ChatGPT x Re-learning: (1) Taylor Series [Column] Science of Thought (No. 68)

🗒️ Introduction | Before reading this column

This column is a long read. And the theme it covers is, quite frankly, "
mathematics itself"—it tackles the concept of the "Taylor series," which is at the high school to early university level, head-on.

It might not be for everyone.
It might not be the kind of material typically covered on note.

However, that is precisely why I wrote this, hoping it would reach those who once stumbled over mathematics and those who want to learn it again. It is also addressed to my former self.

And one more thing.
This column is also a form of experimental learning created together with ChatGPT.

I would be happy if you could feel, even a little, that "AI isn't just for searching and summarizing; it can also dig deep into the 'background' and 'why' of things like this."

Three points of this column

• Mathematical formulas can be deciphered by their 'meaning'!
  → The Taylor series is not just a formula, but a lens that expresses the 'movement of a function'.

• Differentiation is a 'tool for observing change'
  → When you understand the geometric meanings of 'slope' and 'curvature' behind the formulas, the way you see the world changes.

• It clicks when you understand 'why it was necessary'
  → When you understand it from the background of scientific revolutions or the inner workings of calculators, learning turns into realization.

Prologue | Let's start from the 'reasons for stumbling'

◆ Introduction

The first math class in university.
The familiar formula that appears on the blackboard: "This is the Taylor series":

$${f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots}$$

"What is this...?"
I think many people have had their minds go blank.

But please, rest assured.
It is not because you are stupid, nor because you are bad at math.

◆ Why do so many people stumble over the Taylor series?

I have organized some common 'stumbling points' below:

❶ Too many symbols

• $${f'(a)}$$、$${f′′(a)}$$、$${n!}$$、$${(x - a)^n}$$
  → You want to scream, "How can I understand the meaning of all these at once!" right?

❷ Not knowing "what you are doing" in the first place

• "What is the point of doing this?"

• "What is the benefit of this expansion?"

→ If you are forced to memorize it without meaning, it just becomes a "spell".

❸ The meaning of the derivative is vague

• You could "do differentiation" in high school, but what does it actually represent...?

• What is $${f^{(n)}(a)}$$? What does "n-th order derivative" mean?

→ The formula moves forward while the concept remains fuzzy.

◆ "Stumbling" is actually natural

In fact, stumbling here is very natural. This is because Taylor series:

• Has a high level of abstraction

• Has many symbols

• Presents formulas without their underlying meaning

which means it has all the conditions to make it "natural not to understand".

◆ What we will learn together

In this course, we will go on a journey to unravel the "unknowns" one by one.

To do that, we will proceed in the following order:

1. What is a Taylor series?

2. Why do we perform a Taylor expansion?

3. How is the formula derived?

4. What is a derivative in the first place?

5. Why is this expansion necessary? (History and Applications)

In other words, the goal is not to 'memorize' formulas, but to 'understand and be able to use' them effectively.

◆ A final word

Taylor expansion is a method to
'break down the movement of a function using derivatives and replace it with a polynomial'.

It is not just a calculation technique, but a
lens for understanding nature, and also an
eye for discerning the structure within change.

It is okay if it doesn't click right away.
I hope this course can be a guide that stays close to your 'I don't understand'.

Chapter 1 | What is Taylor expansion?

[1. Basic form]

When a function $${f(x)}$$ is Taylor expanded around a point $${a}$$, it takes the following form:

$${f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + \frac{f^{(3)}(a)}{3!}(x - a)^3 + \cdots}$$

This is called a Taylor series. More generally, when expressed using a
general term:

$${f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n}$$

Where:

• $${f^{(n)}(a)}$$: The n-th derivative of function $${f(x)}$$, evaluated at point $${a}$$

• $${n!}$$: factorial (e.g., 3! = 3×2×1 = 6)

[2. Common examples of Taylor expansion (center a=0, Maclaurin expansion)]

• $${e^x}$$:

  • $${e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots}$$

• $${\sin x}$$:

  • $${\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots}$$

• $${\cos x}$$:

  • $${\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots}$$

[3. When is it used?]

• For approximate calculations of functions (such as internal computer operations)

• Analysis of differential equations

• Building approximate models in physics and engineering

• Analyzing the properties (convergence, behavior) of functions

[4. Points to note]

• Even if you perform a Taylor expansion, not all functions necessarily match their series.

• For example, the function $${f(x) = e^{-1/x^2}}$$ (where $${x \ne 0}$$ and $${f(0) = 0}$$) has all derivatives equal to 0, but it is not actually 0. → This is an example where Taylor expansion is completely unreliable.

Chapter 2 | What is the geometric meaning of Taylor expansion?

