plex number, denoted by 1/z or z–1, which
is the inverse of the complex number z = x + i y, i.e. it is the number z–1for which zz–1
= 1. It is given by z = x/(x2+ y2) – i y/(x2+
y2). If z is written in terms of polar coordi-
nates z = r (cos θ + i sin θ) = r exp (iθ) then z–1 is given by (1/r) exp (–iθ) = (1/r) (cos
θ – i sin θ), which exists if r ≠ 0.
inverse ratio
A reciprocal ratio. For ex- ample, the inverse ratio of x to y is the ratio of 1/x to 1/y.inverse square law
A physical law in which an effect varies inversely as the square of the distance from the source pro- ducing the effect. An example is Newton’s law of universal gravitation.inverse trigonometric functions
The in- verse functions of sine, cosine, tangent, etc. For example, the inverse sine of a variable is called the arc sine of x; it is written arc sinx or sin–1x and is the angle (or number)of which the sine is x. Similarly, the other inverse trigonometric functions are: inverse cosine of x (arc cosine, written arc cosx or cos–1x)
inverse tangent of x (arc tangent, written arc tanx or tan–1x)
inverse cotangent of x (arc cotangent, writ- ten arc cotx or cot–1x)
inverse cosecant of x (arc cosecant, written arc cosecx or cosec–1x)
inverse secant of x (arc secant, written arc secx or sec–1x).
inverter gate
See logic gate.involute
/in-vŏ-loot/ The involute of a curve is a second curve that would be ob- tained by unwinding a taut string wrapped around the first curve. The involute is the curve traced out by the end of the string.I/O
See input/output.irrational number
/i-rash-ŏ-năl/ A num- ber that cannot be expressed as a ratio of two integers. The irrational numbers are precisely those infinite decimals that are not repeating. Irrational numbers are of two types:(i) Algebraic irrational numbers are roots of polynomial equations with rational numbers as coefficients. For example, √3 = 1.732 050 8… is a root of the equation x2
= 3. This equation does not have a rational solution since such a solution could be written x = m/n with m2= 3n2, but this is
impossible since 3 divides the left-hand side an even number of times and the right- hand side an odd number of times. (ii) Transcendental numbers are irrational numbers that are not algebraic, e.g. e and π.
irrational term
A term in which at least one of the indices is an irrational number. For example, xy√2and 2xπare irrational terms.irrotational vector
A vector V for which the curl of V vanishes, i.e. ∇ × V = 0. Ex- amples of irrotational vectors include the gravitational force described by Newton’s law of gravity and the electrostatic force governed by Coulomb’s law. If u is the dis- placement of a plane wave in an elastic medium (or a spherical wave a long way from the source of the wave) then the con- dition that u is an irrotational vector means that the wave is a longitudinal wave. When this result is combined with the concept of a SOLENOIDAL VECTOR andHELMHOLTZ’S THEOREMit is useful in ana- lyzing waves in seismology.
If a vector is an irrotational vector then it can be written as –1 times the gradient of a scalar function. In physical applications the scalar function is usually called the scalar potential.
isolated point
A point that satisfies the equation of a curve but is not on the main arc of the curve. For example, the equation y2(x2– 4) = x4has a solution at x = 0 and y= 0, but there is no real solution at any
point near the origin, so the origin is an iso- lated point. See also double point; multiple point.
isolated system
See closed system.isometric paper
/ÿ-sŏ-met-rik/ Paper on which there is a grid of equilaterial trian- gles printed. This type of paper is useful for drawing shapes such as cubes and cuboids, with each face being a parallelogram in the drawing. Using this type of paper it is pos- sible to draw all the dimensions of the fig- ures correctly and to measure them by using the regular grid pattern.isometry
/ÿ-som-ĕ-tree/ A transformation in which the distances between the points remain constant.isomorphic
/ÿ-sŏ-mor-fik/ See homomor- phism.isomorphism
/ÿ-sŏ-mor-fiz-ăm/ See ho- momorphism.isosceles
/ÿ-soss-ĕ-leez/ Having two equal sides. See triangle.issue price
See nominal value.iterated integral
/it-ĕ-ray-tid/ (multipleintegral) A succession of integrations per-
formed on the same function. For example, a DOUBLE INTEGRALor a TRIPLE INTEGRAL.
iteration
/it-ĕ-ray-shŏn/ A method of solv- ing a problem by successive approxima- tions, each using the result of the preceding approximation as a starting point to obtain a more accurate estimate. For example, the square root of 3 can be calculated by writ- ing the equation x2= 3 in the form 2x2= x2+ 3, or x = ½(x + 3/x). To obtain a solution for x by iteration, we might start with a first estimate, x1= 1.5. Substituting this in the equation gives the second estimate, x2 = ½(1.5 + 2) = 1.750 00. Continuing in this way, we obtain:
x3= ½(1.75 + 3/1.75) = 1.732 14 x4= ½(1.732 14 + 3/1.732 14) =
1.732 05
and so on, to any required accuracy. The difficulty in solving equations by iteration is in finding a formula for iteration (algo- rithm) that gives convergent results. In this case, for example, the algorithm xn+1 =
3/xn does not give convergent results.
