The term **IDEAL** (as a noun) was introduced by Richard Dedekind
(1831-1916) in P. G. L. Dirichlet *Vorles. über Zahlentheorie* (ed.
2, 1871) Suppl. x. 452 (OED2).

**IDEAL NUMBER.** Ernst Eduard Kummer (1810-1893) introduced the
term *ideale zahl* in 1846 in *Ber. über die zur Bekanntmachung
geeigneten Verh. d. K. Preuss. Akad. d. Wiss. zu Berlin* 87 (OED2).

**IDEMPOTENT** and **NILPOTENT** were used by Benjamin Peirce
(1809-1880) in 1870:

When an expression raised to the square or any higher power vanishes, it may be callednilpotent;but when, raised to a square or higher power, it gives itself as the result, it may be calledidempotent.

The defining equation of nilpotent and idempotent expressions are respectivelyA= 0, and^{n}A=^{n}A;but with reference to idempotent expressions, it will always be assumed that they are of the form

unless it be otherwise distinctly stated.This citation is excerpted from "Linear Associative Algebra," a memoir read by Benjamin Peirce before the National Academy of Sciences in Washington, 1870, and published by him as a lithograph in 1870. In 1881, Peirce's son, Charles S. Peirce, reprinted it in the

The OED2 shows a 1937 citation with a simplified definition of *idempotent*
in *Modern Higher Algebra* (1938) iii 88 by A. A. Albert: "A matrix
*E*
is called idempotent if *E*^{2} = *E.* [Older dictionaries
pronounce idempotent with the only stress on the second syllable, but newer
ones show a primary stress on the first syllable and a secondary stress
on the penult.]

**IDENTITY** (an equation satisfied by all values of the variable)
is found in 1859 in *Arithmetic & Algebra in their principles and
application,* sixth edition, by Barnard Smith: "Such an expression as
(*x* + 1)^{2} = *x*^{2} + 2*x* + 1, where
one of the quantities, between which the sign equality is placed, results
from performing the operations indicated in the other, is called an Identity"
(OED2).

**IDENTITY** (the element which leaves other elements unchanged)
is found in 1894 in the *Bulletin of the American Mathematical Society*
(OED2).

**IDENTITY ELEMENT** is found in 1902 in *Transactions of the American
Mathematical Society* III. 486: "There exists a left-hand identity element,
that is, an element *i _{l}e* such that, for every element

**IDENTITY MATRIX** is found in "Representations of the General Symmetric
Group as Linear Groups in Finite and Infinite Fields," Leonard Eugene Dickson,
*Transactions
of the American Mathematical Society,* Vol. 9, No. 2. (Apr., 1908).

The term is also found in "Concerning Linear Substitutions of Finite
Period with Rational Coefficients," Arthur Ranum, *Transactions of the
American Mathematical Society,* Vol. 9, No. 2. (Apr., 1908).

**IFF.** On the last page of his autobiography, Paul R. Halmos (1916-
) writes:

My most nearly immortal contributions are an abbreviation and a typographical symbol. I invented "iff", for "if and only if" -- but I could never believe that I was really its first inventor. I am quite prepared to beieve that it existed before me, but I don'tThis quote is fromknowthat it did, and my invention (re-invention?) of it is what spread it thorugh the mathematical world. The symbol is definitely not my invention -- it appeared in popular magazines (not mathematical ones) before I adopted it, but, once again, I seem to have introduced it into mathematics. It is the symbol that sometimes looks like [an empty square], and is used to indicate an end, usually the end of a proof. It is most frequently called the "tombstone", but at least one generous author referred to it as the "halmos".

The earliest citation of "iff" in the OED2 is 1955 in *General Topology*
by J. L. Kelley, who credits the term to Halmos.

The terms **IMAGINARY** and **REAL** were introduced in French
by Rene Descartes (1596-1650) in "La Geometrie" (1637):

...tant les vrayes racines que les fausses ne sont pas tousiours réelles; mais quelquefois seulement imaginaires; c'est à dire qu'on peut bien toujiours en imaginer autant que aiy dit en chàsque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde à celles qu'on imagine. comme encore qu'on en puisse imaginer trois en celle cy, xAn early appearance of the word^{3}- 6xx + 13x - 10 = 0, il n'y en a toutefois qu'une réelle, qui est 2, & pour les deux autres, quois qu'on les augmente, ou diminué, ou multiplié en la façon que ie viens d'éxpliquer, on ne sçauroit les rendre autres qu'imaginaires. [...neither the true roots nor the false are always real; sometimes they are, however, imaginary; namely, whereas we can always imagine as many roots for each equation as I have predicted, there is still not always a quantity which corresponds to each root so imagined. Thus, while we may think of the equation x^{3}- 6xx + 13x - 10 = 0 as having three roots, yet there is just one real root, which is 2, and the other two, however, increased, diminished, or multiplied them as we just laid down, remain always imaginary.] (page 380)