[1. Taylor expansion is 'tracing a curve']

Given the graph of a function $${f(x)}$$, we consider a polynomial that 'traces' the curve at the point $${x = a}$$. At this time:

• The 0th-order term $${f(a)}$$ is → a constant function (horizontal line) passing through the point $${(a, f(a))}$$

• If we use up to the 1st-order term → tangent line (linear function)

• If we use up to the 2nd-order term → parabola (matches the curvature in addition to the tangent line)

• As we proceed to the 3rd, 4th, ... order → even more precise 'curve tracing'

In other words, it is a sequence of polynomials that gradually mimics the graph of the function more accurately around the point $${a}$$ .

[2. Geometric image (in words)]

For example, if we Taylor expand $${f(x) = \sin x}$$ near $${x = 0}$$:

• 1st-order approximation: $${ \sin x \approx x }$$ (straight line)

• 3rd-order approximation: $${ \sin x \approx x - \frac{x^3}{6} }$$ (slightly curved)

• 5th-order approximation: $${\sin x \approx x - \frac{x^3}{6} + \frac{x^5}{120}}$$ (even smoother)

In this way, the higher the order of the Taylor polynomial used, the more closely it hugs the original function.

[3. Relationship with "Differentiation"]

Geometrically speaking, Taylor expansion is increasing the "degree of contact" with the original function.

• 1st order: Point of contact and slope match (tangent line)

• 2nd order: Point of contact, slope, and curvature (2nd derivative) all match

• 3rd order and above: Even "twists" and "rapid changes" are faithfully reproduced

This can be said to be a "mathematical extension" of the local shape of a function.

[4. In terms of a metaphor?]

Taylor expansion is like
"when walking on a curved road, layering straight paths, slightly curved paths, and more winding paths... to get closer to the actual road"
.

[Summary]

The "meaning" behind each term of a Taylor expansion

• 0th-order Taylor term: Only the height at the point (function value) matches

• Adding up to the 1st-order term: The slope matches (the tangent line is the same) -> becomes linear approximation

• Adding up to the 2nd-order term: Even the curvature (how the curve rounds) matches

• Adding up to the nth-order term: The "way of touching" up to the $${n}$$th derivative matches -> becomes a more precise higher-order approximation curve

Chapter 3 | Why perform Taylor expansion?

[Purpose and reasons for using Taylor expansion]

1. Approximation of complex functions

Many functions (exponential functions, trigonometric functions, logarithmic functions, etc.) are difficult to calculate accurately.

→ Replacing with a "polynomial" using Taylor expansion makes calculations easier!

For example:

• $${e^x}$$, $$\sin x$$, and $$\ln(1+x)$$ can be approximated by infinite series

• Computers and calculators use this to determine values during internal calculations

(2) Used for solving differential equations

Many phenomena (physics, chemistry, biology) are expressed by differential equations, but there are many cases where analytical solutions (formulaic forms) cannot be obtained.

→ In such cases, the solution can be expressed in series form using Taylor expansion Example: Approximation of solutions to initial value problems for ordinary differential equations (an extension of Euler's method)

(3) Local analysis and simplification of physical models

When you want to analyze the behavior of a function at a certain point (e.g., $${x = 0}$$ or $${x = a}$$), you look only around that point.

→ By using Taylor expansion, the properties of the function around that point (slope, curvature, etc.) can be mathematically decomposed and described

(4) Approximate calculations in theoretical physics and engineering

• Newtonian mechanics/Quantum mechanics: Potential approximation (harmonic approximation)

• Structural mechanics/Electromagnetism: Linearizing nonlinear terms (also frequently appears in control theory)

(5) Elucidation of mathematical properties

It is used as a tool to delve into the properties of functions, such as how many times a function can be differentiated, how smooth it is, and to what extent the series matches (whether it is an analytic function).

[Summary: Why use Taylor expansion?]

• Simplification of calculations: Approximate functions with polynomials to facilitate manual and numerical calculations

• Numerical analysis: Approximate solutions for differential equations, numerical processing by computers

• Theory construction: Linearization of models and analysis of local behavior

• Understanding functions: Grasping properties such as differentiability, curvature, and smoothness

For example, when you calculate "$${\cos(1.2)}$$" on a smartphone calculator, it is actually performing an approximate calculation based on a Taylor series.

Chapter 4 | Steps to approximate cos(1.2) using a Taylor series

• Premise: Taylor series of $${\cos x}$$ (centered at $${a = 0}$$, Maclaurin series)

$${\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \cdots}$$

Using this, we will find the approximate value for $${x = 1.2}$$.

[Step 1: Calculate the necessary terms (about 4 terms)]

First, let's substitute $${x = 1.2}$$ and calculate them in order.

