Arithmetic With Whole Numbers

Review

• 15 times Table
• 20 times Tables

Students should know or be able to fill in 10 times and 10 addition tables rapidly without a calculator.

Mental Arithmetic: Students should also know and memorize the squares of the whole numbers 1 to 15 including especially the squares of primes 2, 3, 5, 7, 11 and 13.

Decimals Notation and Methods

Decimals and Prime Number Decomposition

• Primes & Composites
  Composite numbers less than 101
  Prime Numbers less than 101
• Primes Factorization
  Unique Prime Factorization Theorem (Cryptic Statement)
• Primes & Composites
  Calculation of Greatest Common Divisors and Least Common Multiples from Prime Factorizations
• Prime Factorization Aids & Prime Factorization Examples (two pages) If a whole number N < 121 is
  not divisible by each prime 2, 3, 5 and 7 < 121 = 11² then N is prime. Alternatively, if N < 121 is composite then it will be a single or repeat multiple at least one of the primes 2, 3, 5 and 7. Whence recognizing whether or not a whole number N < 121 is divisible by 2, 3, 5 or 7 gives a quick test for primality and a quick method for prime factorization. See examples.
  
• Counting Whole No. Factors. How to count and generate all possible factors of a whole number from its prime factorization. Introduces the concept of proper and improper factors - For the latter, is there a more standard terminology?
  
• Divisibility Rules and Remainders for Division by 2, 3, 5, 9 and 11. There are many rules for recognizing when whole numbers are multiples of 2, 3, 4, 5, 6, 7, 8, 9,10 and 11. Those rules are consequences of modulo or remainder arithmetic.
  
• Arithmetic Videos (4 groups - Realplayer is needed to watch them. Will post flash or wmv format later) These videos illustrate prime number factorization methods and uses described in above webpages.
  

  1. Primes, How to Recognize Them. Extras include statement and justification of rules for division by 2, 3, 5, 9 and 11, and the calculation of remainders for division by 2, 3, 5, 9 and 11.

  2. Fractions, Operations With. Addition, Multiplication and Reduction (Simplification) using primes, LCM, GCD. Euclid's Algorithm for computing the GCD of a pair of whole numbers provides a method for simplifying fractions, quickly without using prime decomposition of numerators and denominators.

  3. Greatest Common Divisors, Calculation using Primes or Euclid Algorithm

  4. Least Common Multiples, Calculation using Primes or Greatest Common Divisor

Arithmetic With Decimals & Fractions
(read after fractions below)

• Fractions & Decimals
  
  Decimal and Non-Decimal Fractions: A fraction is decimal when and only when it is equivalent to a fraction whose denominator is a power of ten. The latter occurs when the fraction is equivalent to a fraction whose denominator has twos and fives, and no other primes in its (the denominator's) prime factorization.
  
  A fraction is a decimal when and only when it is equivalent to a proper or improper fraction with numerator given a power of ten. Such fractions are equivalent to fractions in reduced form where the denominator > 1 is a product of 2s and/or 5s.
  
• Long Division Continued - or calculation of places after the decimal point with error bounds for a difference.
  

• Fractions as Decimals.
  
  Long Division method provides a sequence of decimal approximations to a fraction M/N. If the sequence is finite the fraction is decimal. So if the sequence does not terminate, the fraction is non-decimal.
  
  Long division and the Pigeon whole principle implies each non-decimal fraction M/N has an infinite decimal expansion with period of at most N. The decimal expansion in fact represents a sequence of approximations increasing to the fraction, where the k-th approximation ( the decimal expansion to k places after the decimal point) differs from the fraction and all following approximations by at most 1/10^k units.

• 1 = 0.999 Recurring.
  
  The number 1 can be represented exactly by itself. It can also be regarded as the limit of the sequence

  0.9 0.99 0.99 0.999 0.9999

  The decimal expansion 0.999 Recurring in fact represents a sequence of approximations increasing to the fraction, where the k-th approximation ( the decimal expansion to k places after the decimal point) differs from the fraction and all following approximations by at most 1/10^k units.
  

