Math Foundations/A

Available as a playlist https://www.youtube.com/playlist?list=PL5A714C94D40392AB (1-79)

Subtraction and Division
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Number Theory

 * Factors
 * Squares
 * Primes (and Composites)

Subtraction

 * The inverse of addition
 * n-m is only defined (for now) if n>m

Laws of Subtraction

 * n-(m+k)=(n-m)-k
 * n+(m-k)=(n+m)-k
 * n-(m-k)=(n-m)+k
 * n(m-k)=nm-nk
 * (n-m)k=nk-mk

Division

 * The inverse of multiplication
 * n÷m is only defined if m is a factor of n

Laws of Division

 * k÷m + n÷m = (k+n)÷m
 * n÷1 = n
 * (nk)÷(mk)=n÷m
 * n÷m + k÷l = (nk)÷(ml)
 * (n÷m) ÷ (k÷l) = (nl)÷(mk)

Arithmetic and Math Education
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Main Topics

 * Arithmetic - natural numbers, integers, fractions, decimals
 * Geometry - Euclidean geometry, trigonometry, transformations, 3D geometry
 * Algebra - polynomial arithemtic, solving equations, elementary combinatorics, basic number theory
 * Analysis - graphs of functions, exponents and logarithms, circular functions, integrals, and derivatives

Basic Principles

 * CONTENT is king
 * logical structure should always be visible
 * the teacher teachers, the student solves problems
 * calculators and computers are unnecessary ... even evil
 * elaborate real-life scenarios are a waste of time
 * maintain contact with historical mathematics

Arithmetic and Primary Education

 * numbers should introduced as I-strings, then Roman numerals, then Hindu-Arabic
 * Laws should be explained and then practiced
 * subtraction and division should be introduced via natural numbers
 * number theory provides examples for counting (triangular numbers, etc.)
 * the most important subject is fractions

The Hindu-Arabic number system
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Arithmetic with Hindu-Arabic numbers
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Division
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Fraction
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Arithmetic with fractions
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'continued on Math Foundations/A/2