• 0th order term

$${1}$$ → as is $${{\boxed1}}$$

• 1st order term

$${-\frac{(1.2)^2}{2!} = -\frac{1.44}{2} = \boxed{-0.72}}$$

• 2nd order term

$${\frac{(1.2)^4}{4!}}$$

• $${1.2^4 = (1.44)^2 = 2.0736}$$

• $${{4!}=4\times3\times2\times1=24}$$

• $${\frac{2.0736}{24} ≈ \boxed{0.086424}}$$

• 3rd order term

$${\frac{(1.2)^6}{6!}}$$

• $${1.2^6 = (1.2^3)^2 = (1.728)^2 ≈ 2.98598}$$

• $${{6!}=6\times5\times4\times3\times2\times1=720}$$

• $${\frac{2.986}{720} ≈ \boxed{0.00415720}}$$

[Step 2: Add them up]

$${\cos(1.2) ≈ 1 - 0.72 + 0.0864 - 0.00415 = \boxed{0.36225}}$$

[Step 3: Compare with the actual value]

Using a calculator or similar tool,

$${\cos(1.2) ≈ \boxed{0.36235775...}}$$

is displayed, so,

$${\text{Error: } \approx 0.0001}$$

[Summary]

• If you use the Taylor expansion up to 4 terms,an approximation accurate to the 4th decimal place is possible

• In this way,the values of functions can be reproduced by "human hands" without a calculator, just like the inner workings of a scientific calculator

• In physics and engineering, this is highly valued because sufficient accuracy can be obtained with just a few terms

Chapter 5 | Approximating other representative functions using Taylor expansion

1. Approximate $${e^{1.2}}$$ using the Taylor expansion of $${e^x}$$!

[Formula (Maclaurin expansion)]

$${e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots}$$

[Substitute x = 1.2]

Calculate the terms in order:

• $${1}$$

• $${\frac{1.2}{1} = 1.21}$$

• $${\frac{1.2^2}{2} = \frac{1.44}{2} = 0.722}$$

• $${\frac{1.2^3}{6} = \frac{1.728}{6} ≈ 0.2886}$$

• $${\frac{1.2^4}{24} = \frac{2.0736}{24} ≈ 0.086424}$$

Total:

$${e^{1.2} ≈ 1 + 1.2 + 0.72 + 0.288 + 0.0864 = \boxed{3.2944}}$$

Actual value:

$${e^{1.2} ≈ 3.3201}$$

→ The error is about 0.025, so it's quite close! (approx. 99% accuracy with 5 terms)

(2) Approximate $$\ln(1.2)$$ using the Taylor series for $$\ln(1+x)$$!

[Formula (Maclaurin series centered at x = 0)]

$${\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \cdots \quad (|x| < 1)}$$

*Note: This expansion is often used when $${x < 1}$$.
This time, let's calculate $$\ln(1.2)$$ by setting $${x = 0.2}$$.

[Substitute x = 0.2]

• $${0.2}$$

• $${-\frac{0.2^2}{2} = -\frac{0.04}{2} = -0.02}$$

• $${\frac{0.2^3}{3} = \frac{0.008}{3} ≈ 0.00267}$$

• $${-\frac{0.2^4}{4} = -\frac{0.0016}{4} = -0.0004}$$

Total:

$${\ln(1.2) ≈ 0.2 - 0.02 + 0.00267 - 0.0004 = \boxed{0.18227}}$$

Actual value:

$${\ln(1.2) ≈ 0.18232}$$

→ The error is about 0.00005!Very high precision

[Summary: Comparison of 3 functions]

• $${\cos(1.2)}$$

  • Approximate value (4-5 terms): 0.36225

  • Actual value: 0.36236

  • Error: ≈ 0.0001

• $${e^{1.2}}$$

  • Approximate value (4-5 terms): 3.2944

  • Actual value: 3.3201

  • Error: ≈ 0.025

• $${\ln(1.2)}$$

  • Approximate value (4-5 terms): 0.18227

  • Actual value: 0.18232

  • Error: ≈ 0.00005

Chapter 6 | How is the Taylor expansion formula derived?

[1. Basic idea: I want to approximate a function with a polynomial]

Suppose we want to represent a smooth function $${f(x)}$$ around the point $${x = a}$$ as a polynomial of the form

$${f(x) \approx c_0 + c_1(x - a) + c_2(x - a)^2 + c_3(x - a)^3 + \cdots}$$

→ This is the starting point of the "Taylor expansion".

[2. How to determine the coefficients Cn? : Use derivatives to make them match]

We want the function $${f(x)}$$ and this polynomial to match at the point $${x = a}$$ in terms of:

• Value (0th order)

• Slope (1st derivative)

• Curvature (2nd derivative)

• …

and everything else.