We have before had occasion (in the Solution of some Quadratick and Cubick Equations) to make mention of Negative Squares, and Imaginary Roots, (as contradistinguished to what they call Real Roots, whether affirmative or Negative)... These *Imaginary* Quantities (as they are commonly called) arising from *Supposed* Root of a Negative Square, (when they happen) are reputed to imply that the Case proposed is Impossible.The quotation above is from Chapter LXVI (p. 264),

As a way of removing the stigma of the name, the American mathematician
Arnold Dresden (1882-1954) suggested that imaginary numbers be called *normal*
numbers, because the term "normal" is synonymous with perpendicular, and
the y-axis is perpendicular to the x-axis (Kramer, p. 73).

The term **IMAGINARY GEOMETRY** was used by Lobachevsky, who in 1835
published a long article, "Voobrazhaemaya geometriya" (Imaginary Geometry).

The term **IMAGINARY PART** was used by Sir William Rowan Hamilton
in an 1843 paper. He was referring to the vector and scalar portions of
a quaternion [James A. Landau].

The term **IMAGINARY UNIT** was used (and apparently introduced)
by by Sir William Rowan Hamilton in "On a new Species of Imaginary Quantities
connected with a theory of Quaternions," *Proceedings of the Royal Irish
Academy,* Nov. 13, 1843: "...the extended expression...which may be
called an imaginary unit, because its modulus is = 1, and its square is
negative unity."

**IMPLICIT DEFINITION.** In the literature of mathematics, this term
was introduced by Joseph-Diaz Gergonne (1771-1859) in *Essai sur la théorie
des définitions*, Annales de Mathématique Pure et Appliquée (1818) 1-35,
p. 23. (The *Annales* begun to be published by Gergonne himself in
1810.) He also emphasized the contrast between this kind of definition
and the other "ordinary" ones which, according to him, should be called
"explicit definitions". According to his own example, given the words "triangle"
and "quadrilateral" we can define (implicitly) the word "diagonal" (of
a quadrilateral) in a satisfactory way just by means of a *property*
that individualizes it (namely, that of dividing the quadrilateral in two
equal triangles). Gergonne’s observations are now viewed by many
as an anticipation of the "modern" idea of "definition by axioms" which
was so fruitfully explored by Dedekind, Peano and Hilbert in the second
half of the nineteenth century. In fact, still today the axioms of a theory
are treated in many textbooks as "implicit definitions" of the primitive
concepts involved. We can also view Gergonne’s ideas as anticipating,
to a certain extent, the use of "contextual definitions" in Russell’s
theory of descriptions (1905). [Carlos César de Araújo]

**IMPROPER FRACTION** was used in English in 1542 by Robert Recorde
in *The ground of artes, teachyng the worke and practise of arithmetike*:
"An Improper Fraction...that is to saye, a fraction in forme, which in
dede is greater than a Unit."

**IMPROPER DEFINITE INTEGRAL** occurs in "Concerning Harnack's Theory
of Improper Definite Integrals" by Eliakim Hastings Moore, *Trans. Amer.
Math. Soc.,* July 1901.

*Improper integral* appears in the same paper.

**INCENTER** is dated ca. 1890 in MWCD10.

The term **INDEFINITE INTEGRAL** is defined by Sylvestre-François
Lacroix (1765-1843) in *Traité du calcul différentiel et integral*
(Cajori 1919, page 272).

**INDEX.** Schoner, writing his commentary on the work of Ramus,
in 1586, used the word "index" where Stifel had used "exponent" (Smith
vol. 2).

The term **INDUCTION** was first used in the phrase *per modum
inductionis* by John Wallis in 1656 in *Arithmetica Infinitorum.*
Wallis was the first person to designate a name for this process; Maurolico
and Pascal used no term for it (Burton, page 440). [See also *mathematical
induction, complete induction, successive induction.* ]

**INFINITE DESCENT.** Pierre de Fermat (1601-1665) used the term
*method
of infinite descent* (Burton, page 488; DSB).

The term **INFINITELY SMALL** was used by Christian Huygens (1629-1695)
(*DSB*).

