• In symbols, for every a, one has This law was first identified in Brahmagupta’s Brahmasphutasiddhanta in 628 AD, although he wrote it as three separate laws, depending on
whether a is negative, positive, or zero itself, and he used words rather than algebraic symbols.

• There are also situations where addition is “understood”, even though no symbol appears: • A whole number followed immediately by a fraction indicates the sum of the two,
called a mixed number.

• Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real
numbers and complex numbers.

• Properties Commutativity  with blocks Addition is commutative, meaning that one can change the order of the terms in a sum, but still get the same result.

• It is commutative, meaning that the order of the operands does not matter, and it is associative, meaning that when one adds more than two numbers, the order in which addition
is performed does not matter.

• However, it is not obvious how one should extend this version of addition to include fractional numbers or negative numbers.

• This is appropriate not only because the sum of two positive numbers is greater than either, but because it was common for the ancient Greeks and Romans to add upward, contrary
to the modern practice of adding downward, so that a sum was literally higher than the addends.

• Addition of very small numbers is accessible to toddlers; the most basic task, 1 + 1, can be performed by infants as young as five months, and even some members of other animal
species.

• Associativity  with segmented rods Addition is associative, which means that when three or more numbers are added together, the order of operations does not change the
result.

• When given a problem that requires that two items and three items be combined, young children model the situation with physical objects, often fingers or a drawing, and then
count the total.

• For example, Terms  The numbers or the objects to be added in general addition are collectively referred to as the terms,[6] the addends[7][8][9] or the summands;[10]
this terminology carries over to the summation of multiple terms.

• Addition (usually signified by the plus symbol +) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division.

• [36] • Making ten: An advanced strategy uses 10 as an intermediate for sums involving 8 or 9; for example, .

• For example, can be derived from by adding one more.

• Even for the simple case of adding natural numbers, there are many possible interpretations and even more visual representations.

• [33] Different nations introduce whole numbers and arithmetic at different ages, with many countries teaching addition in pre-school.

• Successor  Within the context of integers, addition of one also plays a special role: for any integer a, the integer (a + 1) is the least integer greater than a, also
known as the successor of a.

• Possibly the most basic interpretation of addition lies in combining sets: • When two or more disjoint collections are combined into a single collection, the number of objects
in the single collection is the sum of the numbers of objects in the original collections.

• Nonetheless, in the teaching of arithmetic, some students are introduced to addition as a process that always increases the addends; word problems may help rationalize the
“exception” of zero.

• • One or two more: Adding 1 or 2 is a basic task, and it can be accomplished through counting on or, ultimately, intuition.

• Doubles facts form a backbone for many related facts, and students find them relatively easy to grasp.

• On the other hand, it is usually meaningless to try to add 3 meters and 4 square meters, since those units are incomparable; this sort of consideration is fundamental in dimensional
analysis.

• When addition is used together with other operations, the order of operations becomes important.

• Because of this succession, the value of a + b can also be seen as the bth successor of a, making addition iterated succession.

• [27] Another 1992 experiment with older toddlers, between 18 and 35 months, exploited their development of motor control by allowing them to retrieve ping-pong balls from
a box; the youngest responded well for small numbers, while older subjects were able to compute sums up to 5.

• [2] The addition of two whole numbers results in the total amount or sum of those values combined.

• Since the end of the 20th century, some US programs, including TERC, decided to remove the traditional transfer method from their curriculum.

• Given that addition is associative, the choice of definition is irrelevant.

• [36] • Doubles: Adding a number to itself is related to counting by two and to multiplication.

• With additional experience, children learn to add more quickly by exploiting the commutativity of addition by counting up from the larger number, in this case, starting with
three and counting “four, five.”

• It is also useful in higher mathematics (for the rigorous definition it inspires, see § Natural numbers below).

• [33] Carry  Main article: Carry (arithmetic) The standard algorithm for adding multidigit numbers is to align the addends vertically and add the columns, starting from
the ones column on the right.

• Interpretations Addition is used to model many physical processes.

• [20] Instead of calling both a and b addends, it is more appropriate to call a the augend in this case, since a plays a passive role.

• For example, a child asked to add six and seven may know that and then reason that is one more, or 13.

• [52] Addition of numbers To prove the usual properties of addition, one must first define addition for the context in question.

• Addition requires two numbers in scientific notation to be represented using the same exponential part, so that the two significands can simply be added.

