ample 0/0.
index
(pl. indexes or indices) A number that indicates a characteristic or function in a mathematical expression. For exam- ple, in y4, the exponent, 4, is also known asinclined plane
Incircle: an incircle touches all the sides of its surrounding figure.
the index. Similarly in 3√27 and log
10x, the numbers 3 and 10 respectively are called indices (or indexes).
indirect proof
(reductio ad absurdum) A logical argument in which a proposition or statement is proved by showing that its negation or denial leads to a CONTRADIC-TION. Compare direct proof.
induction
/in-duk-shŏn/ 1. (mathematical induction) A method of proving mathe- matical theorems, used particularly for se- ries sums. For instance, it is possible to show that the series 1 + 2 + 3 + 4 + … has a sum to n terms of n(n + 1)/2. First we show that if it is true for n terms it must also be true for (n + 1) terms. According to the formulaSn= n(n + 1)/2
if the formula is correct, the sum to (n + 1) terms is obtained by adding (n + 1) to this
Sn+1= n(n + 1)/2 + (n + 1)
Sn+1= (n + 1)(n + 2)/2
This agrees with the result obtained by replacing n in the general formula by (n + 1), i.e.:
Sn+1= (n + 1)(n + 1 + 1)/2
Sn+1= (n + 1)(n + 2)/2
Thus, the formula is true for (n + 1) terms if it is true for n terms. Therefore, if it is true for the sum to one term (n = 1), it must be true for the sum to two terms (n + 1). Similarly, if true for two terms, it must be true for three terms, and so on through all values of n. It is easy to show that it is true for one term:
Sn= 1(1 + 1)/2
Sn= 1
which is the first term in the series. Hence the theorem is true for all integer values of n.
2. In logic, a form of reasoning from indi-
vidual cases to general ones, or from ob- served instances to unobserved ones. Inductive arguments can be of the form: F1 is A, F2is A … Fnis A, therefore all Fs are
A (‘this swan has wings, that swan has wings … therefore all swans have wings’); or: all Fs observed so far are A, therefore all Fs are A (‘all swans observed so far are white, therefore all swans are white’). Un- like deduction, asserting the premisses
while denying the conclusion in an induc- tion does not lead to a CONTRADICTION. The conclusion is not guaranteed to be true if the premisses are. Compare deduction.
inelastic collision
/in-i-las-tik/ A collision for which the restitution coefficient is less than one. In effect, the relative velocity after the collision is less than that before; the kinetic energy of the bodies is not con- served in the collision, even though the sys- tem may be closed. Some of the kinetic energy is converted into internal energy. See also restitution, coefficient of.inequality
A relationship between two expressions that are not equal, often writ- ten in the form of an equation but with the symbols > or < meaning ‘is greater than’ and ‘is less than’. For example, if x < 4 then x2 < 16. If y2 > 25, then y > 5 or y < –5. Ifthe end values are included, the symbols ≥ (is greater than or equal to) and ≤ (is less than or equal to) are used. When one quan- tity is very much smaller or greater than another, it is shown by << or >>. For ex- ample, if x is a large number x >> 1/x or 1/x << x. See also equality.
inequation
/in-i-kway-zhŏnz/ Another word for inequality.inertia
/i-ner-shă/ An inherent property of matter implied by Newton’s first law of motion: inertia is the tendency of a body to resist change in its motion. See also inertial mass; Newton’s laws of motion.inertial mass
/i-ner-shăl/ The mass of an object as measured by the property of iner- tia. It is equal to the ratio force/accelera- tion when the object is accelerated by a constant force. In a uniform gravitational field, it appears to be equal to GRAVITA-TIONAL MASS – all objects have the same gravitational acceleration at the same place.
inertial system
A frame of reference in which an observer sees an object that is free of all external forces to be moving at con- stant velocity. The observer is called an in- ertial observer. Any FRAME OF REFERENCEinertial system
that moves with constant velocity and without rotation relative to an inertial frame is also an inertial frame. NEWTON’S LAWS OF MOTIONare valid in any inertial frame (but not in an accelerated frame), and the laws are therefore independent of the velocity of an inertial observer.
