真空嫌忌
アリストテレスは空虚な空間（真空）の存在を否定する。第1の理由は、月より下の世界における物体の運動の観察による。月下の物体は重さ、軽さという物体の本性に起因する上昇と下降の運動を成そうとするが、同時に力によって強制的に運動させられる。空気はその本性から軽くもあり、重くもあるので、空気はその力に押されて運動のきっかけを得ると、上昇や下降の運動を起こし、物体に運動を伝える。第2の理由は、物体の相互変化の観察からである。すべての物体は他の物体から生まれる。あらゆる物体に無からの生成はありえない。もし、生成するものがあるとするならば、生成してくるものの占めるべき空虚な空間が、予めどこかに存在しなくてはならないことになる。アリストテレス「天体論 第3巻 月下の物体について」参照。
Dividing by Nothing
by Alberto Martinez
imageIt is well known that you cannot divide a number by zero. Math teachers write, for example, 24 ÷ 0 = undefined. They use analogies to convince students that it is impossible and meaningless, that “you cannot divide something by nothing.” Yet we also learn that we can multiply by zero, add zero, and subtract zero. And some teachers explain that zero is not really nothing, that it is just a number with definite and distinct properties. So, why not divide by zero? In the past, many mathematicians did.
In 628 CE, the Indian mathematician and astronomer Brahmagupta claimed that “zero divided by a zero is zero.” At around 850 CE, another Indian mathematician, Mahavira, more explicitly argued that any number divided by zero leaves that number unchanged, so then, for example, 24 ÷ 0 = 24. Later, around 1150, the mathematician Bhaskara gave yet another result for such operations. He argued that a quantity divided by zero becomes an infinite quantity.image This idea persisted for centuries, for example, in 1656, the English mathematician John Wallis likewise argued that 24 ÷ 0 = ∞, introducing this curvy symbol for infinity. Wallis wrote that for ever smaller values of n, the quotient 24 ÷ n becomes increasingly larger (e.g., 24 ÷ .001 = 24,000), and therefore he argued that it becomes infinity when we divide by zero.
The common attitude toward such old notions is that past mathematicians were plainly wrong, confused or “struggling” with division by zero. But that attitude disregards the extent to which even formidably skilled mathematicians thoughtfully held such notions. In particular, the Swiss mathematician, Leonhard Euler, is widely admired as one of the greatest mathematicians in history, having made extraordinary contributions to many branches of mathematics, physics, and astronomy in hundreds of masterful papers written even during his years of blindness. Euler’s Complete Introduction to Algebra (published in German in 1770) has been praised as the most widely published book in the history of algebra. We here include the pages, in German and from an English translation, in which Euler discussed division by zero. He argued that it gives infinity.
Euler_German
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
Euler_Algebra_1810
Title page of an English edition of Euler’s Algebra, published in 1810 (translated from a French translation of the German original), and Article 83, p. 34, with footnote additions.
A common and reasonable view is that despite his fame, Euler was clearly wrong because if any number divided by zero gives infinity, then all numbers are equal, which is ridiculous. For example:
if 3 ÷ 0 = ∞, and 4 ÷ 0 = ∞,
then ∞ x 0 = 3, and ∞ x 0 = 4.
Here, a single operation, ∞ x 0, has multiple solutions, such that apparently 3 = 4. This is absurd, so one might imagine that there was something “pre-modern” in Euler’s Algebra, that the history of mathematics includes prolonged periods in which mathematicians had not yet found the right answer to certain problems. However, Euler had fair reasons for his arguments. Multiple solutions to one equation did not seem impossible. Euler argued, for one, that the operation of extracting roots yields multiple results. For example, the number 1 has three cube roots, any of which, multiplied by itself three times, produces 1. Today all mathematicians agree that root extraction can yield multiple results.
So why not also admit multiple results when multiplying zero by infinity?
An alternate way to understand the historical disagreements over division by zero is to say that certain mathematical operations evolve over time. In antiquity, mathematicians did not divide by zero. Later, some mathematicians divided by zero, obtaining either zero or the dividend (e.g., that 24 ÷ 0 = 24). Next, other mathematicians argued, for centuries, that the correct quotient is actually infinity. And nowadays again, mathematicians teach that division by zero is impossible, that it is “undefined.” But ever since the mid-1800s, algebraists realized that certain aspects of mathematics are established by convention, by definitions that are established at will and occasionally refined, or redefined. If so, might the result of division by zero change yet again?
In 2005, I showed students at UT how the computer in our classroom, an Apple iMac, carried out division. I typed 24 ÷ 0, then the enter key. The computer replied “infinity!” Strange that it included an exclamation mark. Some students complained that the computer was an Apple instead of a Windows PC. The following year, a similar Apple computer, a newer model, also answered “infinity,” but without the exclamation mark. In 2010, the same operation on a newer computer in the classroom replied: “DIV BY ZERO.” Yet that same computer has an additional calculator, a so-called scientific calculator, and the same operation on this more sophisticated calculator gave “infinity.” image image
Students’ calculators, such as in their cell phones, gave other results: “error,” or “undefined.” One student’s calculator, a Droid cell phone, answered: “infinity.” None of these answers is an accident, each has been thoughtfully programmed into each calculator by mathematically trained programmers and engineers. Computer scientists confront the basic and old algebraic problem: using variables and arithmetical operations, occasionally computers encounter a division in which the divisor has a value of zero—what should computers do then? Stop, break down?