• Finally, one performs the same addition process as above, except the decimal point is placed in the answer, exactly where it was placed in the summands.

• Part of Charles Babbage’s Difference Engine including the addition and carry mechanisms The abacus, also called a counting frame, is a calculating tool that was in use centuries
before the adoption of the written modern numeral system and is still widely used by merchants, traders and clerks in Asia, Africa, and elsewhere; it dates back to at least 2700–2300 BC, when it was used in Sumer.

• [40] Adding two single-digit binary numbers is relatively simple, using a form of carrying: Adding two “1” digits produces a digit “0”, while 1 must be added to the next column.

• In set theory, addition is then extended to progressively larger sets that include the natural numbers: the integers, the rational numbers, and the real numbers.

• In modern digital computers, integer addition is typically the fastest arithmetic instruction, yet it has the largest impact on performance, since it underlies all floating-point
operations as well as such basic tasks as address generation during memory access and fetching instructions during branching.

• The other popular definition is recursive: • Let n+ be the successor of n, that is the number following n in the natural numbers, so .

• “Full adder” logic circuit that adds two binary digits, A and B, along with a carry input Cin, producing the sum bit, S, and a carry output, Cout.

• It is based on the remark that every integer is the difference of two natural integers and that two such differences, a – b and c – d are equal if and only if a + d = b +
c. So, one can define formally the integers as the equivalence classes of ordered pairs of natural numbers under the equivalence relation Addition of ordered pairs is done component-wise: A straightforward computation shows that the equivalence
class of the result depends only on the equivalences classes of the summands, and thus that this defines an addition of equivalence classes, that is integers.

• [55]) Natural numbers  Further information: Natural number There are two popular ways to define the sum of two natural numbers a and b.

• Rational numbers (fractions)  Addition of rational numbers can be computed using the least common denominator, but a conceptually simpler definition involves only integer
addition and multiplication: • Define As an example, the sum .

• The sum of real numbers a and b is defined element by element: • Define [65] This definition was first published, in a slightly modified form, by Richard Dedekind in 1872.

• The corresponding definition of addition must proceed by cases: Although this definition can be useful for concrete problems, the number of cases to consider complicates proofs
unnecessarily.

• Real numbers  Further information: Construction of the real numbers A common construction of the set of real numbers is the Dedekind completion of the set of rational
numbers.

• Addition is first defined on the natural numbers.

• The influence of the abacus on mathematical thinking was strong enough that early Latin texts often claimed that in the process of adding “a number to a number”, both numbers
vanish.

• An alternate version of this definition allows A and B to possibly overlap and then takes their disjoint union, a mechanism that allows common elements to be separated out
and therefore counted twice.

• One slight improvement is the carry skip design, again following human intuition; one does not perform all the carries in computing , but one bypasses the group of 9s and

• Addition is defined term by term: • Define [69] This definition was first published by Georg Cantor, also in 1872, although his formalism was slightly different.

• [62] Another straightforward computation shows that this addition is the same as the above case definition.

• Giovanni Poleni followed Pascal, building the second functional mechanical calculator in 1709, a calculating clock made of wood that, once setup, could multiply two numbers
automatically.

• [66] The commutativity and associativity of real addition are immediate; defining the real number 0 to be the set of negative rationals, it is easily seen to be the additive
identity.

• [59] This recursive formulation of addition was developed by Dedekind as early as 1854, and he would expand upon it in the following decades.

• [42] Addition is also fundamental to the operation of digital computers, where the efficiency of addition, in particular the carry mechanism, is an important limitation to
overall performance.

• The most common situation for a general-purpose analog computer is to add two voltages (referenced to ground); this can be accomplished roughly with a resistor network, but
a better design exploits an operational amplifier.

• [53] (In mathematics education,[54] positive fractions are added before negative numbers are even considered; this is also the historical route.

• Addition of fractions is much simpler when the denominators are the same; in this case, one can simply add the numerators while leaving the denominator the same: , so .

• As an example, one can consider addition in binary.

• Given a set with an addition operation, one cannot always define a corresponding subtraction operation on that set; the set of natural numbers is a simple example.

• [72][73] That is to say: Using the visualization of complex numbers in the complex plane, the addition has the following geometric interpretation: the sum of two complex numbers
A and B, interpreted as points of the complex plane, is the point X obtained by building a parallelogram three of whose vertices are O, A and B. Equivalently, X is the point such that the triangles with vertices O, A, B, and X, B, A, are congruent.