To do that…

1. Evaluate both sides at $${x = a}$$

2. Differentiate once, twice, etc., in order, and evaluate at $${x = a}$$ again

3. Solve for each $${c_n}$$

[3. Let's actually try it (in the case where the center a = 0)]

$${f(x) = c_0 + c_1 x + c_2 x^2 + c_3 x^3 + \cdots}$$

• 1st time: As is Substitute $${x = 0}$$into it

$${f(0) = c_0}$$
　$${\Rightarrow c_0 = f(0)}$$

• Second time: Differentiate once and $${x = 0}$$ substitute

$${f'(x) = c_1 + 2c_2 x + 3c_3 x^2 + \cdots}$$
　$${\Rightarrow f'(0) = c_1}$$
　$${\Rightarrow c_1 = f'(0)}$$

• Third time: Differentiate twice and $${x = 0}$$ substitute

$${f''(x) = 2c_2 + 6c_3 x + \cdots}$$
　$${\Rightarrow f''(0) = 2c_2}$$
　$${\Rightarrow c_2 = \frac{f''(0)}{2}}$$

• Fourth time: Differentiate three times and $${x = 0}$$ substitute

$${f^{(3)}(x) = 6c_3 + \cdots}$$
　$${\Rightarrow c_3 = \frac{f^{(3)}(0)}{3!}}$$

...

[4. Generalizing]

$${c_n = \frac{f^{(n)}(0)}{n!}}$$

Therefore,

$${f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n}$$

This is the Maclaurin expansion (a special case of Taylor expansion).

[5. Why does the "factorial" appear?]

The factorial n! appears to cancel out the coefficients that emerge when differentiating n times.

For example, let's differentiate $${x^3}$$ three times.

• First derivative

$${\frac{d}{dx} (x^3) = 3x^2}$$

• Second derivative

$${\frac{d^2}{dx^2} (x^3) = \frac{d}{dx}(3x^2)=6x}$$

• Third derivative

$${\frac{d^3}{dx^3} (x^3) =\frac{d^2}{dx^2} (3x^2) = \frac{d}{dx}(6x)=6}$$

$${6 =3\times2\times1= 3!}$$

In other words, when the function $${x^3}$$ is differentiated three times, the value becomes $${3! = 6}$$.
Conversely, if you want to return to the original polynomial from the value obtained by differentiating three times,$${\div 3!}$$ you must perform.

In short, you need to divide by the factorial to correctly reproduce the polynomial.

[Summary]

• Step 1: Want to approximate a function with a polynomial

  • $${f(x) ≈ c_0 + c_1(x-a) + \cdots}$$

• Step 2: Differentiate and match the properties at point $${a}$

  • Match value, slope, and curvature

• Step 3: Differentiating produces factorials

  • Because the coefficients expand according to the number of differentiations

• Step 4: Therefore, the coefficient is $${\frac{f^{(n)}(a)}{n!}}$$

  • All terms take the form of "derivative ÷ factorial"

[Addition] How to derive the Taylor expansion formula for other functions

Example 1: Maclaurin expansion of $${e^x}$$ (center a=0)

Step 1: Examine the function and its derivatives

$${f(x) = e^x}$$

The amazing thing about this function is:

• All derivatives are the same:

$${f'(x) = e^x,\quad f''(x) = e^x,\quad f^{(n)}(x) = e^x}$$

• Therefore, $${f^{(n)}(0) = e^0 = 1}$$

Step 2: Substitute into the formula

$${f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n = \sum_{n=0}^{\infty} \frac{1}{n!} x^n}$$

Conclusion:

$${e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots}$$

This is the proof for "Why does the Taylor expansion of $${e^x}$$ take this form?"!

Example 2: Maclaurin expansion of $${\sin x}$$

Step 1: List the derivatives

$${\begin{aligned} f(x) &= \sin x \\ f'(x) &= \cos x \\ f''(x) &= -\sin x \\ f^{(3)}(x) &= -\cos x \\ f^{(4)}(x) &= \sin x \quad (one cycle here) \end{aligned}}$$

In other words, there is periodicity that loops every four terms!

Step 2: Substitute x = 0

$${\begin{aligned} f(0) &= 0 \\ f'(0) &= 1 \\ f''(0) &= 0 \\ f^{(3)}(0) &= -1 \\ f^{(4)}(0) &= 0 \\ f^{(5)}(0) &= 1 \\ f^{(6)}(0) &= 0 \\ f^{(7)}(0) &= -1 \\ \end{aligned}}$$

→ Even-order derivatives are 0, odd-order derivatives alternate between ±1

Step 3: Substitute into the formula

$${\sin x = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!} x^n}$$

Picking only the non-zero terms:

$${\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots}$$

[Additional Summary] Why does this expansion occur?

