Category Archives: Peano Axioms

Peano Axioms Associativity of Addition

Associativity of Addition is one of the easy induction proofs. A short example is here. http://homepages.math.uic.edu/~libgober/math215/export/215problem.pdf Aitken lectures http://public.csusm.edu/aitken_html/m378/Ch1PeanoAxioms.pdf BYU lectures http://www.math.byu.edu/~andy/math190_Ch1.pdf The following skips associative but does discuss commutative and proof by induction. http://catdir.loc.gov/catdir/samples/cam031/82004206.pdf More difficult to read perhaps: … Continue reading

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Giuseppe Peano 27 August 1858 – 20 April 1932

Peano was born today in 1858.   Peano is best known for the Peano Axioms. The definition of addition by induction was invented by the school teachers Hermmann Grassmann in 1861.  Richard Dedekind wrote his 1888 book that created New Math … Continue reading

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Multiplication of Negative Numbers Recursive Continuation

Recently, the subject of how to teach or justify rules for multiplication of negative numbers has come out.  The following is an approach based on the Peano Axioms. Natural numbers are 0, S(0), S(S(0))), … 1= 0′. S(0) = 1 … Continue reading

Posted in Analtyic Continuation, Analtyic Recursive Continuation, Angela Chen Chronicles Higher Education, dy/dan Dan Meyer, Inductive Definitions, It Ain't No Repeated Addition, James Tanton Math Videos, Justin Reich, Keith Devlin Stanford, Khan Academy, Math Video Series, MTT2k, MTT2k Prize, Mystery Teacher Theatre 2000, Recursion, Recursive Continuation, Recursive Definitions, Recursive Inductive Definition of Multiplication, Recusive Inductive Definition of Addition, Uncategorized | 50 Comments

Peano Axioms Number Line

Peano Axioms for the Number Line.  Think of a number line that starts at 0.  It only has the natural numbers as ticks, so 0, 1, 2, etc.  It has no fractions marked and it has no signs positive or … Continue reading

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Pair of Number Lines and Successor Identities

This post concerns using a pair of number lines to illustrate the successor identities.  At lower grade levels, we can simply use a pair of number lines to “prove” the successor identities.   We actually make them plausible, but at an … Continue reading

Posted in Addition Right Identities, Addition Right Successor Identity, Addition Right Zero Identity, Lesson Plans, Lessons Lower Elementary Grade Levels, Number Line Induction, Number Line Recursion, Pair of Number Line Method of Addition, Recusive Inductive Definition of Addition, Teaching prime notation with number lines, Teaching x and y with Number Lines | 1 Comment

Virginia Standards of Learning Math v Peano Axioms

The difficulty of the Peano Axioms is not greater either in abstraction or in formulas from algebra I.  Natural numbers are defined by the Peano Axioms below. To give away the answer, they are 0,1,2,.. Note the natural numbers are … Continue reading

Posted in Algebra One, Algebra One Word Problems, Carl Lee University of Kentucky, Edmund Landau Peano Axioms, Peano Axioms, Virginia Standards of Learning, Virginia Standards of Learning Algebra One, Virginia Standards of Learning Math | Tagged , , | Leave a comment

re: Alexandre Borovik Why is arithmetic difficult?

Math professor Alexandre Borovik has a post on understanding why arithmetic is difficult to learn.  It is because it has a greater conceptual structure than is realized.  He points out this complexity goes from hidden to explicit by use of … Continue reading

Posted in Addition Proofs, Alexandre Borovik, Christian Spannagel, Folge der natürlichen Zahlen, Peano Axioms for Pre-Service Teachers, PH-Heidelberg Teacher Education Math, Pre-Service Teacher Math Education, Proof and Works | Leave a comment