2013 年，身為數學家，也是科普作者的伊恩．史都華 (Ian Stewart) 出版了《改變世界的 17 個方程式》(The 17 Equations that Changed the World) 一書。近來，我們在 Dr. Paul Coxon 的 Twitter (由數學輔導老師，也是部落客的 Larry Phillips 所註冊) 上發現這個他摘錄書中方程式所成的簡便表格：
1. 畢氏定理 (The Pythagorean Theorem)
這個定理是我們理解幾何的基礎。
它描述平面上一個直角三角形三邊之間的關係：兩個短邊 (a 和 b) 平方相加，可以得到長邊 (c) 的平方。在某些方面，這個關係確實將我們熟知的平面歐氏幾何和曲面的非歐幾何分開來。例如，畫在球面上的直角三角形就不用遵守畢氏定理。
This theorem is foundational to our understanding of geometry. It describes the relationship between the sides of a right triangle on a flat plane: square the lengths of the short sides, a and b, add those together, and you get the square of the length of the long side, c.
This relationship, in some ways, actually distinguishes our normal, flat, Euclidean geometry from curved, non-Euclidean geometry. For example, a right triangle drawn on the surface of a sphere need not follow the Pythagorean theorem.
2. 對數 (Logarithms)
對數是指數函數的反函數。
一個特定底數的對數，告訴你用這個底數來表示這個數所需要的指數 (power)。譬如10°=1，所以底數為 10 的 1 之對數 log1=0；10¹=10，所以 log10=1；10²=100，所以 log100=2。方程式 logab = loga + logb，展示對數最有用的應用：將乘法變成加法。一直到計算機發展之前，這是快速進行大數字相乘最常用的方法，大大地加快在物理、天文，以及工程上的計算。
Logarithms are the inverses, or opposites, of exponential functions. A logarithm for a particular base tells you what power you need to raise that base to to get a number. For example, the base 10 logarithm of 1 is log(1) = 0, since 1 = 100; log(10) = 1, since 10 = 101; and log(100) = 2, since 100 = 102.
The equation in the graphic, log(ab) = log(a) + log(b), shows one of the most useful applications of logarithms: they turn multiplication into addition.
Until the development of the digital computer, this was the most common way to quickly multiply together large numbers, greatly speeding up calculations in physics, astronomy, and engineering.
3. 微積分 (Calculus)
這裏給的公式是微積分上導數的定義，導數是衡量兩個量之間變化關係的比率。
比方說，我們考慮速度，這是位置對時間的導數。假如你每小時用3英里的速度行走，表示你以每個小時3英里改變你的位置。一般而言，多數科學家有興趣的是瞭解事物如何改變，導數和積分，微積分另一個基礎，正是數學家與科學家理解變化的關鍵。
The formula given here is the definition of the derivative in calculus. The derivative measures the rate at which a quantity is changing. For example, we can think of velocity, or speed, as being the derivative of position — if you are walking at 3 miles per hour, then every hour, you have changed your position by 3 miles.
Naturally, much of science is interested in understanding how things change, and the derivative and the integral — the other foundation of calculus — sit at the heart of how mathematicians and scientists understand change.
4. 萬有引力定律 (Law of Gravity)
牛頓的萬有引力定律描述兩個物體間的吸引力F，由常數 G、兩物體的質量 m1 和 m2，以及兩物體之間的距離 r 而決定。
牛頓的定律是科學史非常出色的部份，它幾乎完美地解釋行星為何依照他們所遵循的方式移動。同樣出色的是它的普遍性，不只是適用於地球上的重力如何作用，或是在我們的太陽系，也包括宇宙的任何角落。牛頓的引力有效維持了 200 年，直到被愛因斯坦的廣義相對論取代為止。
Newton’s law of gravitation describes the force of gravity between two objects, F, in terms of a universal constant, G, the masses of the two objects, m1 and m2, and the distance between the objects, r. Newton’s law is a remarkable piece of scientific history — it explains, almost perfectly, why the planets move in the way they do. Also remarkable is its universal nature — this is not just how gravity works on Earth, or in our solar system, but anywhere in the universe.
Newton’s gravity held up very well for two hundred years, and it was not until Einstein’s theory of general relativity that it would be replaced.
