8 Arithmetic with Signed Numbers
2 signed and unsigned numbers as coordinates
3 signed coordinates for maps and planes
4 signed coordinates for regions in space
5 lengths and signs of numbers
6 adding signed numbers
7 negative and additive inverse
8 multiplying signed numbers
9 subtracting signed numbers
10 dividing signed numbers
11 What are real lengths and numbers
Notes
Arithmetic with signed numbers includes arithmetic with integers,
rational numbers (signed fractions) and beyond them, real numbers.
Lesson 1 has just become the last lesson 11. Writing is an iterative
affair. In retrospect, being last appears to the best place for it.
Lesson 2 signed and unsigned numbers as
coordinates introduces signs as prefixes to provide coordinates along
a full number number. This use of signs + and - as prefixes to numbers in
service of providing coordinates for a full line provides an initial
context and motivation for signed numbers.
Optional Reading: Lesson 3 signed coordinates
for maps and planes and Lesson 4 signed coordinates for regions in
space describe the further use of signed numbers in pairs or triplets
to locate points.
Lesson 5 lengths and signs of numbers. Signed
numbers have a sign prefixed to a unsigned part. The latter part may be
called its magnitude, absolute value or length of the signed
number.
Lesson 6 adding signed numbers introduces methods
for adding signed numbers - those with a common signs and those with
diferent signs. To add two or several numbers with a common sign, use the
slogan prefix the common sign to the sum of their lengths. To add
two numbers with unlike signs, prefix the sign of the longest [or
largest] to the difference, the longest length minus the shortest
length. Lesson 6 describes these methods for adding with words and
with algebra, and then gives many, many examples.
Lesson 7 negative and additive inverse for each
number IT, identifies its additive inverse - a second number which when
added to IT gives a result of zero. People who think algebraically may
think x in place of IT. Lesson 7 is preparation for lesson 9. The
accompanying slogan for computing a negative or additive is simple: keep
the length, but change the sign prefixed to it. In that change, a plus +
becomes a minus -, and a minus becomes a plus.
Lesson 8 multiplying signed numbers is based on
the slogan, multiple the signs, multiple the lengths to compute
the produce of two or more sign numbers. Twenty or so multiplication
examples employing integers, proper and improper fractions, and symbols
denoting real numbers are given.
Teachers:
The case of signed mixed numbers is not
covered here, but they can be rewritten as signed improper fractions.
How to multiply mixed numbers without this conversion must wait mastery
of the distributive law and perhaps associated column multiplication
methods to exploit.
The multiply the lengths, multiply the signs
slogan provides a prequel to and slogan and rule multiply the
lengths, add the angles for multiplying complex numbers.
Lesson 9 subtracting signed numbers shows how to
subtract a number by adding its negative or additive inverse. Multiple
examples are given. Those examples are followed by two interpretations of
subtraction, the more-than interpretation [i], that the length of the
difference between two numbers gives the number of units, one is more
than another; and the geometric distance interpretation [ii], that the
length of the difference of two numbers gives the distance between two.
There-in lies a prequel to the discussion of length calculation along a
coordinate line using absolute values.
Lesson 10 dividing signed numbers presents slogan
multiply the signs, divide the lengths to say how to divide signed
numbers.
Lesson 11 What are real lengths and numbers
describes how numbers may describe length and position along straight
lines - number lines. It easy to understand the associated use of whole
numbers and fractions - proper and improper, but it may come as a
surprise that there are points on straight lines whose distance to the
origin is not a whole and/or fractional multiple of a unit length. Thus
more numbers to describe lengths appear. Thus extra numbers - the
irrationals - together with proper and improper fractions form the
unsigned real numbers. The latter provide coordinates along a
half-line.
Remark: If a number is written without a sign, its
sign is deemed to the plus sign. With that convention, the arithmetic
operations described below can also be applied to expressions involving a
mix of signed and unsigned numbers.
A Nuance
Location of Signs: The use of signs + and - in
the super-prefix position, examples +5 and -3,in
the introduction of integers appeared in modern mathematics secondary
and college education 1967-75 say. But the but was not used in practice
with rational numbers given by unsigned fractions (a/b). With the
latter, signs were employed as prefixed position but not in the
superscript position. In the following lessons, signs appear in prefix
position at normal or superscript hieght, or somewhere in between.
Whether or not the symbols + and - serve as number signs or as the
number operations - here addition and subtraction, or calculating a
negative inverse - is usually well indicated by the context, with any
ambiguity being harmless. For example -5 may indicate negative 5 - the
number - or the calculation of negative inverse of 5, a calculation
that has value negative the number.
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