Addition (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Addition" in English language version.

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archive.org

Some authors think that "carry" may be inappropriate for education; van de Walle (2004), p. 211 calls it "obsolete and conceptually misleading", preferring the word "trade". However, "carry" remains the standard term. van de Walle, John (2004). Elementary and Middle School Mathematics: Teaching developmentally (5e ed.). Pearson. ISBN 978-0-205-38689-5.
Devine, Olson & Olson (1991), p. 263. Devine, D.; Olson, J.; Olson, M. (1991). Elementary Mathematics for Teachers (2e ed.). John Wiley & Sons. ISBN 978-0-471-85947-5.
Department of the Army (1961), Section 5.1. Department of the Army (1961). Army Technical Manual TM 11-684: Principles and Applications of Mathematics for Communications-Electronics.
Shmerko, Yanushkevich & Lyshevski (2009), p. 80; Schmid (1974); Schmid (1983). Shmerko, V. P.; Yanushkevich, Svetlana N.; Lyshevski, S. E. (2009). Computer arithmetics for nanoelectronics. CRC Press. Schmid, Hermann (1974). Decimal Computation (1st ed.). Binghamton, NY: John Wiley & Sons. ISBN 0-471-76180-X. Schmid, Hermann (1983) [1974]. Decimal Computation (reprint of 1st ed.). Malabar, FL: Robert E. Krieger Publishing Company. ISBN 978-0-89874-318-0.
Schwartzman (1994), p. 19. Schwartzman, Steven (1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. MAA. ISBN 978-0-88385-511-9.
Schwartzman (1994), p. 212 attributes adding upwards to the Greeks and Romans, saying it was about as common as adding downwards. On the other hand, Karpinski (1925), 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. Schwartzman, Steven (1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. MAA. ISBN 978-0-88385-511-9. Karpinski, Louis (1925). The History of Arithmetic. Rand McNally. LCC QA21.K3.
Stewart (1999), p. 8. Stewart, James (1999). Calculus: Early Transcendentals (4th ed.). Brooks/Cole. ISBN 978-0-534-36298-0.
Smith (2002), p. 130. Smith, Frank (2002). The Glass Wall: Why Mathematics Can Seem Difficult. Teachers College Press. ISBN 978-0-8077-4242-6.
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.
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
The identity of the augend and addend varies with architecture. For ADD in x86 see Horowitz & Hill (2001), p. 679. For ADD in 68k see Horowitz & Hill (2001), p. 767. Horowitz, P.; Hill, W. (2001). The Art of Electronics (2nd ed.). Cambridge University Press. ISBN 978-0-521-37095-0. Horowitz, P.; Hill, W. (2001). The Art of Electronics (2nd ed.). Cambridge University Press. ISBN 978-0-521-37095-0.
Begle (1975), p. 49; Johnson (1975), p. 120; Devine, Olson & Olson (1991), p. 75. Begle, Edward (1975). The Mathematics of the Elementary School. McGraw-Hill. ISBN 978-0-07-004325-1. Johnson, Paul (1975). From Sticks and Stones: Personal Adventures in Mathematics. Science Research Associates. ISBN 978-0-574-19115-1. Devine, D.; Olson, J.; Olson, M. (1991). Elementary Mathematics for Teachers (2e ed.). John Wiley & Sons. ISBN 978-0-471-85947-5.
Ferreirós (1999), p. 223. Ferreirós, José (1999). Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Birkhäuser. ISBN 978-0-8176-5749-9.
Campbell (1970), p. 83. Campbell, Howard E. (1970). The structure of arithmetic. Appleton-Century-Crofts. ISBN 978-0-390-16895-5.
Campbell (1970), p. 84. Campbell, Howard E. (1970). The structure of arithmetic. Appleton-Century-Crofts. ISBN 978-0-390-16895-5.
Ferreirós (1999), p. 135; see section 6 of Stetigkeit und irrationale Zahlen Archived 2005-10-31 at the Wayback Machine. Ferreirós, José (1999). Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Birkhäuser. ISBN 978-0-8176-5749-9.
Ferreirós (1999), p. 128. Ferreirós, José (1999). Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Birkhäuser. ISBN 978-0-8176-5749-9.
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.
Rudin (1976), p. 178. Rudin, Walter (1976). Principles of Mathematical Analysis (3rd ed.). McGraw-Hill. ISBN 978-0-07-054235-8.