• Arithmetic with infinite decimal expansions is done by approximation, with the idea (assumption) that better approximations would yield more and more (an unlimited number) of decimal places in the arithmetic expansions of sums, differences, products and quotients etc. The assumption can be justified by calculus level error analysis (continuity & convergence analysis)
  
• Compound Fractions:
  
  Ratio of Simple Fractions with rules for shifting decimal point in ratios of decimal fractions, and error analysis for long division process.
  
  Ratio of Decimal Fractions - the foregoing continued in a technical manner.
  

• Square Root manipulation and Primes
  
  Square roots can be calculated (written) exactly or approximately with the aid of a calculator. This webpage and its videos show how to "simplify" square roots of whole numbers using the prime number number decomposition of the latter. Such simplification may be useful in exact numerical or algebraic manipulation of formulas and equations. Realplayer is needed to watch them. Will post flash or wmv format later)

Square Root Rule: A number N is prime if it is not divisible by all primes p whose square p² is less than or equal to N. On the other hand if a number N is not prime, it will be divisible by a prime p with p² less than N+1. With a calculator, the best bet is check where all primes p < sqrt(N) starting with the smallest. Here if N = Mq where all primes < p are not divisor of the prime N then all primes < p will not be divisors of M. With the aid of a calculators and rules for divisibility by 2,3, 5, and 11, you can quickly get the prime decomposition of a whole number N.

Videos

1. [Play Video] 5 minutes - Calculation of Squares and Square Roots for Natural Numbers without and with decimal approximations. Exact representation of square roots without approximation requires not using a calculating. That is important in algebra - the statement and derivation of formulas.
   
2. [Play Video] 1¾ minutes - How to Compute Square Roots by Factorization
   
3. [Play Video] 3 minutes - Computational Properties - More on square computation by factorization.
   
4. [Play Video] 3 minutes - Examples of square root computation by factorization.
   
5. [Play Video]3¾ minutes - Examples of square root computation by prime factorization.

In algebra, this simplification rewrites square roots in a standard form, a standard that may lead to a common representation of square roots of whole numbers when they appear in formulas and the derivation or justification of formulas.

Notes and Comments

Most of secondary school mathematics is implied by the needs of calculus. Calculus-implied standards for skill and concept development need to be well-known and applied.

Arithmetic Before and
Besides Technology

The presence of calculators in school and out lessens obviates the need for an immediate and efficient mastery of exact arithmetic with decimals and fractions. But that mastery provides a model for algebra with polynomials and permits exact arithmetic within algebra and the further parts of mathematics which depends on algebra, including trig and calculus.

Electronic gadgets, calculators and GPS included, allow us to figure and calculate in a plug and play manner, using results and making decisions without understanding the underlying reasoning or logic. But mathematics is different. Mathematics is an arts and disciplines in which the thought-based development of skills, concepts and tools (gadgets included) can be explained at the high school level with minimal dependence on plug and play items. There-in lies a model and standard for skill and concept development in other arts and disciplines to strive for, but not attained. In that model, students see the need to follow or apply the steps of a method, one at a time, one after another, carefully and precisely, as an error in one step (if not compensated for by a further errors) leads to incorrect results.

In arithmetic, results and the chains of reason or figuring that leads to them are expected to be repeatable, reproducible, observable and thus verifiable. That expectation provides one model for care and precision in applying methods in and outside of arithmetic, a model independent of the question of whether or not the methods are justifiable. Methods and practices that lead or appear to lead to repeatable. reproducible and hence verifiable results independent of who or what follows them are not necessarily correct in terms of justice, pollution and sustainable development, questions beyond the immediate scope of this index. None the less for an example of repeatable, reproducible, observable and hence verifiable results student may master arithmetic methods, and learn to how to present their result and the chains of reason or figuring that lead to the results in an observable and hence verifiable or correctable manner.

Proper Format for Evaluation of Arithmetic Expressions - very, very important!

The site area LAMP provides a format for the evaluation of arithmetic and algebraic expressions. That format advice extends the advice given for showing work in the lesson  evaluation of an algebraic formulas

Teach students to evaluate arithmetic expressions in a manner that shows and records the steps in the evaluation of sub-expressions to help students and teachers identify errors, or better yet avoid them. 