5. -1 的平方根 (The square root of -1)
數學家總是在擴展數字到底是什麼的概念，從自然數，到負數、分數、和實數。-1 的平方根(經常被寫作 i)，完成這個過程，並且引出複數。
在數學上，複數是非常優雅的。代數能完美依循我們所要的方式運作，任何方程式都有複數解，不一定有實數解：像 x²+4=0 沒有實數解，但它有兩個複數的解：±2i。微積分可以擴展到複數，經由這麼做，我們發現這些數字一些驚人的對稱性和性質，這些性質使得複數在電路和信號處理上變得不可或缺。
Mathematicians have always been expanding the idea of what numbers actually are, going from natural numbers, to negative numbers, to fractions, to the real numbers. The square root of -1, usually written i, completes this process, giving rise to the complex numbers.
Mathematically, the complex numbers are supremely elegant. Algebra works perfectly the way we want it to — any equation has a complex number solution, a situation that is not true for the real numbers : x2 + 4 = 0 has no real number solution, but it does have a complex solution: the square root of -2. Calculus can be extended to the complex numbers, and by doing so, we find some amazing symmetries and properties of these numbers. Those properties make the complex numbers essential in electronics and signal processing.
6. 歐拉的多面體公式 (Euler’s Polyhedra Formula)
多面體是多邊形三維的版本，像立方體就是其中一個例子。多面體的角落就稱為頂點，連接頂點的線段稱為邊，覆蓋它的多邊形就稱為面。
一個立方體有 8 個頂點，12 條邊，和 6 個面。倘若將頂點數和面數加起來，再減去邊數，得到 8+6-12=2。歐拉的公式指出，只要你的多面體是良態的 (well behaved)，將頂點數和面數相加，再減去邊數，你總會得到2。無論你的多面體有 4、8、12、20 或是任意面數，都將為真。歐拉的發現是現在被稱為拓樸不變量的最初例子之一，同類形狀所共享的某些數字和性質彼此都相似。良態的多面體都有 V+F-1=2。這個發現，連同歐拉哥尼斯堡七橋問題 (the Bridges of Konigsburg Problem) 的解法，一起為現代物理學不可或缺的數學分支拓樸學的發展鋪好道路。
Polyhedra are the three-dimensional versions of polygons, like the cube to the right. The corners of a polyhedron are called its vertices, the lines connecting the vertices are its edges, and the polygons covering it are its faces.
A cube has 8 vertices, 12 edges, and 6 faces. If I add the vertices and faces together, and subtract the edges, I get 8 + 6 – 12 = 2.
Euler’s formula states that, as long as your polyhedron is somewhat well behaved, if you add the vertices and faces together, and subtract the edges, you will always get 2. This will be true whether your polyhedron has 4, 8, 12, 20, or any number of faces.
Euler’s observation was one of the first examples of what is now called a topological invariant — some number or property shared by a class of shapes that are similar to each other. The entire class of “well-behaved” polyhedra will have V + F – E = 2. This observation, along with with Euler’s solution to the Bridges of Konigsburg problem, paved the way to the development of topology, a branch of maths essential to modern physics.
The normal probability distribution, which has the familiar bell curve graph to the left, is ubiquitous in statistics.
The normal curve is used in physics, biology, and the social sciences to model various properties. One of the reasons the normal curve shows up so often is that it describes the behaviour of large groups of independent processes.
8. 波動方程式 (Wave Equation)
這是一個微分方程，規範波函數對時間與空間變數的二次微分的方程式，其中 C 代表波的傳遞速率。
波動方程式可描述波的行為，如振動的吉他弦，石塊丟出後池塘產生的漣漪，白熾燈泡產生的光等。波動方程式雖然只是一個微分方程式，解決這個方程所發展出的技巧開啟了理解其他微分方程的大門。
This is a differential equation, or an equation that describes how a property is changing through time in terms of that property’s derivative, as above. The wave equation describes the behaviour of waves — a vibrating guitar string, ripples in a pond after a stone is thrown, or light coming out of an incandescent bulb. The wave equation was an early differential equation, and the techniques developed to solve the equation opened the door to understanding other differential equations as well.
