Interval

Notation

Applications

Resources

An interval is a set containing all the real numbers located between any two specific real numbers on the number line. It is a property of the set of real numbers that between any two real numbers, there are infinitely many more. Thus, an interval is an infinite set. An interval may contain its endpoints, in which case it is called a closed interval. If it does not contain its end-points, it is an open interval. Intervals that include one or the other of, but not both, endpoints are referred to as half-open or half-closed.

Notation

An interval can be shown using set notation. For instance, the interval that includes all the numbers between 0 and 1, including both endpoints, is written 0 ≤x ≤1, and read “the set of all x such that 0 is less than or equal to x and x is less than or equal to 1.” The same interval with the endpoints excluded is written 0 < x < 1, where the less than symbol (<) has replaced the less than or equal to symbol (≤). Replacing only one or the other of the less than or equal to signs designates a half-open interval, such as 0 ≤x < 1, which includes the endpoint 0 but not 1. A shorthand notation, specifying only the endpoints, is also used to

KEY TERMS

Continuous —The property of a function that expresses the notion that it is unbroken in the sense that no points are missing from its graph and no sudden jumps occur in its graph.

designate intervals. In this notation, a square bracket is used to denote an included endpoint and a parenthesis is used to denote an excluded endpoint. For example, the closed interval 0 ≤x ≤1 is written [0,1], while the open interval 0 < x < 1 is written (0,1). Appropriate combinations indicate half-open intervals such as [0,1) corresponding to 0 ≤x < 1.

An interval may be extremely large, in that one of its endpoints may be designated as being infinitely large. For instance, the set of numbers greater than 1 may be referred to as the interval 1 < x < ∞, or simply (1,∞). Notice that when an endpoint is infinite, the interval is assumed to be open on that end. For example the half-open interval corresponding to the non-negative real numbers is [0,∞), and the half-open interval corresponding to the nonpositive real numbers is (-∞,0].

Applications

There are a number of places where the concept of interval is useful. The solution to an inequality in one variable is usually one or more intervals. For example, the solution to 3x + 4 ≤10 is the interval (-∞,2].

The interval concept is also useful in calculus. For instance, when a function is said to be continuous on an interval [a,b], it means that the graph of the function is unbroken, no points are missing, and no sudden jumps occur anywhere between x = a and x = b. The concept of interval is also useful in understanding and evaluating integrals. An integral is the area under a curve or graph of a function. An area must be bounded on all sides to be finite, so the area under a curve is taken to be bounded by the function on one side, the x-axis on one side and vertical lines corresponding to the endpoints of an interval on the other two sides.

See also Domain; Set theory.

Resources

BOOKS

Bittinger, Marvin L, and Davic Ellenbogen. Intermediate Algebra: Concepts and Applications. 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Burton, David M. The History of Mathematics: An Introduction. New York: McGraw-Hill, 2007.

Larson, Ron. Calculus With Analytic Geometry. Boston: Houghton Mifflin College, 2002.

Setek, William M. Fundamentals of Mathematics. Upper Saddle River, NJ: Pearson Prentice Hall, 2005.

J. R. Maddocks

Interval

An interval is a set containing all the real numbers located between any two specific real numbers on the number line. It is a property of the set of real numbers that between any two real numbers, there are infinitely many more. Thus, an interval is an infinite set. An interval may contain its endpoints, in which case it is called a closed interval. If it does not contain its endpoints, it is an open interval. Intervals that include one or the other of, but not both, endpoints are referred to as half-open or half-closed.

Notation

An interval can be shown using set notation. For instance, the interval that includes all the numbers between 0 and 1, including both endpoints, is written 0 ≤ x ≤ 1, and read "the set of all x such that 0 is less than or equal to x and x is less than or equal to 1." The same interval with the endpoints excluded is written 0 < x < 1, where the less than symbol (<) has replaced the less than or equal to symbol (≤). Replacing only one or the other of the less than or equal to signs designates a half-open interval, such as 0 ≤ x < 1, which includes the endpoint 0 but not 1. A shorthand notation, specifying only the endpoints, is also used to designate intervals. In this notation, a square bracket is used to denote an included endpoint and a parenthesis is used to denote an excluded endpoint. For example, the closed interval 0 ≤ x ≤ 1 is written [0,1], while the open interval 0 <: x < 1 is written (0,1). Appropriate combinations indicate half-open intervals such as [0,1) corresponding to 0 ≤ x < 1.