arxiv.org

See Viro (2001) for an example of the sophistication involved in adding with sets of "fractional cardinality". 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.
Baez & Dolan (2001), 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!" 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.
For an example of left and right distributivity, see Loday (2002), p. 15. Loday, Jean-Louis (2002). "Arithmetree". Journal of Algebra. 258: 275. arXiv:math/0112034. doi:10.1016/S0021-8693(02)00510-0.
Compare Viro (2001), p. 2, Figure 1. 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.
Mikhalkin (2006), p. 1. 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.
Akian, Bapat & Gaubert (2005), p. 4. 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.
Mikhalkin (2006), p. 2. 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.
Litvinov, Maslov & Sobolevskii (1999), p. 3. Litvinov, Grigory; Maslov, Victor; Sobolevskii, Andreii (1999). "Idempotent mathematics and interval analysis". arXiv:math/9911126.
Viro (2001), p. 4. 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.

berkeley.edu

math.berkeley.edu

For a version that applies to any poset with the descending chain condition, see Bergman (2005), p. 100 Bergman, George (2005). An Invitation to General Algebra and Universal Constructions (2.3 ed.). General Printing. ISBN 978-0-9655211-4-7.

books.google.com

Enderton (1977), 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." Enderton, Herbert (1977). Elements of Set Theory. Academic Press. ISBN 978-0-12-238440-0.
Musser, Peterson & Burger (2013), p. 87. Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013). Mathematics for Elementary Teachers: A Contemporary Approach. John Wiley & Sons. ISBN 978-1-118-48700-6.
National Research Council (2001), p. 74. National Research Council (2001). Adding It Up: Helping Children Learn Mathematics. National Academy Press. doi:10.17226/9822. ISBN 978-0-309-06995-3.
Mosley (2001), p. 8. Mosley, F. (2001). Using number lines with 5–8 year olds. Vol. 4. Nelson Thornes. ISBN 978-1-874099-95-6.
Musser, Peterson & Burger (2013), p. 89. Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013). Mathematics for Elementary Teachers: A Contemporary Approach. John Wiley & Sons. ISBN 978-1-118-48700-6.
Berg (1967), p. 14. Berg, Lothar (1967). Lauwerier, H. A.; Koiter, W. T. (eds.). Introduction To The Operational Calculus. Vol. 2. North-Holland Publishing. ISBN 978-0-323-16245-6.
Behr & Jungst (1971), p. 59. Behr, Merlyn J.; Jungst, Dale G. (1971). Fundamentals of Elementary Mathematics. Academic Press.
Rosen 2013, See the Appendix I. Rosen, Kenneth (2013). Discrete Maths and Its Applications Global Edition. McGraw Hill. ISBN 978-0-07-131501-2.
Posamentier et al. (2013), p. 71. Posamentier, Alfred S.; Farber, William; Germain-Williams, Terri L.; Paris, Elaine; Thaller, Bernd; Lehmann, Ingmar (2013). 100 Commonly Asked Questions in Math Class. Corwin Press. ISBN 978-1-4522-4308-5.
Musser, Peterson & Burger (2013), p. 90. Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013). Mathematics for Elementary Teachers: A Contemporary Approach. John Wiley & Sons. ISBN 978-1-118-48700-6.
Hempel (2001), p. 7. Hempel, C. G. (2001). Fetzer, J. H. (ed.). The philosophy of Carl G. Hempel: studies in science, explanation, and rationality. Oxford University Press. ISBN 978-0-19-534387-8.
Gschwind & McCluskey (1975), p. 233. Gschwind, Hans W.; McCluskey, E. J. (1975). Design of Digital Computers: An Introduction (2nd ed.). Springer-Verlag.
Enderton (1977), p. 79. Enderton, Herbert (1977). Elements of Set Theory. Academic Press. ISBN 978-0-12-238440-0.
Enderton (1977), p. 79 observes, "But we want one binary operation $+$ , not all these little one-place functions." Enderton, Herbert (1977). Elements of Set Theory. Academic Press. ISBN 978-0-12-238440-0.
Enderton (1977), p. 92. Enderton, Herbert (1977). Elements of Set Theory. Academic Press. ISBN 978-0-12-238440-0.
The verifications are carried out in Enderton (1977, p. 104) and sketched for a general field of fractions over a commutative ring in Dummit & Foote (1999), p. 263. Enderton, Herbert (1977). Elements of Set Theory. Academic Press. ISBN 978-0-12-238440-0. Dummit, D.; Foote, R. (1999). Abstract Algebra (2nd ed.). Wiley. ISBN 978-0-471-36857-1.
Enderton (1977), p. 114. Enderton, Herbert (1977). Elements of Set Theory. Academic Press. ISBN 978-0-12-238440-0.
The intuitive approach, inverting every element of a cut and taking its complement, works only for irrational numbers; see Enderton (1977), p. 117 for details. Enderton, Herbert (1977). Elements of Set Theory. Academic Press. ISBN 978-0-12-238440-0.
Kay (2021), p. 44. Kay, Anthony (2021). Number Systems: A Path into Rigorous Mathematics. CRC Press. ISBN 978-0-429-60776-9.
Musser, Peterson & Burger (2013), p. 101. Musser, Gary L.; Peterson, Blake E.; Burger, William F. (2013). Mathematics for Elementary Teachers: A Contemporary Approach. John Wiley & Sons. ISBN 978-1-118-48700-6.
Enderton (1977), p. 164. Enderton, Herbert (1977). Elements of Set Theory. Academic Press. ISBN 978-0-12-238440-0.