• A similar “wrap around” operation arises in geometry, where the sum of two angle measures is often taken to be their sum as real numbers modulo 2π.

• A circular slide rule In the real and complex numbers, addition and multiplication can be interchanged by the exponential function:[80] This identity allows multiplication
to be carried out by consulting a table of logarithms and computing addition by hand; it also enables multiplication on a slide rule.

• The formula is still a good first-order approximation in the broad context of Lie groups, where it relates multiplication of infinitesimal group elements with addition of
vectors in the associated Lie algebra.

• [95] Convolution is used to add two independent random variables defined by distribution functions.

• Linear combinations combine multiplication and summation; they are sums in which each term has a multiplier, usually a real or complex number.

• The sum of two vectors is obtained by adding their individual coordinates: This addition operation is central to classical mechanics, in which velocities, accelerations and
forces are all represented by vectors.

• Set theory and category theory  A far-reaching generalization of addition of natural numbers is the addition of ordinal numbers and cardinal numbers in set theory.

• Furthermore, since addition preserves the ordering of real numbers, addition distributes over “max” in the same way that multiplication distributes over addition: For these
reasons, in tropical geometry one replaces multiplication with addition and addition with maximization.

• The field of abstract algebra is centrally concerned with such generalized operations, and they also appear in set theory and category theory.

• Generalizations There are many binary operations that can be viewed as generalizations of the addition operation on the real numbers.

• Complex numbers  Addition of two complex numbers can be done geometrically by constructing a parallelogram.

• The sum of two m × n (pronounced “m by n”) matrices A and B, denoted by A + B, is again an m × n matrix computed by adding corresponding elements:[75][76] For example: Modular
arithmetic  Main article: Modular arithmetic In modular arithmetic, the set of available numbers is restricted to a finite subset of the integers, and addition “wraps around” when reaching a certain value, called the modulus.

• [96] In general, convolution is useful as a kind of domain-side addition; by contrast, vector addition is a kind of range-side addition.

• The set of integers modulo 2 has just two elements; the addition operation it inherits is known in Boolean logic as the “exclusive or” function.

• Related operations Addition, along with subtraction, multiplication and division, is considered one of the basic operations and is used in elementary arithmetic.

• If n is not a natural number, the product may still make sense; for example, multiplication by −1 yields the additive inverse of a number.

• These give two different generalizations of addition of natural numbers to the transfinite.