inf
See infimum.inference
/in-fĕ-rĕns/ 1. The process of reaching a conclusion from a set of pre- misses in a logical argument. An inference may be deductive or inductive. See also de- duction; induction.2. See sampling.
infimum
(inf) The greatest lower BOUNDof a set.
infinite number
The smallest infinite number is '0(aleph zero). This is the num- ber of members in the set of integers. A whole hierarchy of increasingly large infi- nite numbers can be defined on this basis. '1, the next largest, is the number of sub- sets of the set of integers. See also aleph; continuum; countable.infinite sequence
See sequence.infinite series
See series.infinite set
A set in which the number ofelements is infinite. For example, the set of ‘positive integers’, z = {1, 2, 3, 4, …}, is in- finite but the set of ‘positive integers less than 20’ is a finite set. Another example of an infinite set is the number of circles in a particular plane. Compare finite set.
infinitesimal
/in-fi-nă-tess-ă-măl/ Infi-nitely small, but not equal to zero. Infini- tesimal changes or differences are made use of in CALCULUS(infinitesimal calculus).
infinity
Symbol: ∞ The value of a quantitythat increases without limit. For example, if y = 1/x, then y becomes infinitely large, or approaches infinity, as x approaches 0. An infinitely large negative quantity is de- noted by –∞ and an infinitely large positive
quantity by +∞. If x is positive, y = –1/x tends to –∞ as x tends to 0.
inflection
See point of inflection.information theory
The branch of prob-ability theory that deals with uncertainty, accuracy, and information content in the transmission of messages. It can be applied to any system of communication, including electrical signals and human speech. Ran- dom signals (noise) are often added to a message during the transmission process, altering the signal received from that sent. Information theory is used to work out the probability that a particular signal received is the same as the signal sent. Redundancy, for example simply repeating a message, is needed to overcome the limitations of the system. Redundancy can also take the form of a more complex checking process. In transmitting a sequence of numbers, their sum might also be transmitted so that the receiver will know that there is an error when the sum does not correspond to the rest of the message. The sum itself gives no extra information since, if the other num- bers are correctly received, the sum can easily be calculated. The statistics of choos- ing a message out of all possible messages (letters in the alphabet or binary digits for example) determines the amount of infor- mation contained in it. Information is mea- sured in bits (binary digits). If one out of two possible signals are sent then the infor- mation content is one bit. A choice of one out of four possible signals contains more information, although the signal itself might be the same.
inner product
Consider a vector space Vover a scalar field F. An inner product on V is a mapping of ordered pairs of vectors in V into F; i.e. with every pair of vectors x and y there is associated a scalar, which is written 〈x,y〉 and called the inner product of x and y, such that for all vectors x, y, z and scalars α
(i) 〈x+y,z〉 = 〈x,z〉 + 〈y,z〉 (ii) 〈x,y〉 = α〈x,y〉
(iii) 〈x,y〉 = 〈y,x〉, where 〈a,b〉 is the com- plex conjugate of 〈a,b〉
(iv) 〈x,x〉 ≥ 0, 〈x,x〉 = 0 if and only if x = 0.
An inner product on V defines a norm on V given by ||x|| = √〈x,x〉. See norm.
input
1. The signal or other form of in-formation that is applied (fed in) to an elec- trical device, machine, etc. The input to a computer is the data and programmed in- structions that a user communicates to the machine. An input device accepts com- puter input in some appropriate form and converts the information into a code of electrical pulses. The pulses are then trans- mitted to the central processor of the com- puter.
2. The process or means by which input is
applied.
3. To feed information into an electrical
device or machine.
See also input/output; output.
input/output
(I/O) The equipment and operations used to communicate with a computer, and the information passed in or out during the communication. Input/ output devices include those used only forINPUT or for OUTPUT of information and those, such as visual display units, used for both input and output.