existential quantifier In mathematical logic, a symbol meaning ‘there is (are)’ It is

usually written ∃. The quantifier is fol- lowed by a variable that it is said to bind. Thus (∃x)F(x) means ‘There is something that has property F’.

expansion

A quantity expressed as a sum of a series of terms. For instance, the expression:

(x + 1)(x + 2) can be expanded to:

x2_{+ 3x + 2}

Often a function can be written as an infinite series that is convergent. The function can then be approximated to any re- quired accuracy by taking the sum of a sufficient number of terms at the beginning of the series. There are general formulae for expanding some types of expression. For example, the expansion of (1 + x)n_is

1 + nx + [n(n – 1)/2!]x2₊

[n(n – 1)(n – 2)/3!] x3_{+ …}

where x is a variable between –1 and +1, and n is an integer. See binomial expan- sion; determinant; Fourier series; Taylor series.

expectation

See expected value.

expected value

(expectation) The value of a variable quantity that is calculated to be most likely to occur. If x can take any of the set of discrete values

{x1,x2,…xn}

which have corresponding probabilities {p1,p2,…pn}, then the expected value is

E(x) = x1p1+ x2p2+ … + xnpn

If x is a continuous variable with a probability density function f(x), then

E(x) =

_–∞_∞

xf(x)dx

explicit

Denoting a function that contains no dependent variables. Compare implicit.

exponent

/eks-poh-nĕnt/ A number or symbol placed as a superscript after an expression to indicate the power to which it is raised. For example, x is an exponent in yx_{and in (ay + b)}x_.

The laws of exponents are used for combining exponents of numbers as follows: Multiplication: xa_xb_{= x}a+b Division: xa_/xb_{= x}a–b Power of a power: (xa₎b_{= x}ab Negative exponent: x–a_{= 1/x}a Fractional exponent: xa/b₌b_√xa

A number raised to the power zero is equal to 1; i.e. x0_{= 1.}

exponential

/eks-pŏ-nen-shăl/ A function or quantity that varies as the power of an- other quantity. In y = 4x_{, y is said to vary}

exponentially with respect to x. The func- tion ex_{(or expx), where e is the base of nat-}

ural logarithms, is the exponential of x. The infinite series

1 + x + x2_{/2! + x}3_{/3! + … + x}n_{/n! + …}

is equal to ex _{and is known as the expo-}

nential series. The exponential form of a complex number is

reiθ_{= r(cos}_{θ + i sinθ)}

See also complex number; Euler’s for- mula; Taylor series.

exponential series

The infinite power series that is the expansion of the function ex_,

namely:

1 + x + x2_{/2! + x}3_{/3! + …}

+ xn_{/n! + …}

This series is convergent for all real- number values of the variable x.

Replacing x by –x gives an alternating series for e–x_:

1 – x + x2_{/2! – x}3_{/3! …}

Series for sinhx and coshx can be ob- tained by combining series for ex_{and e}–x_.

expression

A combination of symbols (representing numbers of other mathematical entities) and operations; e.g. 3x2_{, √(x}2

+ 2), ex_{– 1.}

exterior angle

The angle formed on the outside of a plane figure between the ex- tension of one straight edge beyond a vertex, and the outer side of the other straight edge at that vertex. In a triangle, the exterior angle at one vertex equals the sum of the angles on the insides of the other two vertices, i.e. the sum of the interior oppo- site angles. Compare interior angle.

extraction

The process of finding a root of a number.

extrapolation

The process of estimating a value outside a known range of values. For example, if the speed of an engine is controlled by a lever, and depressing the lever by two, four, and six centimeters gives speeds of 20, 30, and 40 revolutions per second respectively, then one can ex- trapolate from this information and as- sume that depressing it by a further two centimeters will increase the speed to 50 revolutions per second. Extrapolation can be carried out graphically; for example, a graph can be drawn over a known range of values and the resulting curve extended. The further from the known range, the greater will be the uncertainty in the extrapolation. The case in which the graph of the behavior is a straight line is a linear ex- trapolation. Compare interpolation.

extrapolation

β α

γ δ

face

A flat surface on the outside of a solid figure. A cube has six identical faces.

facet

A FACE, or flat side, of a many-sided object.

factor

(divisor) A number by which an- other number is divided. See also common factor.

factorial

The product of all the whole numbers less than or equal to a number. For example, factorial 7, written 7!, is equal to 7× 6 × 5 × 4 × 3 × 2 × 1. Factorial zero is defined as 1.

factorization

The process of changing al- gebraic or numerical expressions from a sum of terms into a product. For example, the left side of the equation 4x2_{– 4x – 8 =}

0 can be factorized to (2x + 2)(2x – 4) mak- ing it easy to solve for x. As the product of the two factors is 0 when either of the fac- tors is 0, it follows that (2x + 2) = 0 and (2x – 4) = 0 will provide solutions, i.e. x = –1 and x = 2.

factor theorem

The condition that (x – a) is a factor of a polynomial f(x) in a variable x if and only if f(a) = 0. For example, if f(x) = x2_{+ x – 6, f(2) = 4 + 2 – 6 = 0 and f(–3)}

= 9 – 3 – 6 = 0, so the factors of f(x) are (x – 2) and (x + 3). The factor theorem is de- rived from the remainder theorem.

Fahrenheit degree

/fa-rĕn-hÿt/ Symbol: °F A unit of temperature difference equal to one hundred and eightieth of the difference between the temperatures of freezing and boiling water. On the Fahrenheit scale water freezes at 32°F and boils at 212°F. To convert from a temperature on the

Fahrenheit scale (TF) to a temperature on

the Celsius scale (TC) the following for-

mula is used: TF= 9TC/5 + 32. The scale is

named for the German physicist (Gabriel) Daniel Fahrenheit (1686–1736).

fallacy

See logic.

family

A set of related curves or figures. For example, the equation y = 3x + c rep- resents a family of parallel straight lines.

farad

/fa-răd, -rad/ Symbol: F The SI unit of capacitance. When the plates of a capacitor are charged by one coulomb and there is a potential difference of one volt between them, then the capacitor has a capacitance of one farad. 1 F = 1 CV–1_{, 1 farad = 1}

coulomb per volt. The unit is named for the British physicist and chemist Michael Fara- day (1791–1867).

Farey sequence

/fair-ee/ The sequence of all fractions in lowest terms whose denom- inators are less than n, where n is the order of the sequence, listed in increasing size. For example, the Farey sequence of order 5 is 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5. The sequence is named for the English mathematician John Farey (1766–1826).

fathom

A unit of length used to measure depth of water. It is equal to 6 feet (1.8288 m).

F distribution

The statistical distribution followed by the ratio of variances, s12/s22,

of pairs of random samples, size n1and n2,

taken from a normal distribution. It is used to compare different estimates of the same variance.

feedback

See cybernetics.

femto-

Symbol: f A prefix denoting 10–15_.

For example, 1 femtometer (fm) = 10–15

meter (m).

Fermat’s last theorem

/fair-mahz/ The

In document The Facts on File Dictionary of Mathematics (Page 93-96)