Works Cited

[‘”Addend” is not a Latin word; in Latin it must be further conjugated, as in numerus addendus “the number to be added”.
2. ^ Some authors think that “carry” may be inappropriate for education; Van de Walle (p. 211) calls it “obsolete and conceptually
History
• Ferreirós, José (1999). Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Birkhäuser. ISBN 978-0-8176-5749-9.
• Karpinski,
Louis (1925). The History of Arithmetic. Rand McNally. LCC QA21.K3.
• Schwartzman, Steven (1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. MAA. ISBN 978-0-88385-511-9.
• Williams, Michael (1985).
A History of Computing Technology. Prentice-Hall. ISBN 978-0-13-389917-7.
Elementary mathematics
• Sparks, F.; Rees C. (1979). A Survey of Basic Mathematics. McGraw-Hill. ISBN 978-0-07-059902-4.
Education
• Begle, Edward (1975). The Mathematics
of the Elementary School. McGraw-Hill. ISBN 978-0-07-004325-1.
• California State Board of Education mathematics content standards Adopted December 1997, accessed December 2005.
• Devine, D.; Olson, J.; Olson, M. (1991). Elementary Mathematics
for Teachers (2e ed.). Wiley. ISBN 978-0-471-85947-5.
• National Research Council (2001). Adding It Up: Helping Children Learn Mathematics. National Academy Press. doi:10.17226/9822. ISBN 978-0-309-06995-3.
• Van de Walle, John (2004). Elementary
and Middle School Mathematics: Teaching developmentally (5e ed.). Pearson. ISBN 978-0-205-38689-5.
Cognitive science
• Fosnot, Catherine T.; Dolk, Maarten (2001). Young Mathematicians at Work: Constructing Number Sense, Addition, and Subtraction.
Heinemann. ISBN 978-0-325-00353-5.
• Wynn, Karen (1998). “Numerical competence in infants”. The Development of Mathematical Skills. Taylor & Francis. ISBN 0-86377-816-X.
Mathematical exposition
• Bogomolny, Alexander (1996). “Addition”. Interactive
Mathematics Miscellany and Puzzles (cut-the-knot.org). Archived from the original on April 26, 2006. Retrieved 3 February 2006.
• Cheng, Eugenia (2017). Beyond Infinity: An Expedition to the Outer Limits of Mathematics. Basic Books. ISBN 978-1-541-64413-7.
• Dunham,
William (1994). The Mathematical Universe. Wiley. ISBN 978-0-471-53656-7.
• Johnson, Paul (1975). From Sticks and Stones: Personal Adventures in Mathematics. Science Research Associates. ISBN 978-0-574-19115-1.
• Linderholm, Carl (1971). Mathematics
• Smith, Frank (2002). The Glass Wall: Why Mathematics Can Seem Difficult. Teachers College Press. ISBN 978-0-8077-4242-6.
• Smith, Karl (1980). The Nature of Modern Mathematics (3rd ed.). Wadsworth.
ISBN 978-0-8185-0352-8.
• Bergman, George (2005). An Invitation to General Algebra and Universal Constructions (2.3 ed.). General Printing. ISBN 978-0-9655211-4-7.
• Burrill, Claude (1967). Foundations of Real Numbers. McGraw-Hill.
LCC QA248.B95.
• Dummit, D.; Foote, R. (1999). Abstract Algebra (2 ed.). Wiley. ISBN 978-0-471-36857-1.
• Gbur, Greg (2011). Mathematical Methods for Optical Physics and Engineering. Cambridge University Press. ISBN 978-0-511-91510-9. OCLC 704518582.
• Enderton,
Herbert (1977). Elements of Set Theory. Academic Press. ISBN 978-0-12-238440-0.
• Lee, John (2003). Introduction to Smooth Manifolds. Springer. ISBN 978-0-387-95448-6.
• Martin, John (2003). Introduction to Languages and the Theory of Computation
(3 ed.). McGraw-Hill. ISBN 978-0-07-232200-2.
• Riehl, Emily (2016). Category Theory in Context. Dover. ISBN 978-0-486-80903-8.
• Rudin, Walter (1976). Principles of Mathematical Analysis (3 ed.). McGraw-Hill. ISBN 978-0-07-054235-8.
• Stewart,
James (1999). Calculus: Early Transcendentals (4 ed.). Brooks/Cole. ISBN 978-0-534-36298-0.
Mathematical research
• Akian, Marianne; Bapat, Ravindra; Gaubert, Stephane (2005). “Min-plus methods in eigenvalue perturbation theory and generalised
Lidskii-Vishik-Ljusternik theorem”. INRIA Reports. arXiv:math.SP/0402090. Bibcode:2004math……2090A.
• Baez, J.; Dolan, J. (2001). Mathematics Unlimited – 2001 and Beyond. From Finite Sets to Feynman Diagrams. p. 29. arXiv:math.QA/0004133. ISBN
3-540-66913-2.
• Litvinov, Grigory; Maslov, Victor; Sobolevskii, Andreii (1999). Idempotent mathematics and interval analysis. Reliable Computing, Kluwer.
• Loday, Jean-Louis (2002). “Arithmetree”. Journal of Algebra. 258: 275. arXiv:math/0112034.
doi:10.1016/S0021-8693(02)00510-0.
• Mikhalkin, Grigory (2006). Sanz-Solé, Marta (ed.). Proceedings of the International Congress of Mathematicians (ICM), Madrid, Spain, August 22–30, 2006. Volume II: Invited lectures. Tropical Geometry and its
Applications. Zürich: European Mathematical Society. pp. 827–852. arXiv:math.AG/0601041. ISBN 978-3-03719-022-7. Zbl 1103.14034.
• Viro, Oleg (2001). Cascuberta, Carles; Miró-Roig, Rosa Maria; Verdera, Joan; Xambó-Descamps, Sebastià (eds.). European
Congress of Mathematics: Barcelona, July 10–14, 2000, Volume I. Dequantization of Real Algebraic Geometry on Logarithmic Paper. Progress in Mathematics. Vol. 201. Basel: Birkhäuser. pp. 135–146. arXiv:math/0005163. Bibcode:2000math……5163V. ISBN
978-3-7643-6417-5. Zbl 1024.14026.
Computing
• Flynn, M.; Oberman, S. (2001). Advanced Computer Arithmetic Design. Wiley. ISBN 978-0-471-41209-0.
• Horowitz, P.; Hill, W. (2001). The Art of Electronics (2 ed.). Cambridge UP. ISBN 978-0-521-37095-0.
• Jackson,
Albert (1960). Analog Computation. McGraw-Hill. LCC QA76.4 J3.
• Rieffel, Eleanor G.; Polak, Wolfgang H. (4 March 2011). Quantum Computing: A Gentle Introduction. MIT Press. ISBN 978-0-262-01506-6.
• Truitt, T.; Rogers, A. (1960). Basics of Analog
Computers. John F. Rider. LCC QA76.4 T7.
• Marguin, Jean (1994). Histoire des Instruments et Machines à Calculer, Trois Siècles de Mécanique Pensante 1642–1942 (in French). Hermann. ISBN 978-2-7056-6166-3.
• Taton, René (1963). Le Calcul Mécanique.
Que Sais-Je ? n° 367 (in French). Presses universitaires de France. pp. 20–28.