doi.org

Mazur (2014), p. 161. Mazur, Joseph (2014). Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers. Princeton University Press. doi:10.2307/j.ctt5hhpnp. ISBN 978-0-691-15463-3. JSTOR j.ctt5hhpnp.
Lewis (1974), p. 1. 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.
National Research Council (2001), p. 74. National Research Council (2001). Adding It Up: Helping Children Learn Mathematics. National Academy Press. doi:10.17226/9822. ISBN 978-0-309-06995-3.
Li & Lappan (2014), p. 204. Li, Y.; Lappan, G. (2014). Mathematics curriculum in school education. Springer. doi:10.1007/978-94-007-7560-2. ISBN 978-94-007-7560-2.
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.
For an example of left and right distributivity, see Loday (2002), p. 15. Loday, Jean-Louis (2002). "Arithmetree". Journal of Algebra. 258: 275. arXiv:math/0112034. doi:10.1016/S0021-8693(02)00510-0.

googleresearch.blogspot.com

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.

harvard.edu

ui.adsabs.harvard.edu

See Viro (2001) for an example of the sophistication involved in adding with sets of "fractional cardinality". 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.
Compare Viro (2001), p. 2, Figure 1. 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.
Akian, Bapat & Gaubert (2005), p. 4. 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.
Viro (2001), p. 4. 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.

jhu.edu

math.jhu.edu

Riehl (2016), p. 100. Riehl, Emily (2016). Category Theory in Context. Dover. ISBN 978-0-486-80903-8.

jstor.org

Mazur (2014), p. 161. Mazur, Joseph (2014). Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers. Princeton University Press. doi:10.2307/j.ctt5hhpnp. ISBN 978-0-691-15463-3. JSTOR j.ctt5hhpnp.
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.

