E

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E, in capitalized form, and e in lower-case, is the fifth letter of the Latin-based alphabet systems, including English. It has several meanings in mathematics.

In pure mathematics:

• Euler's number (e) - a constant that is highly important in higher level mathematics. Also called Napier's Constant or the Natural Base. The exponential function $e^x$ is sometimes written $\exp(x)$ while the logarithmic function $\log_e c$ is often written $\ln c$
• A digit (E) - is the preferred, or default symbol/character of choice to represent the 14th digit of a numbering system of base 15 or higher, such as hexadecimal. Such systems are often the subject of study in the computer sciences and discrete mathematics.
• As a variable used in geometry, e may refer to the eccentricity of an ellipse or other conic section.
• In lowercase, e may take on any arbitrary meaning or value as a variable or constant.
• As a function or transformation, E may take on any arbitrary meaning.

In applied mathematics:

• Expected value (E) - the average expected outcome of a trial in the study of probability theory and statistics.
• In mensuration, the e is an SI prefix pronounced as "exa-", and is to be affixed to a unit of measure such as m (meter) or g (gram), to form the compound measures em (exameter) and eg (exagram). It represents the metric factor of 10^18.
• The elementary matrix (E) - in linear algebra, E typically represents a transformation matrix meeting a particular criteria.
• As already listed, e may refer to the eccentricity of an ellipse, such as those of orbiting bodies such as the planet Earth or the Moon.

In deductive logic and set theory:

• ∃, a backwards E, is a symbol used in logic to denote the existence of an entity in a predicate statement.

In computers and calculators:

• E can be used in place of "×10^" in scientific notation or engineering notation such that $1.9 \times 10^5$ can be equivalently written $1.9 \; \mathrm{E} \; 5$. This is common on older scientific calculator models.
• E can refer to an Error in digital computation, usually caused by a division by zero or the evaluation of a function outside of its domain. It can also written "Err", "Error", "NaN", and various other notations as a default expression returned when a computing error occurs.

In Euclidean Geometry:

• E^2 can refer to the Euclidean space, R^2.