9. 傅立葉變換 (Fourier Transform)
想要理解複雜的波動結構，像是人的說話，傅立葉變換是不可缺少的。
給定一個複雜、凌亂的波動結構，例如人的談話錄音，傅立葉變換允許我們將這凌亂的結構分解成一些簡單波的合成，大大地簡化分析的工作。傅立葉變換是現代信號處理和分析，以及數據壓縮的核心。
The Fourier transform is essential to understanding more complex wave structures, like human speech. Given a complicated, messy wave function like a recording of a person talking, the Fourier transform allows us to break the messy function into a combination of a number of simple waves, greatly simplifying analysis.
The Fourier transform is at the heart of modern signal processing and analysis, and data compression.
10. 納維斯托克斯方程式 (Navier-Stokes Equations)
和波動方程式相同，這也是一個微分方程。
納維斯托克斯方程式描述流體的行為，水在管道的流動，空氣流過機翼，或是煙從點燃的香煙上升起。儘管利用電腦模擬流體運動，我們可以得到納維斯托克斯方程式極佳的近似解，但能否構造出這個方程式數學的精確解，仍然是待解的問題(有百萬美元獎金)。
Like the wave equation, this is a differential equation. The Navier-Stokes equations describes the behaviour of flowing fluids — water moving through a pipe, air flow over an aeroplane wing, or smoke rising from a cigarette. While we have approximate solutions of the Navier-Stokes equations that allow computers to simulate fluid motion fairly well, it is still an open question (with a million dollar prize) whether it is possible to construct mathematically exact solutions to the equations.
11. 馬克士威方程組 (Maxwell’s Equations)
這是由四個描述電 (E) 與磁 (H) 的行為和關係之微分方程所組成的方程組。
馬克士威方程組之於古典電磁學，如同牛頓的運動定律和萬有引力定律之於古典力學，他們是我們解釋電磁學在日常尺度下如何作用的基礎。如要推廣到原子的尺度，就有賴量子力學的修正，這門學問稱為量子電動力學。清楚的是，這些優美的馬克士威方程式是在人類尺度下，電磁學以及光學─光即電磁波─能夠被良好描述的近似方程組。
This set of four differential equations describes the behaviour of and relationship between electricity (E) and magnetism (H).
Maxwell’s equations are to classical electromagnetism as Newton’s laws of motion and law of universal gravitation are to classical mechanics — they are the foundation of our explanation of how electromagnetism works on a day to day scale. As we will see, however, modern physics relies on a quantum mechanical explanation of electromagnetism, and it is now clear that these elegant equations are just an approximation that works well on human scales.
12. 熱力學第二定律 (Second Law of Thermodynamics)
在一個封閉的系統中，熵 (S) 總是保持穩定或逐漸增加，而波茲曼寫下的方程式更賦予了熵統計的意義。
簡單地說，熱力學的熵是度量一個系統的紊亂程度。一個開始時有序，但不平衡的系統，例如，靠近寒冷區域的熱點區域，總是趨向平衡的紊亂狀態，熱會從熱區流向冷區，直到均勻分佈為止。大多數的物理過程都是不可逆的，亦即宇宙的熵會一直變大，這隱含了時間是有方向性的。例如我們將冰塊放入一杯熱咖啡中，我們總是看到冰塊融化，卻未曾見過一杯咖啡生出冰塊而咖啡自己變熱。
This states that, in a closed system, entropy (S) is always steady or increasing. Thermodynamic entropy is, roughly speaking, a measure of how disordered a system is. A system that starts out in an ordered, uneven state — say, a hot region next to a cold region — will always tend to even out, with heat flowing from the hot area to the cold area until evenly distributed.
The second law of thermodynamics is one of the few cases in physics where time matters in this way. Most physical processes are reversible — we can run the equations backwards without messing things up. The second law, however, only runs in this direction. If we put an ice cube in a cup of hot coffee, we always see the ice cube melt, and never see the coffee freeze.
13. 相對論 (Relativity)
愛因斯坦用狹義和廣義相對論從根本上改變了物理的進程，經典的方程式 E=mc² 說明質量與能量的轉換關係。
狹義相對論引進光在真空中的速度是固定不變的，以及不同速度移動的人對時間流逝及空間距離感受並不相同的概念。廣義相對論則認為重力是時間與空間本身的彎曲和摺疊，自牛頓的定律以來，這是我們對重力的理解首次巨大的改變，對於我們了解宇宙的起源、構造，和最終結局，廣義相對論是不可或缺的。
Einstein radically altered the course of physics with his theories of special and general relativity. The classic equation E = mc2 states that matter and energy are equivalent to each other. Special relativity brought in ideas like the speed of light being a universal speed limit and the passage of time being different for people moving at different speeds.