The term **INFINITESIMAL ANALYSIS** was used in 1748 by Leonhard
Euler in *Introductio in analysin infinitorum* (Kline, page 324).

**INFIX (notation)** is found in D. Wood, "A proof of Hamblin's algorithm
for translation of arithmetic expressions from infix to postfix form,"
*BIT,
Nordisk Tidskr. Inform.-Behandl.* 9 (1969).

**INFLECTION POINT** appears in a 1684 paper by Leibniz, according
to Katz (page 528), who has a footnote referring to Struik, Source Book,
page 275.

**INFORMATION THEORY.** The OED2 shows a number of citations for
this term from 1950.

**INJECTION** was used in 1950 by S. MacLane in the *Bulletin of
the American Mathematical Society* (OED2).

**INJECTIVE** was used in 1952 by Eilenberg and Steenrod in *Foundations
of Algebraic Topology* (OED2).

**INNER PRODUCT.** Schwartzman writes (p. 155):

When the German Sanskrit scholar Hermann Günther Grassman (1809-1877) developed the general algebra of hypercomplex numbers, he realized that more than one type of multiplication is possible. To two of the many possible types he gave the namesIn English,innerandouter.The names seem to have been chosen because they are antonyms rather than for any intrinsic meaning.

The term **INNUMERACY** was popularized as the title of a recent
book by John Allen Paulos. The word is found in the *Random House Unabridged
Dictionary,* 2nd ed. (1987), and the word *innumerate* is dated
1959 in MWCD10.

**INTEGER** (as a noun) was used in English in 1571 by Thomas Digges
(1546?-1595) in *A geometrical practise named Pantometria.* An earlier
use exists for "integer number."

**INTEGRABLE** is found in English in 1727-41 in Chambers' *Cyclopaedia*
(OED2).

The word **INTEGRAL** first appeared in print by Jacob (or James
or Jacques I) Bernoulli (1654-1705) in May 1690 in *Acta eruditorum,*
page 218. He wrote, "Ergo et horum Integralia aequantur" (Cajori vol. 2,
page 182; Ball). According to the DSB this represents the first use of
*integral*
"in its present mathematical sense."

However, Jean I (or Johann or John) Bernoulli (1667-1748) also claimed to have introduced the term. According to Smith (vol. I, page 430), "the use of the term 'integral' in its technical sense in the calculus" is due to him.

The the following terms to classify solutions of nonlinear first order
equations are due to Lagrange: *complete solution* or *complete
integral, general integral, particular case* of the general integral,
and *singular integral* (Kline, page 532).

**INTEGRAL CALCULUS.** Leibniz originally used the term *calculus
summatorius* (the calculus of summation) in 1684 and 1686.

Johann Bernoulli introduced the term *integral calculus.*

Cajori (vol. 2, p. 181-182) says:

At one time Leibniz and Johann Bernoulli discussed in their letters both the name and the principal symbol of the integral calculus. Leibniz favored the nameAccording to Smith (vol. 2, page 696), Leibniz in 1696 adopted the termcalculus summatoriusand the long letter [long S symbol] as the symbol. Bernoulli favored the namecalculus integralisand the capital letterIas the sign of integration. ... Leibniz and Johann Bernoulli finally reached a happy compromise, adopting Bernoulli's name "integral calculus," and Leibniz' symbol of integration.

According to Stein and Barcellos (page 311), the term *integral calculus*
is due to Leibniz.

The term "integral calculus" was used by Leo Tolstoy in *Anna Karenina,*
in which a character says, "If they'd told me at college that other people
would have understood the integral calculus, and I didn't, then ambition
would have come in."

**INTEGRAL DOMAIN** is found in A. A. Albert, "Integral domains of
rational generalized quaternion algebras," *Bull. Am. Math. Soc.*
40 (1934).

**INTEGRAL EQUATION.** According to Kline (page 1052) and Cajori
1919 (page 393), the term *integral equation* is due to Paul du Bois-Reymond
(1831-1889), *Jour. für Math.,* 103, 1888, 288. However, Euler used
a phrase which is translated *integral equation* in the paper "De
integratione aequationis differentialis," *Novi Commentarii Academiae
Scientarum Petropolitanae 6, 1756-57* (1761) [James A. Landau].