Put quality first: Substitution need to be made and recorded on at at time and one after another in a manner that makes the work simpler and records for verification or correction by the student, peers there-of, or a teacher. 

The first objective of junior high school learning and teaching would be an operational command of arithmetic methods. The first aim is for student to learn to do arithmetic with explanation optional, yet given where it helps students to obtain that operation command. Students should learn to do their work in a clear and legible, step-by-step development and record of a method, so they, their peers and their tutors or instructors may follow to check or verify each step. Here quality is more important that quantity.  Proper notation is a tool for recording and developing solutions on paper in an observable, repeatable, reproducible manner that communicates the steps or reasoning in a solution. That provides a measure of success.

Explanations are optional. The development of skills and concept via numerical example will be easier for students to follow in the first instance. But the further development of skill and concepts with the aid of algebraic notation and some points to a further level of comprehension and a higher level of explanation. To reach that level, algebra skill and comprehension is required. See the math guide to algebra.

Advanced Material
Start of Number Theory

The pages

Origins of Counting or Tallying
Adding Wholes
Multiplying Wholes

Distributive Law Preamble
Distributive Law for Wholes

Consequences - Justifying Decimal place value, column methods for multiplication & introducing place value column methods for addition

develop properties of whole numbers from assumptions implicit in the primary school development of number sense and skills.

Rote Learning Should be Alive and Well.

Fractions - Operational Command

• Fraction Starter Lesson. Rules to simplify, multiply, divide, simplify, compare, add or subtract covered and explained in part. Real player videos are included.

  Students who have seen fractions before can be given an operational command of fractions through the following steps: (i) Learn how to simplify fractions by canceling common factors in enumerators and denominators; (ii) Learn how to multiply fractions but with an emphasis on postponing multiplication in favor of factoring the numerator and denominators of products in order to cancel and simplify; (iii) Learn how to divide fractions by turning divisions into multiplication by a reciprocal, and then applying the efficient product simplification methods in step; (iv) learn how to add and subtract fractions with like denominators and how to simplify the sum; (v) learn how to add and subtract fractions with unlike denominators and the role of least common denominators in reducing the amount of simplification needed in sums.

  In the foregoing, prime decompositions can be introduced to aid simplification and to aid the computation of least common denominators and greatest common divisors. Teach students to look for factors of whole number among those primes whose square is less than or equal to the whole number in question. If none those of those primes are factors, the whole number in question is prime. Calculators and knowledge of all primes less than 50 are sufficient to quickly generate the prime number decomposition of all numbers < 2500. The link to prime numbers can be postponed.)
  
  The foregoing program provides fraction skills, an operational command, but does not develop develop fraction sense. The site area Solving Linear Equations with and without Stick Diagrams develops fractions skills and sense by illustrating and demanding fractional operations on line segments - the sticks - while also introducing and/or consolidating algebra skills and sense.

  
  

Fraction Sense (Linear)

The following twelve lessons cover fraction skills sequentially and carefully with arithmetic operations developed or extracted from operations on line segments. The twelve lessons may be read in private for skill and concept consolidation (perfection) by serious students. Tutors and teachers may employ the ideas in these lessons to fill gaps in the knowledge or memory of primary school mathematics skills of their students.

1 What is a Fraction - Physical Meaning and an optional on first reading, Algebraic Perspective
2 Multiplication I - unit fraction of unit fraction
3 Multiplication II - unit fraction of simple fraction
4 Multiplication III - simple fraction of a simple fraction
5 Equivalent Fractions - with videos on fraction simplification
6. Mixed Numbers - Improper Fractions and Equivalent Mixed Numbers
7 Comparison - How raising fractions over a common denominator leads to direct comparison (and justifies cross multiplication comparison methods). 8 Addition I - Addition (and Subtraction) with like Denominators.
9 Addition II - Addition (and Subtraction) with unlike Denominators
10 Addition III -Addition (and Subtraction) Efficiently with the use of least common denominators with video indicating how not using latter may be less efficient. (The efficiency methods here are not always the best - sometimes properties of decimals with denominators that are multiples of 2 and 5 give exceptions to the efficienct addition and subtraction methods indicated here.
11 Multiplication IV - Efficient ways to Multiply Fractions. See how Multiplying Fractions with cancellation of common factors done first (recommended) or not, lessens the simplification to be done later.
12 Division Division of Fractions and Compound Fractions. The question of how many times a length goes into longer length may be answered physically by dividing the longer length into segments, each of which has the same length as the shorter one. There may be fraction left over. When the two lengths in question are proper or improper fractions of a unit length, the physical question of how to divide can be answered via arithmetic operations.