General relativity describes gravity as a curving and folding of space and time themselves, and was the first major change to our understanding of gravity since Newton’s law. General relativity is essential to our understanding of the origins, structure, and ultimate fate of the universe.
14. 薛丁格方程式 (Schrodinger’s Equation)
這是在量子力學上最主要的方程式，如同廣義相對論在最大尺度上說明我們的宇宙，這個方程則是支配原子和次原子粒子的行為。
現代量子力學是歷史上非常成功的科學理論所有我們做的實驗結果都和量子力學的預測完全一致。最現代的技術也需要量子力學，舉凡核能、肇基於半導體的電腦，以及雷射等都建立在量子現象上。
This is the main equation in quantum mechanics. As general relativity explains our universe at its largest scales, this equation governs the behaviour of atoms and subatomic particles.
Modern quantum mechanics and general relativity are the two most successful scientific theories in history — all of the experimental observations we have made to date are entirely consistent with their predictions. Quantum mechanics is also necessary for most modern technology — nuclear power, semiconductor-based computers, and lasers are all built around quantum phenomena.
15. 資訊理論 (Information Theory)
這裏給出的方程式是為了夏農資訊熵 (Shannon information entropy)。
如同前述，熱力學的熵是對紊亂的一種度量，這裏指的是對訊息資訊量的度量，一本書，一張網路上寄送的 JPEG 圖片，或是任何可用符號表示的事物。訊息的夏農熵說的是在不漏失內容的情形下，訊息可以被壓縮多少的下限。夏農的熵度量引起資訊理論的數學研究，他的成果是今日我們如何在網路上溝通的核心。
The equation given here is for Shannon information entropy. As with the thermodynamic entropy given above, this is a measure of disorder. In this case, it measures the information content of a message — a book, a JPEG picture sent on the internet, or anything that can be represented symbolically. The Shannon entropy of a message represents a lower bound on how much that message can be compressed without losing some of its content.
Shannon’s entropy measure launched the mathematical study of information, and his results are central to how we communicate over networks today.
16. 混沌理論 (Chaos Theory)
這個方程式是梅的二次多項式映射 (May’s logistic map)，它描述一種通過時間演變的過程 ─x的下一個時間世代 xτ+1─ 由方程式的右邊給出，依賴x目前的世代xτ。
k 是一個選擇的常數，對於某些k值，映射會顯示混沌的行為：如果從某些特殊的初始值x開始，過程將演化出一種結果，如果由其他的初始值開始，甚至非常非常靠近第一個值，過程將演化出完全不同的結果。我們所見的混沌行為，對初始條件非常敏感，在許多領域都是如此。天氣是個典型的例子，大氣條件的一個微小改變可以導致幾天後完全不同的天氣系統，這個概念最常提及的說法就是「蝴蝶效應」。
This equation is May’s logistic map. It describes a process evolving through time — xt+1, the level of some quantity x in the next time period — is given by the formula on the right, and it depends on xt, the level of x right now. k is a chosen constant. For certain values of k, the map shows chaotic behaviour: if we start at some particular initial value of x, the process will evolve one way, but if we start at another initial value, even one very very close to the first value, the process will evolve a completely different way.
We see chaotic behaviour — behaviour sensitive to initial conditions — like this in many areas. Weather is a classic example — a small change in atmospheric conditions on one day can lead to completely different weather systems a few days later, most commonly captured in the idea of a butterfly flapping its wings on one continent causing a hurricane on another continent.
17. 布萊克休斯方程式 (Black-Scholes Equation)
另一個微分方程，布萊克休斯方程式描述金融專家和商人如何找到衍生性商品的價格。
衍生性商品，基於某些潛在資產的金融商品，例如股票，一個現代金融體系的主要部份。布萊克-休斯方程式允許金融專家利用衍生性商品的特性和潛在資產來計算這些金融商品的價值。
Another differential equation, Black-Scholes describes how finance experts and traders find prices for derivatives. Derivatives — financial products based on some underlying asset, like a stock — are a major part of the modern financial system.
The Black-Scholes equation allows financial professionals to calculate the value of these financial products, based on the properties of the derivative and the underlying asset.