• $${e^x}$$

  • Derivative pattern: all are $${e^x}$$

  • Values at $${x = 0}$$: all are 1

  • Terms appearing in expansion: all orders appear

• $${\sin x}$$

  • Derivative pattern: cycles every 4

  • Values at $${x = 0}$$: even orders are 0, odd orders are ±1

  • Terms appearing in expansion: only odd orders appear alternately

The reason these expansions are born is all due to:

• "Examine the derivative"

• "Look at the value at that point"

• "Substitute into the formula"

It comes from this three-step logic.

Chapter 7 | What is a derivative?

[1. What is a derivative?]

For example, suppose there is a function $${f(x)}$$.

• How much does this function increase or decrease near a certain point $${x}$$?

• The derivative $${f'(x)}$$ expresses this as a "slope".

[2. Mathematical definition]

The derivative is defined by the following limit:

$${f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}}$$

This means,

• the difference in function values just a little bit ahead of $${x}$$ (at a point advanced by $${h}$$)

• dividing that by $${h}$$ to find the average rate of increase

• and bringing that $${h}$$ infinitely close to zero

→ In other words, finding the "instantaneous rate of change (instantaneous slope)"!

[3. Image of a derivative]

• The original function $${f(x)}$$ represents "position" or "height relative to $${x}$$"
  → The corresponding derivative $${f'(x)}$$ represents the "speed of change (slope)"

• If the graph of the function is a curve, then
  → The derivative represents the slope at each point on that graph

• Example: Distance as a function of timeIn the case of $${f(t)}$$,
  → The derivative $${f^′(t)}$$ becomes "speed" (velocity)

[4. Examples of common derivatives]

• Original function $${x^2}$$:
  → Derivative $${f^′}$$: $${2x}$$ (slope of a parabola)

• Original function $${sin x}$$:
  → Derivative $${f^′}$$: $${cos x}$$

• Original function $${e^x}$$:
  → Derivative $${f^′}$$: $${e^x}$$ (it doesn't change!)

• Original function $${ln x}$$:
  → Derivative $${f^′}$$: $${1/x}$$

[5. Why is it important?]

Derivatives are used in all kinds of fields, such as the following:

• Physics (velocity, acceleration)

• Economics (changes in revenue, marginal utility)

• Medicine (analysis of rates of change)

• Mathematics (finding the maximum and minimum of functions)

• Computer science (optimization, machine learning)

[Summary]

• What is a derivative?: A function that represents the rate of change (slope function)

• Mathematical definition: $${f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}​}$$

• Intuitive meaning: "How much is it increasing?" "What is the slope?"

• Practical use: Essential for phenomena such as change, increase/decrease, and optimization

Chapter 8 | What is the background behind the birth of Taylor expansion?

[1. Historical background: 17th-18th century Europe]

The Taylor series was born in late 17th to early 18th century Europe,
that is, the era immediately after the invention of calculus by Newton and Leibniz era.

Keywords of this era:

• We want to describe natural phenomena with mathematical formulas!

• We want to approximate and predict curves and motion with mathematics!

• We want to understand the behavior of functions more deeply!

[2. Purpose: To understand the 'local behavior' of functions]

Scientists and mathematicians of the time were struggling with 'how to handle unknown or complex functions?'
Although they had the tool of differentiation, they wanted to know more precisely 'how a function changes near a certain point'.

→ As a solution, the idea of 'approximating with polynomials' emerged.

[3. Development Flow]

● Isaac Newton

• Invented differential calculus

• 'Expansion by infinite series' (e.g., approximating sin x and cos x with polynomials)

• However, the description remained purely geometric and intuitive

● Gottfried Leibniz

• Organized calculus notation

• Logical systematization that emphasized the structure of functions and change

● James Gregory

• Attempted expansions such as arctan x (1670s)

● Brook Taylor (1685–1731)

• In 1715, presented the Taylor series in his book 'Methodus Incrementorum Directa et Inversa' (Direct and Inverse Method of Increments)

• For the first time, the general formula for expanding an arbitrary function into a polynomial using n-th order derivatives and factorials was explicitly stated

→ This "idea of expanding a function using derivatives" later came to be called the "Taylor series"

[4. However, it was not highly regarded at first...]

Brook Taylor's work did not receive much attention initially, and it spread after
Swiss mathematicians like Leonhard Euler and Lagrange developed and popularized it about 20 years later.

Euler applied this to all kinds of functions,

• function approximation

• numerical analysis

• modeling physical phenomena

and utilized it for such purposes.

