := ∑ k = 0 ∞ x k k! = 1 + x + x 2 2 + x 3 6 + x 4 24 + ⋯ Since the radius of convergence of this power series is infinite, this definition is, in fact, applicable to all complex numbers z ∈ C (see below for the extension

Scientific notation

Scientific notation is a way of expressing numbers that are too big or too small to be conveniently written in decimal form. It is used by scientists and engineers, in part because it can simplify certain arithmetic operations. On scientific calculators it is known as "SCI" display mode. In scientific notation all numbers are written in the form m × 10n, where the exponent n is an integer, the coefficient m is any real number; the integer n is called the order of magnitude and the real number m is called the significand or mantissa. However, the term "mantissa" may cause confusion because it is the name of the fractional part of the common logarithm. If the number is negative a minus sign precedes m. In normalized notation, the exponent is chosen so that the absolute value of the coefficient is at least one but less than ten. Decimal floating point is a computer arithmetic system related to scientific notation. Any given real number can be written in the form m×10^n in many ways: for example, 350 can be written as 3.5×102 or 35×101 or 350×100.

In normalized scientific notation, the exponent n is chosen so that the absolute value of m remains at least one but less than ten. Thus 350 is written as 3.5×102. This form allows easy comparison of numbers, as the exponent n gives the number's order of magnitude. In normalized notation, the exponent n is negative for a number with absolute value between 0 and 1; the 10 and exponent are omitted when the exponent is 0. Normalized scientific form is the typical form of expression of large numbers in many fields, unless an unnormalized form, such as engineering notation, is desired. Normalized scientific notation is called exponential notation—although the latter term is more general and applies when m is not restricted to the range 1 to 10 and to bases other than 10. Engineering notation differs from normalized scientific notation in that the exponent n is restricted to multiples of 3; the absolute value of m is in the range 1 ≤ |m| < 1000, rather than 1 ≤ |m| < 10. Though similar in concept, engineering notation is called scientific notation.

Engineering notation allows the numbers to explicitly match their corresponding SI prefixes, which facilitates reading and oral communication. For example, 12.5×10−9 m can be read as "twelve-point-five nanometers" and written as 12.5 nm, while its scientific notation equivalent 1.25×10−8 m would be read out as "one-point-two-five times ten-to-the-negative-eight meters". A significant figure is a digit in a number; this includes all nonzero numbers, zeroes between significant digits, zeroes indicated to be significant. Leading and trailing zeroes are not significant because they exist only to show the scale of the number. Therefore, 1,230,400 has five significant figures: 1, 2, 3, 0, 4; when a number is converted into normalized scientific notation, it is scaled down to a number between 1 and 10. All of the significant digits remain, thus 1,230,400 would become 1.2304 × 106. However, there is the possibility that the number may be known to six or more significant figures, in which case the number would be shown as 1.23040 × 106.

Thus, an additional advantage of scientific notation is that the number of significant figures is clearer. It is customary in scientific measurements to record all the known digits from the measurements, to estimate at least one additional digit if there is any information at all available to enable the observer to make an estimate; the resulting number contains more information than it would without that extra digit, it may be considered a significant digit because it conveys some information leading to greater precision in measurements and in aggregations of measurements. Additional information about precision can be conveyed through additional notations, it is useful to know how exact the final digit are. For instance, the accepted value of the unit of elementary charge can properly be expressed as 1.6021766208×10−19 C, shorthand for ×10−19 C. Most calculators and many computer programs present large and small results in scientific notation invoked by a key labelled EXP, EEX, EE, EX, E, or ×10x depending on vendor and model.

Because superscripted exponents like 107 cannot always be conveniently displayed, the letter E is used to represent "times ten raised to the power of" and is followed by the value of the exponent. In this usage the character e is not related to the mathematical constant e or the exponential function ex. Although the E stands for exponent, the notation is referred to as E-notation rather than exponential notation; the use of E-notation facilitates data entry and readability in textual communication since it minimizes keystrokes, avoids reduced font sizes and provides a simpler and more concise display, but it is not encouraged in some publications. In most po