Signed Real Numbers
operational development

Real or Signed Numbers may be introduced as Coordinates with signs 
in 2D and then 1D
(Using maps & coordinates to introduce rectangular coordinates)

Unsigned numbers and absolute or relative lengths may be used in ordered pairs to locate points on rectangular map, with say the origin at a bottom-left corner. If that small map is extended or placed on a larger rectangular map in which the origin of the small map is not at the bottom, left corner of the larger one, the small map ordered pair, coordinate system may be extended through the use of negative numbers written in the manner ^-5 with negative signs in the super-prescript before a whole number 5. Here positive numbers, for example 6, may be employed in place of and used like unsigned numbers.

Issue: The use of signs + and - in the super-prescript position before whole numbers and decimals appeared in my mathematics education, but was not used in practice with unsigned fractions (a/b) to generate signed fractions. With the latter, signs were employed in prefix position and not in prefix, superscript position). The superscript placement of signs, positive and negative, prefixes appears to be optional. It stems from or before the Modern Mathematics curricula of the mid-1950s. 

Arithmetic with Signed (Real) Numbers 

See how to develop an operational command of arithmetic signed numbers (integers, rational, decimals and in general). This may be met or illustrated first with integers, then rational numbers and decimals. Here decimal arithmetic is done exactly or approximately

Additive Inverse - Negative of a Number A:

For A = sign(A) length (A) is nonzero, the negative of A is -A = co-sign(A) length(A) = the additive inverse of A. If A is 0, the negative of A is 0 and additive inverse of A is zero. (Saying how to calculate A defines it.)

Addition of Signed Numbers:

The sum of two signed (a.k.a real) numbers A and B is given as follows

• If A and B have the same sign then
  
  A+B = (common sign)( Magnitude(A) + Magnitude(B))
  = (common sign)(sum of the addend's magnitudes)
  
  Here the magnitudes are unsigned real numbers given by decimal or fractions etc.
  
• If A and B have opposite signs and are equal in magnitude (length) then A and B
  are additive inverses with B = -A and -A = B, and
  
  A+B = 0
  
• If A and B have opposite signs and unequal in magnitude (length) then
  
  A+ B = (sign of Biggest)( Biggest - Smallest)
  = (sign of longest) (Longest - Shortest)

If sign(A) is + or +1 then co-sign(A) is - or -1. And if sign(A) is - or -1 then co-sign(A) is + or +1.

Subtraction

The rule B - A = B + (-A) allows all subtractions of a signed number A to be expressed (rewritten) as additions involving the negative inverse of A.

Product of Signs:

(+)(+) = +
(+)(-) = -
(-)(+) = -
(-)(-) = +

Multiplication of Signed Numbers:

Next if A and B are signed numbers, their product

AB = (sign A)(sign B) [(length of A)] [(Length of B)]

                     = [(sign A)(sign B)] [(magnitude of A)(magnitude of B)]

Call this the multiply the signs, multiply the lengths for multiplication pf pairs of signed numbers. Take the product AB to be zero if A or B is zero.

Remark:  If we define the multiplication of arrows V by signed numbers A  as follows

The the law of signs for multiplication is consistent with the associativity of this "scalar" or signed real number multiplication of arrows:   

A times (B V) = (AB) times V

whenever A and B are signed numbers and V is an arrows.   This property illuminates the law of signs and provides a geometric motivation for it.  

Signed Real Numbers
operational development continued

Multiplicative Inverse (Reciprocal):

If A is nonzero, then the multiplicative inverse (a.k.a reciprocal) of A is

Division

The rule B/A = B (1/A) allows division involving a signed number A to be expressed (rewritten) as products involving the multiplicative inverse of A.

Comparisons of Signed Numbers:  

Greater in Magnitude Comparison: 

The magnitude (or length) of the signed Numbers  -10, +5, -1, 0, +3 can be compared. We see that -10 has the largest magnitude, namely 10, while 0 has  the small magnitude and that is 0. Here -10 is greater in magnitude than say +5 while 5 is greater than + 3 in magnitude.

Less Than Comparison and the LESS THAN sign < 

Examples:

• Observe  15 = 10 + 5 or 10 = 15 -5. Here 10 is 5 less than 15. We say 10 is 5 LESS THAN 10, 
  and write 10 < 15 (by 5)

• Observe  2 = -4 + 6 or -4 = 2 -6.   Here -4 is 6 LESS than -2, and we write -4 < 2  (by 6)

• Observe -8 = -15 + 7, or -15 = -8  - 7.  So -15 is 7 less than -8, and we write  -15 < -8 (by 7)

The by N part  in parentheses gives the difference. The part  is optional. 

 Definition (Algebraic Form): a first signed number A is less than a second signed number B and we write  A < B by when   A = B - C for some positive number C

More Than Comparison and the MORE THAN sign > 

Examples

• Observe  15 = 10 + 5. Here 15 is 5 more than 10. We say 15 is 5 more than 10, 
  and write 15 > 10

• Observe  2 = -4 + 6.  Here 2 is 6 more than -4, and we write 2 > -4 

• Observe -8 = -15 + 7. So -8 is 7 more than -15, and write -8 > -15

Definition (Algebraic Form):In general a first number A is more than a second number B and we write  A > B when the first number A is given by the second number B  plus a positive number C.  That is,  when A = B + C exceeds B by a positive number C.

Remark (Name Change Suggestion): Instead of calling the sign >,  the greater than sign, teachers and students should call it the more than sign. That may help because primary and junior high school students learn to compare unsigned number by magnitude and not by the more positive idea. The name change is consistent with calling the sign <, the less than sign. See below.  (The webpage Reference: Rename the Greater Than Sign written earlier suggests calling > the more positive sign instead of greater than sign.  However the phrases (i)  -10 is +4 more positive than -14 and (ii) -10 is greater than -14 are  as appealing to my ear as the phrase  -10 is +4 more than -14. 

To Do: Add or link to a lesson explaining how to use the more than or more positive than concept to manipulate inequalities - to obtain properties of inequalities - how they are preserved or reversed under addition of terms and multiplication by signed numbers.

Arithmetic - Ages 10+
1. Deciml Place Value - fun
2. Decimals for Tutors
3. Prime Factors - quickly
4. Fractions + Ratios
5. Arith with units - science