[5. Significance for the modern era]

Taylor expansion is

• elucidating the local behavior of functions

• numerical calculation (calculators, programming)

• analyzing function properties (smoothness, convergence)

• applications in physics, engineering, economics, etc.

and has become an indispensable "staple in the mathematical toolbox".

[Summary: Why was Taylor expansion born?]

• Scientific Revolution
  → A desire was born to accurately describe the laws of nature (motion, light, waves, etc.) using mathematics

• Invention of calculus
  → It became possible to handle the "change" of functions, and the concepts of differentiation and derivatives were established (Newton, Leibniz)

• Need for polynomial approximation
  → A movement spread to express complex functions in an "easy-to-handle form (polynomials)" that is easier to calculate

• Taylor's Contribution
  → Systematized a method for generally expanding functions using derivatives and factorials

• Subsequent Development
  → Figures like Euler popularized it in both theoretical and applied aspects, leading to the foundations of modern mathematics and physics

Chapter 9 | Why did the need for polynomial approximation arise?

It stems from a strong, practical motivation to make complex natural phenomena and unknown functions manageable through manual calculation.

Let's look at the background a little more deeply below.

[1. The Connection Between Mathematics and Natural Science]

During the 17th and 18th centuries, the Scientific Revolution, the following things were happening:

• Wanting to determine planetary orbits (Kepler, Newton)

• Wanting to predict the motion of projectiles and pendulums

• Wanting to express light and sound waves with mathematical formulas

• Needing accurate trigonometric function values for navigation, astronomy, and map-making

However, with the mathematics of that time, there were almost no means to produce accurate values for functions.

[2. Polynomials are 'Calculable']

Functions like trigonometric and exponential functions are difficult to calculate by hand on their own. But polynomials are:

• Fine with just addition and multiplication

• Easy to differentiate (just bring down the exponent)

• Suitable for manual calculation and table creation!

In other words, the idea is that no matter how complex a function is, 'if you approximate it with a polynomial, you can handle it realistically!'.

[3. Explosion of Practical Needs]

What the engineers, astronomers, and navigators of that time needed were:

• Tables (numerical tables)

• Prediction (approximation)

• Processing via mental or written arithmetic

For example:

• Want to get an accurate value for $$\sin(17°)$$ → Cannot be done directly
  → Use a table of values approximated by polynomials!

[4. Advantages of Polynomial Approximation]

• Ease of calculation
  → Processing is simple because it only requires addition, subtraction, and multiplication

• Simple differentiation and integration
  → Polynomials can be differentiated and integrated any number of times, and the operations are clear

• Easy to tabulate
  → The values of each term can be expanded and listed as numbers (easy to convert into numerical tables)

• Can reproduce functions locally
  → Even the behavior of complex functions can be handled by approximating them in parts

[5. Practical Application Examples (18th Century)]

• Nautical almanacs and astronomical tables: Predicting the positions of stars and the phases of the moon

• Architecture and mechanical design: Modeling how forces are applied

• Acoustics and optics: Expressing wave interference and reflection with formulas

• Medicine and physiology: Recording and approximating pulse and respiration since that time

For all of these, the "calculation of function values" was essential, and polynomials were the easiest to handle for manual calculation.

[Summary]

• Development of science and technology
  → Increased need to express phenomena with mathematical formulas to predict and control them

• Difficulty of calculating functions
  → Exponential functions, trigonometric functions, etc., were difficult to handle with manual calculation

• Ease of handling polynomials
  → In all aspects of calculation, analysis, and approximation, polynomials are extremely easy to use.

• Practical needs
  → There was a demand for approximation methods that could be used immediately in navigation, astronomy, mechanics, and surveying.

In other words, Taylor series and polynomial approximations were practical inventions born not from the 'beauty of mathematics,' but from the need to 'solve real-world problems!' that is what they were.

Chapter 10 | How 'log tables' and 'sin tables' (trigonometric function tables) were created

'Log tables' and 'sin tables' are truly the 'crystallization of mathematics and technology.' In an era without calculators or computers, people created these through manual calculation and mathematical approximation.

[1. Why were 'numerical tables' necessary?]

From the 17th to the 19th century, the following calculations were required in fields such as navigation, astronomical observation, surveying, architecture, and engineering:

• Finding length or distance from angles (sin, cos, tan)

• Efficiently performing multiplication and division of large numbers (log)

However, at that time, there were no computers or calculators, so everything had to be done using 'pen and paper,' 'abacus,' or 'tables (lists of numbers).'

[2. How were trigonometric function tables (sin tables, cos tables) created?]

Method 1: Geometric construction (early period)

A method existing since ancient times, where angles were marked using a unit circle or right-angled triangle with a ruler and compass, and values were obtained by measuring lengths. → However, the accuracy was not very high (manual work).