b. From Enderton (p. 138): “…select two sets K and L with card K = 2 and card L = 3. Sets of fingers are handy; sets of apples are preferred by textbooks.”
c. ^
Lewis, Rhys (1974). “Arithmetic”. First-Year Technician Mathematics. Palgrave, London: The MacMillan Press Ltd. p. 1. doi:10.1007/978-1-349-02405-6_1. ISBN 978-1-349-02405-6.
d. ^ “Addition”. www.mathsisfun.com. Retrieved 2020-08-25.
e. ^ Devine
et al. p. 263
f. ^ Mazur, Joseph. Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers. Princeton University Press, 2014. p. 161
g. ^ Department of the Army (1961) Army Technical Manual TM 11-684: Principles and
Applications of Mathematics for Communications-Electronics . Section 5.1
h. ^ Jump up to:a b Shmerko, V.P.; Yanushkevich [Ânuškevič], Svetlana N. [Svitlana N.]; Lyshevski, S.E. (2009). Computer arithmetics for nanoelectronics. CRC Press. p. 80.
i. ^
Jump up to:a b Schmid, Hermann (1974). Decimal Computation (1st ed.). Binghamton, NY: John Wiley & Sons. ISBN 0-471-76180-X. and Schmid, Hermann (1983) [1974]. Decimal Computation (reprint of 1st ed.). Malabar, FL: Robert E. Krieger Publishing Company.
ISBN 978-0-89874-318-0.
j. ^ Jump up to:a b Weisstein, Eric W. “Addition”. mathworld.wolfram.com. Retrieved 2020-08-25.
k. ^ Hosch, W.L. (Ed.). (2010). The Britannica Guide to Numbers and Measurement. The Rosen Publishing Group. p. 38
l. ^ Jump
up to:a b Schwartzman p. 19
m. ^ Karpinski pp. 56–57, reproduced on p. 104
n. ^ Schwartzman (p. 212) attributes adding upwards to the Greeks and Romans, saying it was about as common as adding downwards. On the other hand, Karpinski (p. 103) writes
that Leonard of Pisa “introduces the novelty of writing the sum above the addends”; it is unclear whether Karpinski is claiming this as an original invention or simply the introduction of the practice to Europe.
o. ^ Karpinski pp. 150–153
p. ^
Cajori, Florian (1928). “Origin and meanings of the signs + and -“. A History of Mathematical Notations, Vol. 1. The Open Court Company, Publishers.
q. ^ “plus”. Oxford English Dictionary (Online ed.). Oxford University Press. (Subscription or participating
institution membership required.)
r. ^ See Viro 2001 for an example of the sophistication involved in adding with sets of “fractional cardinality”.
s. ^ Adding it up (p. 73) compares adding measuring rods to adding sets of cats: “For example,
inches can be subdivided into parts, which are hard to tell from the wholes, except that they are shorter; whereas it is painful to cats to divide them into parts, and it seriously changes their nature.”
t. ^ Mosley, F (2001). Using number lines
with 5–8 year olds. Nelson Thornes. p. 8
u. ^ Li, Y., & Lappan, G. (2014). Mathematics curriculum in school education. Springer. p. 204
v. ^ Bronstein, Ilja Nikolaevič; Semendjajew, Konstantin Adolfovič (1987) [1945]. “2.4.1.1.”. In Grosche, Günter;
Ziegler, Viktor; Ziegler, Dorothea (eds.). Taschenbuch der Mathematik (in German). Vol. 1. Translated by Ziegler, Viktor. Weiß, Jürgen (23 ed.). Thun and Frankfurt am Main: Verlag Harri Deutsch (and B.G. Teubner Verlagsgesellschaft, Leipzig). pp.
115–120. ISBN 978-3-87144-492-0.
w. ^ Kaplan pp. 69–71
x. ^ Hempel, C.G. (2001). The philosophy of Carl G. Hempel: studies in science, explanation, and rationality. p. 7