loc.gov

catalog.loc.gov

Karpinski (1925), pp. 56–57, reproduced on p. 104 Karpinski, Louis (1925). The History of Arithmetic. Rand McNally. LCC QA21.K3.
Schwartzman (1994), p. 212 attributes adding upwards to the Greeks and Romans, saying it was about as common as adding downwards. On the other hand, Karpinski (1925), 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. Schwartzman, Steven (1994). The Words of Mathematics: An Etymological Dictionary of Mathematical Terms Used in English. MAA. ISBN 978-0-88385-511-9. Karpinski, Louis (1925). The History of Arithmetic. Rand McNally. LCC QA21.K3.
Karpinski (1925), pp. 150–153. Karpinski, Louis (1925). The History of Arithmetic. Rand McNally. LCC QA21.K3.
Truitt & Rogers (1960), pp. 1, 44–49, 2, 77–78. Truitt, T.; Rogers, A. (1960). Basics of Analog Computers. John F. Rider. LCC QA76.4 T7.
Karpinski (1925), pp. 102–103. Karpinski, Louis (1925). The History of Arithmetic. Rand McNally. LCC QA21.K3.
Textbook constructions are usually not so cavalier with the "lim" symbol; see Burrill (1967), p. 138 for a more careful, drawn-out development of addition with Cauchy sequences. Burrill, Claude (1967). Foundations of Real Numbers. McGraw-Hill. LCC QA248.B95.
Burrill (1967), p. 140. Burrill, Claude (1967). Foundations of Real Numbers. McGraw-Hill. LCC QA248.B95.

nap.edu

National Research Council (2001), p. 74. National Research Council (2001). Adding It Up: Helping Children Learn Mathematics. National Academy Press. doi:10.17226/9822. ISBN 978-0-309-06995-3.

ncl.ac.uk

catless.ncl.ac.uk

Neumann (1987). 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.

openstax.org

Moebs, William; et al. (2022). "1.4 Dimensional Analysis". University Physics Volume 1. OpenStax. ISBN 978-1-947172-20-3.

ru.nl

Ferreirós (1999), p. 135; see section 6 of Stetigkeit und irrationale Zahlen Archived 2005-10-31 at the Wayback Machine. Ferreirós, José (1999). Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Birkhäuser. ISBN 978-0-8176-5749-9.

springer.com

link.springer.com

Lewis (1974), p. 1. 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.

theguardian.com

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.

uu.se

math.uu.se

See Viro (2001) for an example of the sophistication involved in adding with sets of "fractional cardinality". 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.
Compare Viro (2001), p. 2, Figure 1. 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.
Viro (2001), p. 4. 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.

web.archive.org

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.
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.
Neumann (1987). 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.
Ferreirós (1999), p. 135; see section 6 of Stetigkeit und irrationale Zahlen Archived 2005-10-31 at the Wayback Machine. Ferreirós, José (1999). Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Birkhäuser. ISBN 978-0-8176-5749-9.

worldcat.org

Gbur (2011), p. 1. Gbur, Greg (2011). Mathematical Methods for Optical Physics and Engineering. Cambridge University Press. ISBN 978-0-511-91510-9. OCLC 704518582.
Gbur (2011), p. 300. Gbur, Greg (2011). Mathematical Methods for Optical Physics and Engineering. Cambridge University Press. ISBN 978-0-511-91510-9. OCLC 704518582.

search.worldcat.org

Gbur (2011), p. 1. Gbur, Greg (2011). Mathematical Methods for Optical Physics and Engineering. Cambridge University Press. ISBN 978-0-511-91510-9. OCLC 704518582.
Gbur (2011), p. 300. Gbur, Greg (2011). Mathematical Methods for Optical Physics and Engineering. Cambridge University Press. ISBN 978-0-511-91510-9. OCLC 704518582.

zbmath.org

See Viro (2001) for an example of the sophistication involved in adding with sets of "fractional cardinality". 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.
Compare Viro (2001), p. 2, Figure 1. 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.
Mikhalkin (2006), p. 1. 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.
Mikhalkin (2006), p. 2. 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 (2001), p. 4. 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.