The word “scale” has several different meanings.

Generally it means size.

Which natural phenomenon end up on which type of scale?

scientific notation

orders of magnitude

significant digits

Sequence = a progression of numbers

Series = the sum of a sequence (a scalar)

Recursive sequence, each value related to the one before:

• arithmetic - add a constant to the previous number, $1, 3, 5, 7, 9,\cdots$
• linear = arithmetic
• algebraic = arithmetic
• geometric - multiply the previous number by a constant, $1, 2, 4, 8, 16,\cdots$
• exponential - raise the previous number by an exponent, $2, 4, 16, 256, 65536,\cdots$
• logarithmic

Non-recursive sequence:

• exponential - square an arithmetic sequence, $1^2, 2^2, 3^2, 4^2, 5^2,\cdots \rightarrow 1, 4, 9, 16, 25, \cdots$

The above examples use x 1,2,3,4,5 y 1,3,5,7,9

x 1,2,3,4,5 y 1,2,4,8,16

Same examples with x and y constant amount constant ratio

arithmetic sequence grows by a constant amount geometric sequence grows by a constant ratio

geometric sequence

initial value ratio

\begin{align} \text{initial value } c &= 120 \\ \text{ratio } r &= 0.6 \\ x &= 0,1,2,3,4,5 \\ y &= cr^x \\ x=0 &\implies y = (120)0.6^0 = 120 \\ x=1 &\implies y = (120)0.6^1 = 72 \\ x=2 &\implies y = (120)0.6^2 = 43.2 \\ x=3 &\implies y = (120)0.6^3 = 25.92 \\ x=4 &\implies y = (120)0.6^4 = 15.552 \\ x=5 &\implies y = (120)0.6^5 = 9.3312 \\ \end{align}

\begin{align} \text{initial value } c &= 1 \\ \text{ratio } r &= 2 \\ x &= 0,1,2,3,4,5 \\ y &= cr^x \\ x=0 &\implies y = (1)2^0 = 1 \\ x=1 &\implies y = (1)2^1 = 2 \\ x=2 &\implies y = (1)2^2 = 4 \\ x=3 &\implies y = (1)2^3 = 8 \\ x=4 &\implies y = (1)2^4 = 16 \\ x=5 &\implies y = (1)2^5 = 64 \\ \end{align}

\begin{align} y &= cr^x \\ x &= \{ 0,1,2,3,4,5 \} \\ \text{initial value } c &= 1 \\ \text{ratio } r &= 2 \\ x=0 &\implies y = (1)2^0 = 1 \\ x=1 &\implies y = (1)2^1 = 2 \\ x=2 &\implies y = (1)2^2 = 4 \\ x=3 &\implies y = (1)2^3 = 8 \\ x=4 &\implies y = (1)2^4 = 16 \\ x=5 &\implies y = (1)2^5 = 64 \\ \end{align}

\begin{align} a_0+a_1+a_2+\cdots \end{align}

\begin{align} \sum\limits_{i=0}^\infty a_n = a_0+a_1+a_2+a_3+a_4+\cdots \end{align}

\begin{align} \sum\limits_{n=1}^\infty \frac{1}{n} = \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\cdots \end{align}

https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)

Terms used in both music and math:

• harmonic progression - referring to chord changes
• scale - multiple kinds of scales in music: major, minor, 12-tone, pentatonic, etc.

opposites

• exponential growth accellerates
• logarithmic growth slows

software system efficiency - logarithmic growth of resource usage is a good thing, it “scales”

Warning

growth of bacteria shows exponential growth

but when plotted on a graph with a logarithmic scale, you get a straight line

transformations

the exponentiation of an arithmetic sequence results in a geometric sequence, why?

the logarithm of the geometric sequence results in the arithmetic sequence

geometric sequence is discrete

exponential function is continuous

Ray Kurzweil: Second Half of the Chessboard

Khan: Intro to Geometric Sequences https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:sequences/x2f8bb11595b61c86:introduction-to-geometric-sequences/v/geometric-sequences-introduction

Khan: Intro to Logarithms https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:logs/x2ec2f6f830c9fb89:log-intro/a/intro-to-logarithms

Khan: Forms of linear equations review https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:forms-of-linear-equations/x2f8bb11595b61c86:summary-forms-of-two-variable-linear-equations/a/forms-of-linear-equations-review

Khan: Systems of Equations https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:systems-of-equations/x2f8bb11595b61c86:solving-systems-elimination/v/king-s-cupcakes-solving-systems-by-elimination?modal=1

Wikipedia: Quadratic Equation https://en.wikipedia.org/wiki/Quadratic_equation

Wikipedia: Geometric progression https://en.wikipedia.org/wiki/Geometric_progression

Stack Overflow: Logarithmic slider with matplotlib https://mail.google.com/mail/u/0/#inbox/QgrcJHrnvrNsgvhwfZxwXLlPhDPbxVSnkjB

“Be careful, if you want to have 10^3 as initial value you have to pass in valinit=3 not 10**3. Same for valmax and valmin. You can use log10(desired_value) if you can not easily type it.”