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YOU are better than YOU think. Show yourself how: |
-/[]\- Logic
Mastery Do not leave here without it - Logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck. |
-/[]\- After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6; For online automated help in senior high school maths & calculus, visit quickmath.com For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com With overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck. |
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Explore collaborative whiteboards from groupboard, twiddla or scriblink. |
Decimal Expansions
Real Numbers & Arithmetic
Related pages: Fractions Fractions as Decimals 1 = 0.999 Recurring Unsigned Reals Numbers
Some infinite decimal expansion converge or approximate fractions M/N. But there are infinite decimal expansions for points on a coordinate (half) line not given by a fraction M/N, decimal or not. The number p and the square root of prime numbers can be represented by infinite decimal expansion. These points on the line are not equal to any fraction.
Notes and Examples: Physically and geometrically, the number p is given by the circumference of a circle of diameter 1 while the square root of 2 is given by the hypotenuse of an isosceles right triangle with sides of length 1. These assumption that all lengths which occur in diagrams are proper or improper fractions of a unit length in the diagram leads to contradictions or conclusions inconsistent with the properties of whole numbers and fractions. That the fact that the square root of 2 was not a fraction was a shock to the Pythagorean religion, two thousand or so year ago (300 BC ?) due to their assumption that all lengths or measures were rational (that is fractional) multiples of each other. By placing coordinates along a half-line to measure say distance from the origin of the half-line, or the lengths of line segments that start at the origin, we may assume that all points on the half-line or all line segments starting at its origin correspond to whole numbers or fractions. The fact we have need infinite decimal expansion to measure or digitally represent the lengths of some segments is a bit of a surprise. Geometrically, that need comes from the assumption that all line segments can be approximated using a multiples of a unit length suing whole numbers, fractions and then non-repeating decimal expansion.
Unsigned Real Numbers
Each unsigned finite and infinite decimal expansion represents an unsigned real number.
Here and infinite decimal expansions ending in a recurring 9 are identified with and replaced by a decimal fraction.
We assume each finite and infinite decimal expansion gives the length of a line segment with one end at the origin of a half-line as a multiple of a unit length.
Here finite decimal expansions give decimal fractions; repeating decimal expansions give or correspond to whole numbers or simple fractions; and non-repeating decimal expansions correspond to all other points, numbers or line segment lengths.
A real given by a finite decimal expansion equals a proper or improper decimal fraction or a natural number. A real given by a infinite periodic decimal expansion equals or identified with a fraction that is not decimal - the proof that the infinite periodic decimal expansion represents a fraction may be found elsewhere.
Pure mathematics students will learn that while simple fractions are dense in the real half-line drawn above, there are far more, unaccountably many, non-repeating infinite decimal expansion than there are finite or repeating infinite decimal expansions.
www.whyslopes.com
Number Theory
Start of Number Theory
Origins of Counting or Tallying
Number Theory
Continued
Related Site Pages:
For online automated help in senior high school maths & calculus, visit quickmath.com For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com With overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck
Food for thought: Try the Twiddla Whiteboard. In principle, it allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio. Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice..