278 interval

If a function fhas values f(a) and f(b) at locations a

and b, the linear interpolation estimates the value of

the function at x, between aand b, as:

If the graph of fis smooth and the interval [a,b] is

small, then linear interpolation generally gives a good

approximation of the true value. If, however, the inter-

val is large, it is less likely that this will remain the

case. For example, given that 32= 9 and 34= 81, linear

interpolation suggests that 33(which equals 27) is

approximately 9 + (81 –9) =45.

Improved methods of interpolation take into

account other data values. For example, J

OSEPH

-L

OUIS

L

AGRANGE

’s interpolation method, called L

AGRANGE

’

S

FORMULA

, estimates function values by fitting a

POLY

-

NOMIAL

to all the known data values. Alternatively, if it

is suspected, for example, that the function is exponen-

tial, that is, of the form y= bx, then linear interpolation

works well for the logarithm of the function: the rela-

tionship log y= blog xis linear.

EXTRAPOLATION

is the process of estimating func-

tion values outside of the range of values observed.

See also

REGRESSION

.

interval Any single segment of the number line is

called an interval. More precisely, an interval is a set of

numbers containing all real numbers between two

given numbers. The given numbers are called end-

points, and they might, or might not, be regarded as

part of the interval. For example, the set of all real

numbers between, and including, 2 and 5 is an interval;

as is the set of all real numbers strictly greater than 9

but less than or equal to 23.

An interval is called closed if both endpoints are

included. Square brackets are used to indicate that this

is the case. For example,

[a,b]

denotes the set of all real numbers xthat satisfy a≤x≤b.

On a number line, the endpoints are marked by blacked-

in circles to indicate that they are included.

An interval is open if neither endpoint is included.

Parentheses are used to indicate that this is the intent.

For example:

(a,b)

denotes the set of all real numbers xthat satisfy a< x<

b. On a number line, the endpoints are marked by open

circles to indicate that they are not included.

If a variable xsatisfies either a≤x< bor a< x≤b

then it is said to be located in a half-open interval

denoted, respectively:

[a,b) or (a,b]

On a number line, a blacked-in circle marks the end-

point that is included in the interval and an open circle

the endpoint that is excluded.

With this notational system, the two intervals

described in the opening paragraph are thus denoted

[2,5] and (9,23]. Some texts use reverse square brackets

instead of parentheses to denote the exclusion of end-

points—thus an open interval is denoted ]a,b[, and a

half-open interval as [a,b[, for example.

The length of an interval is the distance between its

endpoints. For example, the interval (9, 23] has length

14, as do the intervals [9, 23], [9, 23), and (9, 23).

Incorporating the symbol ∞for infinity allows one

to denote unbounded intervals. For example, [a, ∞)

denotes the set of all real numbers xlarger than or

equal to a; (–∞,a) denotes the set of all real numbers x

strictly less than a; and so on. Sometimes mathemati-

cians write (–∞,∞) for the set of all real numbers. (One

typically does not place a square bracket next to the

infinity symbol, although it is possible to define a num-

ber system that allows for inclusion of ∞as a valid

number for consideration.)

An interval can consist of a single point (the inter-

val [1,1] for example) or can be the empty set (the

interval [4,1] for example). The intersection of any two

intervals is an interval, but the union of two intervals

might or might not be an interval.

See also

CARDINALITY

;

INTERMEDIATE

-

VALUE

THEOREM

.

invariant A property, quantity, or relationship that is

not changed by collection of specific operations or

transformations, or by how that property is observed,

is called an invariant. For example, the distance

between two points in a plane does not change under a

ROTATION

. Thus distance is invariant of rotations. (It is

not, however, invariant of

DILATION

s, for instance.) An

3 – 2

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4 – 2