Method 2: Numerical approximation using Taylor series, etc. (17th century onwards)

For example, expanding $${\sin x}$$ as follows:

$${\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots}$$

Using this formula, calculate accurately to the 4th to 6th decimal place! → Convert $${\sin(10°)}$$ etc. to radians, substitute into this formula, and
add the terms one by one by manual calculation to create a table

Method 3: Interpolation (filling in intermediate values)

10° and 11° are calculated directly → Intermediate values like 10.5° are estimated using '
linear interpolation' or 'difference tables'.

[3. How were logarithmic tables (log tables) created?]

Background:

• Multiplication can be converted to addition, and division to subtraction

• Extremely useful for astronomy, navigation, surveying, etc.!

Method 1: Original definition by Napier (1614)

• Define log_b x = y as 'y that satisfies b^y = x'

• Initially approximated by the idea of mapping geometric series (geometric progressions) to arithmetic series (arithmetic progressions)

Method 2: Common logarithms (base 10) and natural logarithms (base e)

• Since Euler, the concept of the natural logarithm ln x has become widespread

• Create approximate values using Taylor expansion, etc.:

$${\ln(1 + x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots \quad (|x| < 1)}$$

→ Using $${\log_{10} x = \frac{\ln x}{\ln 10}​}$$create all logarithmic values by manual calculation

Method 3: Interpolation and correction using the method of finite differences

• High-precision values are calculated manually using approximation formulas

• Values in between are interpolated

• Utilize 'higher-order difference tables' and 'correction terms' to improve accuracy

[4. The intensity of the actual work]

• Thousands to tens of thousands of items were calculated by hand by mathematicians and assistants over many years

• To prevent typos in transcription or printing, multiple people cross-checked the calculations

• The accuracy reached 6 to 10 decimal places in some cases!

[5. Famous collections of mathematical tables]

• Napier's Logarithm Table(1614)

  • Creator: John Napier

  • Content: First logarithm table

• Briggs' Logarithm Table

  • Creator: Henry Briggs

  • Content: Common logarithms

• Bouvard's Trigonometric Table(19th Century)

  • Creator: French Navy

  • Content: For navigation and surveying

• 7-digit Logarithm Tables and Trigonometric Tables

  • Creators: Revised by many mathematicians

  • Content: Became standard tables used worldwide

[6. How were they used?]

For example, multiplication:

$${47.2 \times 318 = ?}$$

→ Using logarithm tables:

$${log⁡_{10}47.2≈1.6749,　log⁡_{10}318≈2.5024}$$
$${\Rightarrow {Adding them:} 4.1773}$$
$${\Rightarrow 10^{4.1773} ≈ 15,050}$$

*Actual answer: $${47.2\times 318 = 14,989.6}$$
→ Very small error!

[Summary]

• Why were they needed?: There were no calculation tools, so numerical calculations had to be done by hand

• How were they made?: Taylor series, geometry, method of finite differences, and manual calculation

• Who made them?: Successive mathematicians such as Napier, Briggs, and Euler

• How were they used?: Utilized for multiplication, trigonometric calculations, astronomical navigation, and surveying

Chapter 11 | The Relationship Between Taylor Series and the Scientific Revolution

The relationship between Taylor series and the Scientific Revolution goes beyond mere history of mathematics; it is deeply connected to the "modern scientific thinking style" of expressing, approximating, and predicting natural phenomena using mathematical formulas.

Below, I will organize and explain this from historical, ideological, and technical perspectives.

[1. What was the Scientific Revolution?]

The "Scientific Revolution" that occurred in 17th-century Europe marked a shift from a world that had previously relied on theology and Aristotelian views of nature to one based on:

• Understanding nature based on experimentation and observation

• Description of natural phenomena through mathematics

• Exploration of universal laws (e.g., universal gravitation)

was a major turning point.

[2. The idea of 'rewriting' nature with mathematical formulas]

For example:

• Galileo: Observed falling motion -> converted into a mathematical model of constant acceleration motion

• Kepler: Derived planetary orbits from data -> geometric expression known as elliptical orbits

• Newton: Described laws of motion and universal gravitation using differential equations

Thus, the idea that 'nature is something that can be expressed by equations' was born.

[3. Taylor series = the 'ultimate tool' for this idea]

Because:

• Functions (= descriptions of natural phenomena) can be mathematically 'broken down' and reconstructed

• It is highly compatible with differentiation (= change), the central method of the scientific revolution

• Models that predict overall behavior from local behavior can be created

→ In other words, Taylor series were incorporated as a 'calculation engine' at the core of the scientific revolution of observation, approximation, and prediction.