y. ^ R. Fierro (2012) Mathematics for Elementary School Teachers. Cengage
Learning. Sec 2.3
z. ^ Moebs, William; et al. (2022). “1.4 Dimensional Analysis”. University Physics Volume 1. OpenStax. ISBN 978-1-947172-20-3.
aa. ^ Wynn p. 5
bb. ^ Wynn p. 15
cc. ^ Wynn p. 17
dd. ^ Wynn p. 19
ee. ^ Randerson, James
(21 August 2008). “Elephants have a head for figures”. The Guardian. Archived from the original on 2 April 2015. Retrieved 29 March 2015.
ff. ^ F. Smith p. 130
gg. ^ Carpenter, Thomas; Fennema, Elizabeth; Franke, Megan Loef; Levi, Linda; Empson,
Susan (1999). Children’s mathematics: Cognitively guided instruction. Portsmouth, NH: Heinemann. ISBN 978-0-325-00137-1.
hh. ^ Jump up to:a b Henry, Valerie J.; Brown, Richard S. (2008). “First-grade basic facts: An investigation into teaching
and learning of an accelerated, high-demand memorization standard”. Journal for Research in Mathematics Education. 39 (2): 153–183. doi:10.2307/30034895. JSTOR 30034895.
ii. ^ Beckmann, S. (2014). The twenty-third ICMI study: primary mathematics
study on whole numbers. International Journal of STEM Education, 1(1), 1-8. Chicago
jj. ^ Schmidt, W., Houang, R., & Cogan, L. (2002). “A coherent curriculum”. American Educator, 26(2), 1–18.
kk. ^ Jump up to:a b c d e f g Fosnot and Dolk p. 99
ll. ^
“Vertical addition and subtraction strategy”. primarylearning.org. Retrieved April 20, 2022.
mm. ^ “Reviews of TERC: Investigations in Number, Data, and Space”. nychold.com. Retrieved April 20, 2022.
nn. ^ Rebecca Wingard-Nelson (2014) Decimals
and Fractions: It’s Easy Enslow Publishers, Inc.
oo. ^ Dale R. Patrick, Stephen W. Fardo, Vigyan Chandra (2008) Electronic Digital System Fundamentals The Fairmont Press, Inc. p. 155
pp. ^ P.E. Bates Bothman (1837) The common school arithmetic.
Henry Benton. p. 31
qq. ^ Truitt and Rogers pp. 1;44–49 and pp. 2;77–78
rr. ^ Ifrah, Georges (2001). The Universal History of Computing: From the Abacus to the Quantum Computer. New York: John Wiley & Sons, Inc. ISBN 978-0-471-39671-0. p. 11
ss. ^
Jean Marguin, p. 48 (1994); Quoting René Taton (1963)
tt. ^ See Competing designs in Pascal’s calculator article
uu. ^ Flynn and Overman pp. 2, 8
vv. ^ Flynn and Overman pp. 1–9
ww. ^ Yeo, Sang-Soo, et al., eds. Algorithms and Architectures
for Parallel Processing: 10th International Conference, ICA3PP 2010, Busan, Korea, May 21–23, 2010. Proceedings. Vol. 1. Springer, 2010. p. 194
xx. ^ Karpinski pp. 102–103
yy. ^ The identity of the augend and addend varies with architecture. For
ADD in x86 see Horowitz and Hill p. 679; for ADD in 68k see p. 767.
zz. ^ Joshua Bloch, “Extra, Extra – Read All About It: Nearly All Binary Searches and Mergesorts are Broken” Archived 2016-04-01 at the Wayback Machine. Official Google Research
Blog, June 2, 2006.
aaa. ^ Neumann, Peter G. (2 February 1987). “The Risks Digest Volume 4: Issue 45”. The Risks Digest. 4 (45). Archived from the original on 2014-12-28. Retrieved 2015-03-30.
bbb. ^ Enderton chapters 4 and 5, for example, follow
this development.
ccc. ^ According to a survey of the nations with highest TIMSS mathematics test scores; see Schmidt, W., Houang, R., & Cogan, L. (2002). A coherent curriculum. American educator, 26(2), p. 4.
ddd. ^ Baez (p. 37) explains the