There are several standard techniques, such as NEWTON’S METHOD, for obtaining convergent algorithms. Iterative calcula- tions, although often tedious for manual computation, are widely used in comput- ers.
j
An alternative to i for the square root of –1. The use of j rather than i in complex numbers is particularly common among electrical engineers.job
A unit of work submitted to a com- puter. It usually includes several programs. The information necessary to run a job is input in the form of a short program writ- ten in the job-control language (JCL) of the computer. The JCL is interpreted by theOPERATING SYSTEMand is used to identify the job and describe its requirements to the operating system. See also batch process- ing.
joule
/jool/ Symbol: J The SI unit of energy and work, equal to the work done where the point of application of a force of one newton moves one meter in the direction of action of the force. 1 J = 1 N m. The joule is the unit of all forms of energy. The unit is named for the British physicist James Prescott Joule (1818–89).Julia set
/joo-lee-ă/ A set of points in the complex plane defined by iteration of acomplex number. One takes the expression z2+ c, where z and c are complex numbers,
and calculates it for a given value of z and takes the result as a new starting value of z. This process can be repeated indefinitely and three possibilities occur, depending on the initial value for z. One is that the value tends to zero with successive iterations. Another is that the value diverges to infin- ity. There is, however, a set of initial values of z for which successive iterations give val- ues that stay in the set. This set of values is a Julia set and it can be represented by points in the complex plane. The Julia set is the boundary between values that have an attractor at zero and values that have an
ATTRACTORat infinity. The actual form of
the Julia set depends on the value of the constant complex number c. Thus, if c = 0, the iteration is z → z2. In this case the Julia
set is a circle with radius 1. There is an in- finite number of Julia sets showing a wide range of complex patterns. The set is named for the French mathematician Gas- ton Julia (1893–1978). See also Mandel- brot set.
kelvin
Symbol: K The SI base unit of ther- modynamic temperature. It is defined as the fraction 1/273.16 of the thermody- namic temperature of the triple point of water. Zero kelvin (0 K) is absolute zero. One kelvin is the same as one degree on the Celsius scale of temperature. The unit is named for the British theoretical and ex- perimental physicist Baron William Thom- son Kelvin (1824–1907).Kendall’s method
A method of measur- ing the degree of association between two different ways of ranking n objects, using two variables (x and y), which give data (x1,y1),…,(xn,yn). The objects are rankedusing first the xs and then the ys. For each of the 2n(n – 1)/2 pairs of objects a score is assigned. If the RANKof the jth object is
greater (or less) than that of the kth, re- gardless of whether the xs or ys are used, the score is plus one. If the rank of the jth is less than that of the kth using one vari- able but greater using the other, the score is minus one. Kendall’s coefficient of rank correlation τ = (sum of scores)/½n(n – 1). The closer τ is to one, the greater the degree of association between the rankings. The method is named for the British statistician Maurice Kendall (1907–83). See also Spearman’s method.
Kepler’s laws
/kep-lerz/ Laws of plane- tary motion deduced in about 1610 by the German astronomer Johannes Kepler (1571–1630) using astronomical observa- tions made by the Danish astronomer Tycho Brahe (1546–1601):(1) Each planet moves in an elliptical orbit with the Sun at one focus of the ellipse. (2) The line between the planet and the Sun sweeps out equal areas in equal times.
(3) The square of the period of each planet is proportional to the cube of the semi- major axis of the ellipse.
Application of the third law to the orbit of the Moon about the Earth gave support to Newton’s theory of gravitation.
keyboard
A computer input device that a human user can operate to type in data in the form of alphanumeric characters. It has a standard QWERTYkey layout with some additional characters and function keys. See alphanumeric; input device.kilo-
Symbol: k A prefix denoting 103. Forexample, 1 kilometer (km) = 103 meters
(m).
kilogram
/kil-ŏ-gram/ (kilogramme) Sym- bol: kg The SI base unit of mass, equal to the mass of the international prototype of the kilogram, which is a piece of plat- inum–iridium kept at Sèvres in France.kilogramme
/kil-ŏ-gram/ An alternative spelling of kilogram.kilometer
/kil-ŏ-mee-ter, kă-lom-ĕ-ter/ (km) A unit of distance equal to 1000 me- tres, approximating to 0.62 mile.kilowatt-hour
/kil-ŏ-wot/ Symbol: kwh A unit of energy, usually electrical, equal to the energy transferred by one kilowatt of power in one hour. It is the same as the Board of Trade unit and has a value of 3.6 × 106joules.kinematics
/kin-ĕ-mat-iks/ The study of the motion of objects without considera- tion of its cause. See also mechanics.kinetic energy
/ki-net-ik/ Symbol: T Thework that an object can do because of its motion. For an object of mass m moving with velocity v, the kinetic energy is mv2/2.
This gives the work the object would do in coming to rest. The rotational kinetic en- ergy of an object of moment of inertia I and angular velocity ω is given by Iω2/2.
See also energy.