[4. Concrete applications in the scientific revolution]

• Physics (Newton)
  → Used Taylor series when approximating solutions to equations of motion. Also, changes in potential energy are expressed using Taylor series
  

• Astronomy (after Kepler)
  → Captured changes in celestial orbits as minute fluctuations, and approximated them in a Taylor-series-like manner using perturbation theory

• Optics and Acoustics
  → Handled changes in wave functions and refractive indices by expanding them around a certain point (utilized in the analysis of interference and refraction)

• Surveying and Mapping
  → Describing the curved surfaces of the Earth and terrain using differential geometry, where local approximation calculations utilize Taylor series.

In this way, the Taylor series served as a key to "reconstructing the natural world through mathematical formulas," concretely supporting the Scientific Revolution.🔬✨

[5. Changes in Thought Brought About by the Taylor Series]

• Previously: Nature was something mysterious and incomprehensible.
  → After the Scientific Revolution: Nature came to be perceived as something that can be "approximated" using mathematical formulas.

• Previously: Phenomena were understood as a whole. →
  After the Scientific Revolution: A perspective emerged where the whole could be predicted from the parts (the infinitesimal).

• Previously: Philosophy and theology explained nature. →
  After the Scientific Revolution: Mathematics and physics came to explain nature.

[6. Summary: Taylor Series x Scientific Revolution]

• Historical Background: During the Scientific Revolution of the 17th century, a new mathematics called calculus was developed.

• Demand: There was a growing, strong need to quantitatively predict and analyze natural phenomena.

• Role of the Taylor Series: It functioned as a "theory of approximation" that used the tool of differentiation to decompose and reconstruct functions.

• Scientific Significance: It became a "bridge" for deriving universal laws from local observations.

• Subsequent Impact: It influenced every field, including physics, engineering, astronomy, statistics, and economics.

[In a Nutshell]

The Taylor series was the "ultimate approximation device" that responded to the
idea sought by the Scientific Revolution of "approximating and predicting nature through mathematical formulas" from both
computational and theoretical perspectives.

Afterword | The "incomprehensibility" of those days, I understand it now.

Over 30 years ago, when I was a first-year university student in the Faculty of Engineering, I encountered

“Taylor series” in a basic mathematics course.

At first, I couldn't understand it at all.

Honestly, the textbooks back then were unkind.

The professor from the Department of Mathematics
would assign problems with the stance that it was obvious to understand everything from the start.

It was an era without the internet, ChatGPT,
or any easy-to-understand reference books.

The symbols lined up on the blackboard were like “spells,” and
even when told I just needed to memorize them, I didn't understand why.

For me back then, unless I could grasp whether math was “simply solvable” and
what meaning these formulas actually had,
it wouldn't sink in...

But, after becoming a working professional, I realized something.

“It wasn't my fault that I didn't understand.”
“I just hadn't encountered the ‘meaning’ and ‘background’ yet.”

I wrote this column series
because I wanted to reach people like “the me of those days.”
Now is the time to try “math understood through meaning” once more.

And there is one more thing I want to convey here.

As the title suggests, this content
was built together through dialogue with ChatGPT,
repeatedly using trial and error for the structure, analogies, chapter organization, and how to break down terminology.

In fact, the titles of each chapter
are also the prompts I asked ChatGPT.

In other words, this is “re-learning using AI”,
a new form of learning. It's not a textbook. It's not just a search.
It's the feeling of unraveling your own “I don't understand” while engaging in dialogue.

I would be happy if you could feel even a little bit of the potential of such “Re-learning x ChatGPT.”

💬 If this column has been even a little helpful for your “I want to learn again”—
if you'd like, I would be very encouraged if you could share your thoughts via
Like , Follow, or Comment. Let's overcome “I don't understand” together!

Profile | Kazuomi Matsunaga

🧭Thinking Navigator

Biography: Graduated from Kyushu University, Faculty of Engineering (Materials Engineering) => Engineer at a metal materials manufacturer in the Chukyo region (Production Technology) => Advertising sales for a local free paper in Nagasaki Prefecture => University staff member (← Now here). A person in his 50s living in Nagasaki Prefecture. Both children have left the nest as university students, and I am living with my wife.
I am a writer who lets my thoughts play daily at the intersection of “questions” and “curiosity.” Recently, I have been writing about my own unique perspectives on themes such as education, stories, psychology, AI, and design, centered on the serial column “Science of Thought.” I am exploring “free design of knowledge” that goes beyond existing frameworks. My motto is “Imagination opens up the world.”
In my professional life, while working in university and education-related fields, I value carefully observing “the moment when people come to ‘like’ something” and “scenes where questions sprout.”
🧭 I would be happy if I could deliver articles that leave you with a “new perspective” after reading.
📮 Thoughts and messages of support are always welcome!