Arithmetic Properties of Real Numbers 

 Let k be a unit vector for a straight  line with origin at a point 0 (or an origin chosen and labeled by 0)

Then Each real number  a can be identified with  vector ak with tail at the origin. 

Assumption B1. There is a set IR of real numbers a, each of which can be written as a decimal with a + or - prefix. Some of which can also be written as signed fractions of the form

where a and b are whole numbers with b non-zero.  

The following properties (laws) of arithmetic with real numbers are assumed as axioms and algebraically described. In them, Below  Z, W and V denote real numbers, or the result of calculations that yield a real numbers.

• Commutative Law for Addition:   Z + W = W + Z 
   
• Commutative Law for Multiplication:  Z W  = W Z 
   
• Distributive Law:    Z (W + V) = Z W + ZV   
                                 (W + V)Z  =WZ + VZ 
  
• Additive Identity Exists: There is a real number denote by 0  with the property 0 + Z = Z
  
• Multiplicative Identity Exist: There is a real number  1 = 1 such that 1 Z = Z.
  
• Reciprocals (Multiplicative Inverses) Exist for nonzero real numbers: If Z has length s > 0 and Z = ts where t= +1if Z is positive and -1 if Z is negative THEN then W = t*(1/s) implies WZ = 1 and we write W = 1/Z. 
  
  Note: the length s is given by a fraction a/b  then 1/s = b/a = the reciprocal of the fraction a/b.  Each finite decimal s can be written as a/b where b is a power of ten. Now for lengths s > 0 given by an infinite decimal expansion, the decimal expansion s_m of s to m decimal places has a reciprocal 1/s_m which can be computed to m decimal places  as well.  An convergence argument is needed to define 1/s  as the limiting value of 1/s_m as m approaches infinity. Alternatively, the unique measurement assumption implies 1k equal a positive multiple p of  (s k). Therefore 1k =  p (s k) = (ps) k. Therefore 1 = ps.  It follows that p  =  the number of times s goes into 1. So p = 1/s by definition. 
  
  Remark: Greater in this matter is left to advanced calculus or a first course in real analysis. 
  
• Negatives (Additive Inverses) Exist for all real numbers:  If Z = a + ib = [a, b] then W = (-a) + i(-b) has the property that W+ Z = 0.
  
• Zero Product Law Holds: If Z and W have lengths r and s both greater than 0 then their product has length rs > 0 
  
  By the methods of decimal arithmetic for the exact or approximate multiplication of whole numbers and then decimals, the product of two positive numbers or length is positive. Alternatively, the area of a rectangle of with non-zero lengths s and r as sides is nonzero. 
  
  The contra-positive of this law is of most interest for finding solutions of equations via factorization. See the site chapters on logic to learn more).
  
• Linear Comparison Law Holds: If Z and W are signed real numbers then Z-W is positive, zero or negative.

Key Subsets of the real numbers

N = set of natural numbers = {0, 1, 2, 3, ... }. It has the characteristic property that 0 belongs to N and if m belongs to N, then so does m+1.

W = set of whole numbers = {1, 2, 3, ... }. It has the characteristic property that 1 belongs to W and if m belongs to N, then so does m+1.

I = the set of integers 

  = {a in IR | a = +m or -m for some m in N}
 
Q = the set of rational numbers

    = { a in IR | a = +(p/q) or -(p/q) for some numbers p and   }
                                    q in N with q nonzero

Polar Coordinates for Real Numbers:

The number +10 is said to have polar coordinates (10, 0 degrees). The number -4 has polar coordinates (4, 180 degrees).

In general, the polar coordinates of a non-zero signed number A is given by (length (A), 0 degrees) if A is positive, and by (length(A), 180 degrees) if A is negative.  To learn more, see the site discussion of polar coordinates. 

Axioms (Assumed Patterns)

The Logic chapters 1 to 5 in Three Skills for Algebra end with a discussion of Islands and Divisions of Knowledge. That provides a metaphor for the organization of mathematics and the possibility of having different starting points for its development. 

In the modern mathematics curriculum, the existence of the real numbers and the satisfaction of above properties are often stated as assumptions or axioms   to provide a simple starting point for the further development of secondary and college mathematics.  The modern mathematics curricula of the 1950s and their continuation or diluted echo today in course design and delivery departed from pure mathematics to provide a diagram-based development of trigonometry, analytic geometry  calculus and their applications, and did not rejoin the development until after calculus in courses that very few would see. Moreover, primary school mathematics relied or relies on manipulative and physical concepts to develop number skills and sense for students.

Site pages have shown or indicated how the existence of real numbers and their arithmetic properties can be derived from common practices with and assumptions about numbers and geometry with maps, unit lengths, vectors and coordinates. That represent a deliberate departure from pure mathematics. There-in lies an extrinsic or nearly-extrinsic approach sufficient to make real and even complex numbers and their arithmetic properties clear and accessible. The assumption of those properties or their derivation allows mathematics course design and delivery to merge with a modern mathematics curriculum path. 

The development of complex numbers and their arithmetic properties may be found elsewhere in this site

Origin of the Axioms - Arithmetic Properties of Real Numbers:

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