Arithmetic with Signed Fractions (or
Rational Numbers)
Before or besides this page, see video-Based
Lessons on Integers.
The lessons provide a hands-on, thought based
development of rules for integer arithmetic, and described three
roles for integers (0 and signed whole numbers).
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Role I: Integer are first introduced as
coordinates for points on a line, where adjacent points are a
unit distance apart.
-
Role II. Integers then serve as multipliers in
the definition of integer multiples of a unit movement, integer
multiples that can be added and multiplied by whole numbers and
then integers.
-
Role III. Integers themselves may describe
movements, how many steps to the left or right, along a
straight line, and so can be identified with movement, integer
multiples of a unit movement, now called a step. That third
role or identification leads allows integers to be added and
multiplied.
The foregoing may make the the following
easier to understand and explain.
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Signed Numbers as Coordinates:
Unsigned fractions, whole numbers and mixed numbers may be used as
coordinate on a half-line or ray.
We may also use fractions, whole numbers and mixed numbers with signs
prefixed to them as coordinates along a full-line:
The positive sign is optional if we identify positively signed numbers
with unsigned numbers.
In the discussion below, we identify each unsigned numbers with itself
prefixed by a plus sign.
Sign of a Signed Number - Sign Function
The sign of a signed number has the value + or -. For example
sign(-3) = - sign(+10) =
+ and sign (7) = +.
sign (+½) = +, sign (-¾) = -
and sign(3¼) = +
The sign of zero is + (or -) as you like.
Length of a signed number - the Length Function
The length of a signed number is the distance of the point it gives as a
coordinate to the origin, that is the point given by 0. That distance is
an unsigned number.
In particular:
length(-3) = 3 length(+10) =
10 and length (7) = 7.
length (+½) =½, length (-¾) =
¾ and length(3¼) = 3¼
length(0) = 0.
Property of the Length and Sign functions
Each signed number equals is sign prefixed to its length.
-¾ = sign (-¾ )length(-¾ )
as sign (-¾ ) = - and length(-¾ ) = ¾
)
Arithmetic with Signed Numbers
Addition and Subtraction
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.
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.)
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.
Multiplication
Product of Signs:
Take
(+)(+) = +
(+)(-) = -
(-)(+) = -
(-)(-) = +
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.
Think of multiplying A and B as an operation on B, then multiplying
with A multiples the length of B by the length of A while keeping the
sign of B if A is positive and changing the sign of B if A is
negative.
Understanding and Defining Division
Division of Signed Numbers:
In general, if A and B are signed numbers with B nonzero, then A divided
by B is
A÷ B = (sign A)(sign B) [(length of A)÷(Length of B)]
Check that in revisited examples A to B above. That suggest the following
rule or convention for the division of signs.
(+) ÷(+) = (+)(+) = +
(+)÷(-) = (+)(-) = -
(-)÷(+) = (-)(+) = -
(-)÷(-) = (-)(-) = +
Now if we switch from using the division sign to compound fraction
notation, we get the following formulas
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A÷ B =
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A
B
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=
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(sign A)(sign B)
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length (A)
length (B)
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Here length (A) and length (B) will be given by whole numbers, fractions
or mixed numbers. So
when computed is an unsigned number - fraction, whole number or mixed
number prefixed by a the positive or negative sign given by the
product (sign A)(sign B)
Exercise: Show B times A÷ B gives A. So
the question what times B gives A has our expression for A÷
B as an answer.
Working with Arrows (Optional)
If we define the multiplication of arrows V by signed numbers
A as follows
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A times V = {
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the vector of length (length A)(Length V) with the
same direction as V if A is positive and the opposite direction
if A is negative.
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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. The definition of products of signed
numbers could also be presented after the definition of products
of signed numbers and arrows was defined. There-in lies a
longer route, but one that might appeal to some as more natural.
Leading Questions: How many times does the first arrow
go into the second collinear arrow?
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Example
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First Arrow
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Second Arrow
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No of Times
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A
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= = =>
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= = = = = =>
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2 or +2
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B
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<= = =
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= = = = = =>
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-2
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C
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= = >
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<= = = = = =
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-3
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D
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<= =
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< = = = = = =
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3 or +3
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Leading Question: How many times does a arrow of length q
divide or go into a arrow of length p when (i) they are collinear with
the same direction; and (ii) they are collinear with the opposition
directions.
Answer for (i) is p/q
Answer for (ii) is - (p/q).
Now each signed number may be identified with an collinear arrow in the
positive or negative direction of a coordinate axis.
Example
Revisited
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First Arrow
or Number
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Second Arrow
or Number
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No of Times
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A
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= = => (+3)
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= = = = = => (+6)
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2 or +2
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B
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<= = = (-3)
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= = = = = => (+6)
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-2
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C
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= = > (+ 2)
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<= = = = = = (-6)
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-3
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D
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<= = (-2)
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< = = = = = = (-6)
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3 or +3
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