The term **INTEGRAL GEOMETRY** is due to Wilhelm Blaschke (1885-1962),
according to the University of St. Andrews website.

**INTEGRAND** was used in 1897 by H. F. Baker in *Abel's Theorem*:
"The integrand of the Abelian integral *u,* is single-valued on the
Riemann surface" (OED2).

**INTEGRATING FACTOR** occurs in "Three Particular Systems of Lines
on a Surface," Luther Pfahler Eisenhart, *Transactions of the American
Mathematical Society,* Vol. 5, No. 4. (Oct., 1904).

The term **INTEGRATION BY PARTS** appears in the 1902 *Encyclopaedia
Britannica* in the article "Infinitestimal Calculus" and also appears
in the first edition of the OED.

The method was invented by Brook Taylor and discussed in *Methodus
incrementorum directa et inversa* (1715).

**INTERMEDIATE VALUE THEOREM** appears in 1937 in *Differential
and Integral Calculus,* 2nd. ed. by R. Courant [James A. Landau].

The term **INTERPOLATION** was introduced into mathematics by John
Wallis (DSB; Kline, page 440).

**INTERSECTION** (in set theory) is found in *Webster's New International
Dictionary* of 1909.

**INTRINSICALLY CONVERGENT SEQUENCE** is the term used by Courant
for "Cauchy sequence" in *Differential and Integral Calculus,* 2nd.
ed. (1937) [James A. Landau].

The term **INVARIANT** is due to James Joseph Sylvester (1814-1897),
according to Cajori (1919, page 345) and Kline (page 927), who supplies
the reference *Coll. Math. Papers,* I, 273.

According to Kramer (p. 388), Galois used the adjective "invariant" referring to a normal subgroup.

**INVERSE** (as a noun, the element which produces the identity element)
appears in 1900 in *Ann. Math.* (OED2).

The term **INVERSE GAUSSIAN DISTRIBUTION** was coined in 1948 by
M. C. K. Tweedie, according to Gerard Letac.

**INVERSE VARIATION** is found in 1881 in *Elements of Algebra*
by G. A. Wentworth [James A. Landau].

**INVERTIBLE** is found in the phrase "invertible elements of a monoid
*A*"
in 1956 in *Fundamental Concepts of Algebra* ii. 27 by C. Chevelley
(OED2).

The term **INVOLUTION** is due to Gérard Desargues (1593-1662) (Kline,
page 292).

**IRRATIONAL** is used in English by Robert Recorde in 1551 in *The
Pathwaie to Knowledge*: "Numbres and quantitees surde or irrationall."

The term **IRREDUCIBLE INVARIANT** was used by Arthur Cayley (1821-1895).

**ISOGRAPHIC** is the word used by Ernest Jean Philippe Fauquede
Jonquiéres (1820-1901) to describe the transformations he had discovered,
later called *birational transformations* (DSB).

**ISOMETRIC** is found in English in an 1855 dictionary. *Isometrische
Abbildung* (isometric mapping) is found in the 1944 edition of Hausdorff's
*Grundzuge
der Mengenlehre* and may occur in the first 1914 edition [Gerald A.
Edgar].

**ISOMETRY** (equality of measure) is found in English in a 1909
*Webster*
dictionary. Aristotle used the word *isometria.*

In its modern sense, *isometry* occurs in English in 1941 in *Survey
of Modern Algebra* by MacLane and Birkhoff: "An obvious example is furnished
by the symmetries of the cube. Geometrically speaking, these are the one-one
transformations which preserve distances on the cube. They are known as
'isometries,' and are 48 in number" (OED).

**ISOMORPHISM** was used by Walter Dyck (1856-1934) in 1882 in *Gruppentheoretische
Studien* (Katz, page 675).

**ISOSCELES** was used in English 1551 by Robert Recorde in *The
Pathwaie to Knowledge*: "There is also an other distinction of the name
of triangles, according to their sides, whiche other be all equal...other
els two sydes bee equall and the thyrd vnequall, which the Greekes call
*Isosceles,*
the Latine men *aequicurio,* and in english tweyleke may they be called."

In English, an isosceles triangle was called an *equicrure* in
1644 and an *equicrural triangle* in 1650 (OED2). These are the earliest
uses for the alternate term of Latin origin in the OED2.

The term **ITERATED FUNCTION SYSTEM** was coined by Michael Barnsley,
according to an Internet website.