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Home < Arithmetic and Number Theory Skills < 6 Fractions and Ratios << 21 Working With Signs

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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).

  • 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.

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

A÷ B =

A
B

= (sign A)(sign B)

length (A)
length (B)

Here length (A) and length (B) will be given by whole numbers, fractions or mixed numbers. So

length (A)
length (B)

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

A times V = { 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.

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?

Example First Arrow Second Arrow No of Times
A = = => = = = = = => 2 or +2
B <= = = = = = = = => -2
C = = > <= = = = = = -3
D <= = < = = = = = = 3 or +3

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 		First Arrow
or Number 		Second Arrow
or Number 		No of Times
A = = => (+3) = = = = = => (+6) 2 or +2
B <= = = (-3) = = = = = => (+6) -2
C = = > (+ 2) <= = = = = = (-6) -3
D <= = (-2) < = = = = = = (-6) 3 or +3

Secondary Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges with local curriculum control. Study how to include site content - its skill development how-TOs and innovations into present or future lesson plans - some reading required.

Road Safety Messages and Questions: When and why should you face traffic when walking along a road or cycle path? Is it a good idea to hang limbs outside of cars etc? What gives more protection in a crash: a car, motorbike or bicycle? See too, the BBC-Belgium story Texting and Driving - texting & the impossible test - the article links to a gruesome utube video on the subject

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and some in Pattern Based Reason may slowly lead to greater precision in reading, applying and writing laws.

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The 8 Most Popular Site Inlinks

20 Times Table - the most popular site page - popular pages - unexpected.
Fractions & Ratios - with lesson on raising terms to introduce & justify times, division & comparison as well addition & subtraction
Parent Center - See below
Volume 1, Elements of Reason - Intro to all site books.
What is a Variable - best for ages 13+
Written work formats for Arithmetic and Algebra - a skill method and standard!
Complex Numbers Visually - best for ages 13+
Natural Logs, Exponentials, Powers, Roots

Division of Labour: This site offers advice and directions with pointers to resources elsewhere, if known, when they help or lessen the need to write more.

Parent Center: Help your child or teen learn:

Parent-friendly Work Booklets for ages 3+ to 13 Use these or others to check or build skills. Other booklets are available but these booklets allow parents unsure of themselves in mathematics to help their children. The selection acquired in Canada is published in the USA. So it has a US orientation. In retrospect, the selection shows parents what to check with the booklets or by other ways, the choice is theirs. But in retrospect, the selection does not cover integral and fractions liquid weights and measures - ask the publishers to correct that! For ages 9 to 12 say, parents may compensate by showing boys and girls how to use weights or mass, and further measures in food preparation. Beyond that children may be shown how to measure and calculate angles, lengths and areas [proportional amounts too] directly or by using maps and plans drawns to scale. Learning how to gather and measure all the ingredients, pots and pans for a dish or a meal, along with cleaning up sets the stage for like activities or experiments in science courses, and in developing organizational skills, gives boys and girls a head start. Good luck. At the other extreme, more comprehensive than light, if your motto is McCainian: drill, drill, drill then Toronto mathematician and actor John Mighton's jump math organization has jump math workbooks for at least grades 3 to 8 for at-home and in-school use - training sessions for teachers available. Jump math has been expanding to cover older students. Jump Math Samples: plus Fractions for Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8 [unread - likely to be good]. and

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Skills with take home value - A few ideas

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Arithmetic and Number Theory Skills

Algebra Starter Lessons

1 Working With Sets
2 Formula Forward Use - Evaluation
3 Solving Linear Equations - Skip first step with students able to solve 1 eqn in 1 unknown.
4 Computation Rules and Function Notation
5 Real Numbers
6 More Less Greater Than Inequalities and Comparison
7 Axioms Logic and Equivalent Equations
8 Unifying Theme For Algebra
9 Proportionality Backwards and Forwards
10 Examples of Algebraic Reasoning
A Origins of Counting and Figuring Methods
B Real Numbers Extrinsic Development


Site coverage of formuala evaluation format, of computation rules and axioms, and of the forward and backward use of formulas and proportionality relations lessens the amount of natural talent needed to understand and explain algebra.

Geometry - maps plans trigonometry vectors

1 Maps Plans Measurement
2 Euclidean Geometry - Constructions + extras
3 Cartesian and Polar Coordinates
4 Lines and Slopes Take 1
5 What is Similarity
6 Trigonometry first steps
7 Complex Numbers
8 Unit-Circle Trigonometry
9 Lines and Slopes Take 2 with tangent function
10 Intersecting Straight Lines and Transversals
11 Parallel Straight Lines and Transversals
12 Function Translating and Rescaling
13 Vectors
14 Degrees to Radians and Radians to Degrees
15 Arc or Inverse Trigonometric Function

Pre-Teen and young teen mastery of skills and practices which should be common with map-plans-diagrams drawn to scale, contour interpretation included, has actual or potential take-home value for daily- and adult-life in solving routine problems. Elevating some practices to principles, axioms or postualates, provides a base for analytic and Euclidean geometry, an analytic view of similarity, and an efficient mastery of trigonometry and complex numbers. Right triangle trigonometry provide an analytic alternative to solving geometric problems by drawing diagrams to scale.

More Algebra

Natural-Logarithms Exponentials Powers Roots
Five Polynomial Operations
Quadratics Geometrically
Functions
5 Factored Polynomial Sign Analysis Examples
Rewriting algebraic substitution as function substitutions

The first topic leads to a full high school level theory for the forward and backward mastery of growth and decay models and for definition, range and domains of radicals, roots and powers. The next two topics make quadratics and polynomials easier to learn and teach. Site coverage of functions turns vertical and horizontal line rules into computation methods for evaluating functions.

70 Calculus Starter Lessons

Calculus Lessons Elsewhere:

  1. How to Ace Calculus: Street Wise Guide - Mostly Text.

  2. Flash Video for Calculus Phobics

They cover basic topics in ways likely to complement your notes, your textbooks and site material. When Goldilocks trespassed in the house of the three bears, she found three bowls of porridge, two not to her liking, and one just right. Different bears have different tastes. As invited guest here and elsewhere, if one or more explanations is not to liking, try another. It may be better or just right.

Unsolicited Advice

Learning to do and high marks if it comes to easy is often deceptive - light rather than deep. For that reason, students with learning difficulties determined not to let it get in their way may go deeper and farther than those with none. High marks, if the come easy, may be deceptive - provide a too light and not a deep mastery. That could have been your problem in secondary school, one that leads to comprehension shock or difficulties in calculus and more generally in the first year of college. Bon Appetite.

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Home < Arithmetic and Number Theory Skills < 6 Fractions and Ratios << 21 Working With Signs

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Logic-Reason for all
Careful Thinking
Chains of Reason
Mathematical Induction
Responsibility
Bodies-of-Knowledge

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

Geometry
1 Maps + Plans Use
2 Euclidean Geometry
3 Rct +Polr Coordinates
4 Lines-Slopes [I]
5. What is Similarity 		Algebra Starters - the base
1. Better Work Format
2. Solve Linear Eqns
3. Computation Rules
4. Axioms, Item 3 Viewpnt
5. Formulas Backwards 		More Algebra
Logarithms-ax & m/nth roots
Five Polynomial Operations
Quadratics Geometrically
Functions || Vectors too
Arith. Skill Check+Answers 		Calculus Prep/Preview
What is a Variable
Why study slopes
Why factor polynomials
Complex Numbers
Limits + Continuity

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