Mat 391,    Some comments on the definition of addition

N =  The Counting or Natural numbers are the numbers 1, 2, 3, …

W = The set of Whole numbers is the union of the set of Natural Numbers with the number zero that makes our Hindu-Arabic Place Value System work.

A Binary operation on a set S is a rule or correspondence that takes a subset of SXS to S.  Addition, subtraction, multiplication, and division are all binary operations on the real numbers.  The sketch below is intended to remind you of our class discussion of how we can geometrically represent these four binary operations taking ordered pairs in various subsets of the Cartesian plane to the number line. 

For example, addition defined on the natural numbers, +:WxW W takes all ordered pairs of whole in the 1st quadrant and its axes of the Cartesian plane to the their corresponding whole number sum on the number line.  Subtraction is defined only on those whole number pairs (a,b) in the plane for which a<b. This is the set of ordered pairs of natural numbers in the first quadrant of the plane below the line y=x to their difference a-b on the number line.  The geometric representation of addition on the whole numbers picks up the non-negative x and y-axis, etc.

Binary operations can be represented in two dimensions, +:(NxN) x N or in three dimensions, +: (XxY)xZ.  We do not cover three-dimensional functions in these classes so see the relations and functions discussion on my web page.

                  The order ed pair (2,3) under the addition operation, 2+3=5 goes uniquely to the number 5 on the number line as shown below.

                                               

The subtraction operation (a,b) b-a is defined only on the subset of real numbers for which a<b. The ordered pair, (2,3) under subtraction( 3-2=1) goes uniquely to the number 1 on the number line (not shown). The order ed pair (2,3) under the multiplication operation, 3*2=6 goes uniquely to the number 6 (not shown).  The division operation is defined only on the set of whole numbers for which a divided by b is an integer. The division operation on the set of real numbers, (a,b) b/a is defined one the real umbers only if b  ≠ 0.  Defined on the nonzero real numbers, division takes (2,3) to the rational number 2/3.

Other ways to visualize the result of a binary operation is to make a table or sketch a binary function machine that takes inputs a and b and returns their sum, a+b.

                           Table      Binary Function Machine

Definition whole number addition: For Whole Numbers a, b, and c, a+b=n(AUB), where a=n(A), b=n(B) provided A B= , where zero is defined as the number of elements in the empty set, 0=n( ). Students in the lower grades better understand the empty set if it symbolized by a set of empty brackets instead of the symbol . All student can see that { } denotes a set with zero elements in it that they can compare to an empty box that has nothing in it.  The set (box) exists; it just does not contain anything.  The numbers a and b are called addends, and c is called the sum.

Classroom Models for Addition:

                           Set model:                                 application of the formal definition of addition given above.  That is, put (dump) the elements of any two distinct sets of elements into a single set (pile) and count how many elements there are in the combined set..  This model works with any finite number of distinct sets but some care needs to be made when using set notation to denote two sets.  For example, if one table has an apple on it and another table has a different apple, then you have 1+1=2 apples if you put them all on one table.  But, if Mary belongs to list of Ann’s friends and Mary is also on the list of Bob’s friends, then the two lists are not distinct if the same Mary’s name is on both lists.  The sum of the number of names on the two lists would be less than the number of people on a single list of Mary or Bob’s friends by at least one.  In this case, Mary plus Mary equals just one person, Mary.  If Mary Smith is on Ann’s list and Mary Jones is on Bob’s list, then both the first and last names should be used to distinguish between the two Marys.

                  Number line model:        start walking or moving from zero and take "a-many" steps in the direction designated then continue moving  "b-many" more steps in the same direction. In the past, it was most often the case that number lines were shown horizontally or across with positive values going from a left to right direction ( )or vertically with the positive direction up ( ) as is the case for many thermometers.  With increasingly many teachers implementing the Standards into their classrooms, it is nice to see that future students will not be conditioned that horizontal or vertical are the only "correct" orientations that hold for number lines or that all rectangles and triangles etc. must be shown only with a horizontal base.  A student can twist a ruler all around in space.  Therefore, the association of the ruler with the number line makes it as sensible that a number line can be oriented in any direction, diagonally, / or \ as well as up and across and that positive can be whatever direction it is designated to be.  It always bothers me when I see first year calculus students have trouble finding pressure or density values of a fluid where the distance down from the top of the fluid is taken to be the positive direction.  I am anxious to see elementary school textbooks display number lines and rectangles and triangles, etc. in all sorts of directions.

Use your imagination below and see two number lines not in a strictly horizontal or vertical direction.  Each shows that our blue cricket, always starting at the starting point zero, first jumps over 2 steps and then jumps over three more steps to cover 5 steps all total in the designated direction.

Basic single digit plus a single digit addition algorithms (models):

                  Counting On:          start counting from the number with the greater (≥) value, eg.  4 + 7 = 7 + 1+1+1+1=11.

                  Doubles: start with the smaller (≤) number, decompose larger number into sum of the smaller number plus number left over, double the value of the smaller number and add what's left.  4 + (4+3) = (4+4) +3 = 8+3=eleven

                  Making ten:             start with the larger (≥) number and take what is necessary to complete the count to ten from the other

                           Properties:  Whole number addition satisfies the following four number properties: closure, existence of an identity element, commutativity and associativity.  That is, for all whole numbers a, b, c, addition satisfies the properties of:

         C         Closure                      a + b is a unique whole number.

         I              Identity                     there is a unique element 0 such that a + 0 = 0 + a = a.

                                                               This says that zero uniquely identifies every number with itself under addition.

         C             Commutative           a + b = b + a,                     the order of adding numbers doesn't matter.  You can add numbers in any order and you will get the same result.

         A             Associative              a + (b + c) = (a + b) + c    how numbers are grouped to add doesn't matter. You can group numbers in any order under addition and you will get the same result.

         Put together, the front letters say that addition satisfies C CA which some students say help them remember the four addition properties. C CA is expanded to C CADZ to say that multiplication satisfies the first four properties of addition as well as the distributive (lunch bag rule) and zero properties for all numbers, namely, a(b+c)=ab+ac and a*0=0*a=0 for all whole (real ) numbers a, b and c. We explain the latter comments when we discuss multiplication.

Definition of subtraction: Subtraction is defined in terms of addition. 

                           For whole numbers a, b, c, a‑b=c if and only if  (there exists a unique whole number c such that) b+c=a. The number a is the minuend, b is the subtrahend, and c is the difference.

                           Subtraction  “undoes” addition and is called the inverse operation of addition.

                           Subtraction satisfies none of the properties of addition and has no specific properties of its own.

Classroom Models for Subtraction –

                  set or chip model:   (Take Away model)  literally remove items from a non-empty set of objects and count what's left over.

                  comparison model: take two sets and match elements or objects one by one then count all the elements for which there is nothing to pair with it.

                  missing addend:                 (application of definition of subtraction) to subtract b from a, find [?] such that [?]+b=a, i.e. a-b=[?] iff [?]+b=a.

                  number line:                         the b-cricket starts at the number a on the number line and jumps backwards b units.

                                                                        Note, for subtraction to be defined in W, b must be less than or equal to a.  If b is not less than or equal to a, the cricket backs up past the edge of the cliff and disappears.