Notes
Chapter 4: Systems Based on Numbers
Section 1: The Notion of Numbers
https://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-1--the-notion-of-numbers
![A note for mathematicians [about numbers]](/nks/img/thumbnails/notes-4-1--a-note-for-mathematicians-about-numbers--textonly.png "A note for mathematicians [about numbers]")
Section 2: Elementary Arithmetichttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-2--elementary-arithmetic
![Substitution systems [and digit sequences]](/nks/img/thumbnails/notes-4-2--substitution-systems-and-digit-sequences--textonly.png "Substitution systems [and digit sequences]")
![Leading digits [in numbers]](/nks/img/thumbnails/notes-4-2--leading-digits-in-numbers--textonly.png "Leading digits [in numbers]")
![General powers [of numbers]](/nks/img/thumbnails/notes-4-2--general-powers-of-numbers--textonly.png "General powers [of numbers]")
![Relation [of powers] to substitution systems](/nks/img/thumbnails/notes-4-2--relation-of-powers-to-substitution-systems--textonly.png "Relation [of powers] to substitution systems")
![Implementation [of 3/2 system]](/nks/img/thumbnails/notes-4-2--implementation-of-32-system--textonly.png "Implementation [of 3/2 system]")
![Reconstructing initial conditions [in the 3n+1 problems]](/nks/img/thumbnails/notes-4-2--reconstructing-initial-conditions-in-the-3n1-problems--textonly.png "Reconstructing initial conditions [in the 3n+1 problems]")
![A reversible [3n+1 problem] system](/nks/img/thumbnails/notes-4-2--a-reversible-3n1-problem-system--textonly.png "A reversible [3n+1 problem] system")
![History [of reversal-addition systems]](/nks/img/thumbnails/notes-4-2--history-of-reversal-addition-systems--textonly.png "History [of reversal-addition systems]")
Substitution systems [and digit sequences]
Multiplicative digit sequences
Multiples of irrational numbers
Relation [of powers] to substitution systems
Other uniformly distributed sequences
Implementation [of 3/2 system]
Reconstructing initial conditions [in the 3n+1 problems]
A reversible [3n+1 problem] system
Section 3: Recursive Sequenceshttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-3--recursive-sequences
![Computation of [recursive] sequences](/nks/img/thumbnails/notes-4-3--computation-of-recursive-sequences--textonly.png "Computation of [recursive] sequences")
![Properties of [recursive] sequences](/nks/img/thumbnails/notes-4-3--properties-of-recursive-sequences--textonly.png "Properties of [recursive] sequences")
![History [of recursive sequences]](/nks/img/thumbnails/notes-4-3--history-of-recursive-sequences--textonly.png "History [of recursive sequences]")
Computation of [recursive] sequences
Properties of [recursive] sequences
Section 4: The Sequence of Primeshttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-4--the-sequence-of-primes
![Properties [of number theoretic sequences]](/nks/img/thumbnails/notes-4-4--properties-of-number-theoretic-sequences--textonly.png "Properties [of number theoretic sequences]")
![Spectra [of number theoretic sequences]](/nks/img/thumbnails/notes-4-4--spectra-of-number-theoretic-sequences--textonly.png "Spectra [of number theoretic sequences]")
Section 5: Mathematical Constantshttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-5--mathematical-constants
![[Computing] square roots](/nks/img/thumbnails/notes-4-5--computing-square-roots--textonly.png "[Computing] square roots")
![Runs of digits [in numbers]](/nks/img/thumbnails/notes-4-5--runs-of-digits-in-numbers--textonly.png "Runs of digits [in numbers]")
![Leading digits [in numbers]](/nks/img/thumbnails/notes-4-5--leading-digits-in-numbers--textonly.png "Leading digits [in numbers]")
![History [of continued fractions]](/nks/img/thumbnails/notes-4-5--history-of-continued-fractions--textonly.png "History [of continued fractions]")
Section 6: Mathematical Functionshttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-6--mathematical-functions
![Differential equations [for sine sums]](/nks/img/thumbnails/notes-4-6--differential-equations-for-sine-sums--textonly.png "Differential equations [for sine sums]")
![Musical chords [from sine sums]](/nks/img/thumbnails/notes-4-6--musical-chords-from-sine-sums--textonly.png "Musical chords [from sine sums]")
![Substitution systems [and sine sums]](/nks/img/thumbnails/notes-4-6--substitution-systems-and-sine-sums--textonly.png "Substitution systems [and sine sums]")
Differential equations [for sine sums]