historical development, in “stark contrast” with the set theory presentation: “Apparently, half an apple is easier to understand than a negative apple!”
eee. ^ Begle p. 49, Johnson p. 120, Devine et al. p. 75
fff. ^ Enderton p. 79
ggg. ^ For
a version that applies to any poset with the descending chain condition, see Bergman p. 100.
hhh. ^ Enderton (p. 79) observes, “But we want one binary operation +, not all these little one-place functions.”
iii. ^ Ferreirós p. 223
jjj. ^ K.
Smith p. 234, Sparks and Rees p. 66
kkk. ^ Enderton p. 92
lll. ^ Schyrlet Cameron, and Carolyn Craig (2013)Adding and Subtracting Fractions, Grades 5–8 Mark Twain, Inc.
mmm. ^ The verifications are carried out in Enderton p. 104 and sketched
for a general field of fractions over a commutative ring in Dummit and Foote p. 263.
nnn. ^ Enderton p. 114
ooo. ^ Ferreirós p. 135; see section 6 of Stetigkeit und irrationale Zahlen Archived 2005-10-31 at the Wayback Machine.
ppp. ^ The intuitive
approach, inverting every element of a cut and taking its complement, works only for irrational numbers; see Enderton p. 117 for details.
qqq. ^ Schubert, E. Thomas, Phillip J. Windley, and James Alves-Foss. “Higher Order Logic Theorem Proving and
Its Applications: Proceedings of the 8th International Workshop, volume 971 of.” Lecture Notes in Computer Science (1995).
rrr. ^ Textbook constructions are usually not so cavalier with the “lim” symbol; see Burrill (p. 138) for a more careful,
drawn-out development of addition with Cauchy sequences.
sss. ^ Ferreirós p. 128
ttt. ^ Burrill p. 140
uuu. ^ Conway, John B. (1986), Functions of One Complex Variable I, Springer, ISBN 978-0-387-90328-6
vvv. ^ Joshi, Kapil D (1989), Foundations
of Discrete Mathematics, New York: John Wiley & Sons, ISBN 978-0-470-21152-6
www. ^ Gbur, p. 1
xxx. ^ Lipschutz, S., & Lipson, M. (2001). Schaum’s outline of theory and problems of linear algebra. Erlangga.
yyy. ^ Riley, K.F.; Hobson, M.P.;
Bence, S.J. (2010). Mathematical methods for physics and engineering. Cambridge University Press. ISBN 978-0-521-86153-3.
zzz. ^ Cheng, pp. 124–132
aaaa. ^ Riehl, p. 100
bbbb. ^ The set still must be nonempty. Dummit and Foote (p. 48) discuss
this criterion written multiplicatively.
cccc. ^ Rudin p. 178
dddd. ^ Lee p. 526, Proposition 20.9
eeee. ^ Linderholm (p. 49) observes, “By multiplication, properly speaking, a mathematician may mean practically anything. By addition he may
mean a great variety of things, but not so great a variety as he will mean by ‘multiplication’.”
ffff. ^ Dummit and Foote p. 224. For this argument to work, one still must assume that addition is a group operation and that multiplication has an
identity.
gggg. ^ For an example of left and right distributivity, see Loday, especially p. 15.
hhhh. ^ Compare Viro Figure 1 (p. 2)
iiii. ^ Enderton calls this statement the “Absorption Law of Cardinal Arithmetic”; it depends on the comparability
of cardinals and therefore on the Axiom of Choice.
jjjj. ^ Enderton p. 164
kkkk. ^ Mikhalkin p. 1
llll. ^ Akian et al. p. 4
mmmm. ^ Mikhalkin p. 2
nnnn. ^ Litvinov et al. p. 3
oooo. ^ Viro p. 4
pppp. ^ Martin p. 49
qqqq. ^ Stewart
p. 8
rrrr. ^ Rieffel and Polak, p. 16
ssss. ^ Gbur, p. 300
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