An interval may be extremely large, in that one of its endpoints may be designated as being infinitely large. For instance, the set of numbers greater than 1 may be referred to as the interval 1 < x < ∞, or simply (1,∞). Notice that when an endpoint is infinite, the interval is assumed to be open on that end. For example the half-open interval corresponding to the nonnegative real numbers is [0,∞), and the half-open interval corresponding to the nonpositive real numbers is (-∞,0].

Applications

There are a number of places where the concept of interval is useful. The solution to an inequality in one variable is usually one or more intervals. For example, the solution to 3x + 4 ≤ 10 is the interval (-∞,2].

The interval concept is also useful in calculus . For instance, when a function is said to be continuous on an interval [a,b], it means that the graph of the function is unbroken, no points are missing, and no sudden jumps occur anywhere between x = a and x = b. The concept of interval is also useful in understanding and evaluating integrals. An integral is the area under a curve or graph of a function. An area must be bounded on all sides to be finite, so the area under a curve is taken to be bounded by the function on one side, the x-axis on one side and vertical lines corresponding to the endpoints of an interval on the other two sides.

See also Domain; Set theory.

Resources

books

Bittinger, Marvin L, and Davic Ellenbogen. Intermediate Algebra: Concepts and Applications. 6th ed. Reading, MA: Addison-Wesley Publishing, 2001.

Gowar, Norman. An Invitation to Mathematics. New York: Oxford University Press, 1979.

Larson, Ron. Calculus With Analytic Geometry. Boston: Houghton Mifflin College, 2002.

J. R. Maddocks

KEY TERMS

Continuous

—The property of a function that expresses the notion that it is unbroken in the sense that no points are missing from its graph and no sudden jumps occur in its graph.

in·ter·val / ˈintərvəl/ • n. 1. an intervening time or space: after his departure, there was an interval of many years without any meetings the intervals between meals were very short.2. a pause; a break in activity: an interval of mourning. ∎ Brit. an intermission separating parts of a theatrical or musical performance. ∎  Brit. a break between the parts of an athletic contest: leading 3-0 at the interval.3. a space between two things; a gap. ∎  the difference in pitch between two musical sounds.PHRASES: at intervals1. with time between, not continuously: the light flashed at intervals.2. with spaces between: the path is marked with rocks at intervals.DERIVATIVES: in·ter·val·lic / ˌintərˈvalik/ adj.

interval. The ‘distance’ between 2 notes is called an ‘interval’, i.e. the difference in pitch between any 2 notes. The ‘size’ of any interval is expressed numerically, e.g. C to G is a 5th, because if we proceed up the scale of C the 5th note in it is G. The somewhat hollow-sounding 4th, 5th, and octave of the scale are all called perfect. They possess what we may perhaps call a ‘purity’ distinguishing them from other intervals. The other intervals, in the ascending major scale, are all called major (‘major 2nd’, ‘major 3rd’, ‘major 6th’, ‘major 7th’).

If any major interval be chromatically reduced by a semitone it becomes minor; if any perfect or minor interval be so reduced it becomes diminished; if any perfect or major interval be increased by a semitone it becomes augmented.

Enharmonic intervals are those which differ from each other in name but not in any other way (so far as modern kbd. instruments are concerned). As an example take C to G♯ (an augmented 5th) and C to A♭ (a minor 6th).

Compound intervals are those greater than an octave, e.g. C to the D an octave and a note higher, which may be spoken of either as a major 9th or as a compound major 2nd.

Inversion of intervals is the reversing of the relative position of the 2 notes defining them. It will be found that a 5th when inverted becomes a 4th, a 3rd becomes a 6th, and so on. It will also be found that perfect intervals remain perfect (C up to G a perfect 5th; G up to C a perfect 4th, etc.), while major ones become minor, minor become major, augmented become diminished, and diminished become augmented.

Every interval is either concordant or discordant. The concordant comprise all perfect intervals and all major and minor 3rds and 6ths; the discordant comprise all augmented and diminished intervals and all 2nds and 7ths. It therefore follows that all concordant intervals when inverted remain concordant and all discordant intervals remain discordant.

Musical examples of intervals are shown above.

interval period between two events XIII; space between two things XV. ult. — L. intervallum orig. space between ramparts, f. INTER- + vallum (see WALL), but the earliest forms, entrewal, entervale, intervalle, are — OF. entreval(e), later -valle (mod. intervalle).