The Alternative Fraction Starter Lesson Fractions in a Nutshell may be best viewed as a review or summary of the foregoing twelve lessons.

Notes and Comments

• The mastery of mathematics from algebra to calculus demands fraction sense and skills fully and efficiently.
• How to add, compare, subtract, multiply and divide fractions efficiently without the use of the calculator is a must for algebra and for justifying some operations with decimals.
• Students need an operational command in which arithmetic operations alone or in combination are recorded step by step in an observable manner for the sake of verification (repeatable and reproducible results) or correction.

Conversion to decimals provides approximate methods for arithmetic with fractions, suitable perhaps for use in daily life, but insufficient for the numerical and algebraic skills and concepts required by calculus.

The site area Solving Linear Equations with and without Stick Diagrams strengthens fractions skills and sense by illustrating and demanding fractional operations on line segments - the sticks - while also introducing and/or consolidating algebra skills and sense.

The number theory area of this site contains two further essays

• What is a Fraction
• Compound Fractions

which complement and duplicate
other site lessons.

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)]

AB = [(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.

Signed Real Numbers
geometric development

Coordinates, relative or absolute?

Coordinates may be given relative to a choice of unit length and direction (a unit vector) along the coordinate axes of a map. Or, equivalently, coordinates may be given relative to a choice of unit length and a choice of positive direction for the coordinate axes. In both cases, these relative coordinates are ordered pairs of signed numbers.

Coordinates may also be given absolutely relative to a choice of unit length and choice of positive direction for each coordinate axis. For example, a point in a planar map may be determined by absolute coordinates

[⁺5 cm, ^-6 cm]

where here the unit of length is the centimeter cm. Implicit here (a first example) is the multiplication of the unit length by a signed number. Implicit here (another first example) is a multiplication of the unit and unit vectors along the coordinate axes by signed numbers. 

Addition of vectors (displacements) in the plane and more specifically collinear vectors in a line, their multiplication by signed numbers (coordinates) and their representation as signed numbers multiplies of a unit vector implies is or consistent with definition of the addition and multiplication of the signed number multipliers alone, apart from their role in representing collinear vectors as multiples of a given unit vector.  

Details and Theory

• Unsigned Reals Numbers - use of unsigned decimals as coordinates.
  
• Signed Coordinates - Introduction of real numbers by prefixing signs to hitherto unsigned numbers.
  
• Plane Vectors - Navigation - use of arrows or vectors in describing piecewise linear paths in the plane; Head-to-tail addition; Associativity of in place head-to-tail addition.
  
• Horizontal Vectors & Adding Vector Multiples of unit vectors]. Addition of horizontal, more generally collinear, vectors that represent displacements, AND properties of this addition - commutativity included.
  

•  Adding Signed Numbers. The addition of signed numbers A and B is defined so the addition of multiples A and B of a  vector  equals the multiple A+B the vector.

•  Multiplying Signed Numbers. The product or multiplication of signed numbers is defined so the multiplication by signed number A of a signed number multiple B  of a vector is equals the multiple AB of that vector.

• Distributive Law for Reals. The sum of collinear vectors given by multiples A and B of a nonzero k should not change if k = c m where m is another vector. Two methods of expressing the sum as multiple of c lead to the distributive property  (A+B)C = AC + BC for signed real numbers.

• [Real Numbers Axioms] The foregoing considerations imply a superset of the real number axioms assumed in modern mathematics curricula (or derived in a context free manner in pure mathematics.)

• Modular or Remainder Arithmetic for real numbers- Here is real number generalization of modular or remainder arithmetic for whole numbers. 

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