Musical chords [from sine sums]
Section 7: Iterated Maps and the Chaos Phenomenonhttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-7--iterated-maps-and-the-chaos-phenomenon
![Exact iterates [in iterated maps]](/nks/img/thumbnails/notes-4-7--exact-iterates-in-iterated-maps--textonly.png "Exact iterates [in iterated maps]")
![Problems with computer experiments [in chaos theory]](/nks/img/thumbnails/notes-4-7--problems-with-computer-experiments-in-chaos-theory--textonly.png "Problems with computer experiments [in chaos theory]")
![Mathematical perspectives [on iterated map complexity]](/nks/img/thumbnails/notes-4-7--mathematical-perspectives-on-iterated-map-complexity--textonly.png "Mathematical perspectives [on iterated map complexity]")
![Higher-dimensional generalizations [of iterated maps]](/nks/img/thumbnails/notes-4-7--higher-dimensional-generalizations-of-iterated-maps--textonly.png "Higher-dimensional generalizations [of iterated maps]")
![[Iterated maps from] bitwise operations](/nks/img/thumbnails/notes-4-7--iterated-maps-from-bitwise-operations--textonly.png "[Iterated maps from] bitwise operations")
Exact iterates [in iterated maps]
Problems with computer experiments [in chaos theory]
Mathematical perspectives [on iterated map complexity]
Information content of initial conditions
Higher-dimensional generalizations [of iterated maps]
Section 8: Continuous Cellular Automatahttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-8--continuous-cellular-automata
![Implementation [of continuous cellular automata]](/nks/img/thumbnails/notes-4-8--implementation-of-continuous-cellular-automata--textonly.png "Implementation [of continuous cellular automata]")
![History [of continuous cellular automata]](/nks/img/thumbnails/notes-4-8--history-of-continuous-cellular-automata--textonly.png "History [of continuous cellular automata]")
![Properties [of addition continuous cellular automata]](/nks/img/thumbnails/notes-4-8--properties-of-addition-continuous-cellular-automata--textonly.png "Properties [of addition continuous cellular automata]")
![Additive [continuous cellular automaton] rules](/nks/img/thumbnails/notes-4-8--additive-continuous-cellular-automaton-rules--textonly.png "Additive [continuous cellular automaton] rules")
Implementation [of continuous cellular automata]
History [of continuous cellular automata]
Properties [of addition continuous cellular automata]
Section 9: Partial Differential Equationshttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-9--partial-differential-equations
![Origins of the [partial differential] equations](/nks/img/thumbnails/notes-4-9--origins-of-the-partial-differential-equations--textonly.png "Origins of the [partial differential] equations")
![Nonlinearity [in PDEs]](/nks/img/thumbnails/notes-4-9--nonlinearity-in-pdes--textonly.png "Nonlinearity [in PDEs]")
![[PDEs in] higher dimensions](/nks/img/thumbnails/notes-4-9--pdes-in-higher-dimensions--textonly.png "[PDEs in] higher dimensions")
![Singular behavior [in PDEs]](/nks/img/thumbnails/notes-4-9--singular-behavior-in-pdes--textonly.png "Singular behavior [in PDEs]")
![Existence and uniqueness [in PDEs]](/nks/img/thumbnails/notes-4-9--existence-and-uniqueness-in-pdes--textonly.png "Existence and uniqueness [in PDEs]")
![Equation for the background [in my PDEs]](/nks/img/thumbnails/notes-4-9--equation-for-the-background-in-my-pdes--textonly.png "Equation for the background [in my PDEs]")
![Numerical analysis [of PDEs]](/nks/img/thumbnails/notes-4-9--numerical-analysis-of-pdes--textonly.png "Numerical analysis [of PDEs]")
![Implementation [of my PDEs]](/nks/img/thumbnails/notes-4-9--implementation-of-my-pdes--textonly.png "Implementation [of my PDEs]")
![[PDEs involving] different powers](/nks/img/thumbnails/notes-4-9--pdes-involving-different-powers--textonly.png "[PDEs involving] different powers")
Ordinary differential equations
Origins of the [partial differential] equations
Existence and uniqueness [in PDEs]
Equation for the background [in my PDEs]
Section 10: Continuous Versus Discrete Systemshttps://www.wolframscience.com/nks/chap-4--systems-based-on-numbers--notes#sect-4-10--continuous-versus-discrete-systems
![History [of continuous versus discrete mathematics]](/nks/img/thumbnails/notes-4-10--history-of-continuous-versus-discrete-mathematics--textonly.png "History [of continuous versus discrete mathematics]")
![[The word]](/nks/img/thumbnails/notes-4-10--the-word-calculus--textonly.png "[The word]")