On 21 September 1997, the USS Yorktown battleship was testing “Smart Ship” technologies on the coast of Cape Charles, Virginia. At one point, a crew member entered a set of data that mistakenly included a zero in one field, causing a Windows NT computer program to divide by zero. This generated an error that crashed the computer network, causing failure of the ship’s propulsion system, paralyzing the cruiser for more than a day.
These issues show that we are unjustified in assuming that we are lucky enough to live in an age when all the basic operations of mathematics have been settled, when the result of division by zero in particular cannot change again. Instead, when we look at pages from old math books, such as Euler’s Algebra, we should be reminded that some parts of mathematics include operations and concepts involving ambiguities that admit reasonable disagreements. These are not merely mistakes, but instead, plausible alternative directions that mathematics has previously taken and still might take. After all, other operations that seemed impossible for centuries, such as subtracting a greater number from a lesser, or taking roots of negative numbers, are now common. In mathematics, sometimes the impossible becomes possible, often with good reason.
Want to know more about negative numbers?
Alberto A. Martinez, Negative Math: How Mathematical Rules Can Be Positively Bent
Posted April 12, 2011
More Blog, Discover
https://notevenpast.org/dividing-nothing/
[非表示]
ビジュアルエディターがウィキペディア日本語版にも導入されました（詳細）。
リーマン球面
Question book-4.svg
この記事は検証可能な参考文献や出典が全く示されていないか、不十分です。
出典を追加して記事の信頼性向上にご協力ください。（2015年9月）
リーマン球面は、複素平面で包んだ球面（ある形式の立体射影による ― 詳細は下記参照）として視覚化できる。
数学においてリーマン球面（リーマンきゅうめん、英語: Riemann sphere）は、無限遠点を一点追加して複素平面を拡張する一手法であり、ここに無限遠点
1/0 = ∞
は、少なくともある意味で整合的かつ有用である。 19 世紀の数学者ベルンハルト・リーマンから名付けられた。 これはまた、以下の通りにも呼ばれる。
複素射影直線と言い、CP1 と書く。
拡張複素平面と言い、 {\displaystyle \mathbb {\hat {C}} } または C ∪ {∞} と書く。
純代数的には、無限遠点を追加した複素数全体は、拡張複素数として知られる数体系を構成する。無限を伴う算術は、通常の代数規則すべてに従う訳ではないので、拡張複素数全体は体を構成しない。しかしリーマン球面は、幾何学的また解析学的に無限遠においてさえもよく振舞い、リーマン面とも呼ばれる 1-次元複素多様体をなす。
複素解析において、リーマン球面は有理型関数の洗練された理論で重要な役割を果たす。 リーマン球面は、射影幾何学や代数幾何学では、複素多様体、射影空間、代数多様体の根源的な事例として常に登場する。 リーマン球面はまた、量子力学その他の物理学の分野等、解析学と幾何学に依存する他の学問分野においても、有用性を発揮している。
拡張複素数[編集]
拡張複素数 (extended complex numbers) は複素数 C と ∞ からなる。拡張複素数の集合は C ∪ {∞} と書け、しばしば文字 C に追加の装飾を施して表記される。例えば
{\displaystyle {\hat {\mathbf {C} }},\quad {\overline {\mathbf {C} }},\quad {\text{or}}\quad \mathbf {C} _{\infty }.}
幾何学的には、拡張複素数の集合はリーマン球面 (Riemann sphere) （あるいは拡張複素平面 (extended complex plane)）と呼ばれる。
演算[編集]
複素数の加法は任意の複素数 z に対して
{\displaystyle z+\infty =\infty }
と定義することで拡張され、乗法は任意の 0 でない複素数 z に対して
{\displaystyle z\cdot \infty =\infty }
とし、∞ ⋅ ∞ = ∞ と定義することで拡張される。∞ + ∞, ∞ – ∞, 0 ⋅ ∞ は未定義のままであることに注意せよ。複素数とは違って、拡張複素数は体をなさない。∞ は乗法逆元をもたないからだ。それでもなお、C ∪ {∞} 上の除法を次のように定義するのが習慣である。0 でないすべての複素数 z に対して
{\displaystyle z/0=\infty \quad {\text{and}}\quad z/\infty =0,}
∞/0 = ∞ そして 0/∞ = 0。商 0/0 および ∞/∞ は定義されないままである。
有理関数[編集]
任意の有理関数 f(z) = g(z)/h(z) （言い換えると、f(z) は複素係数の z の多項式関数 g(z) と h(z) であって共通因子をもたないようなものの比である）はリーマン球面上の連続関数に拡張できる。具体的には、 {\displaystyle z_{0}} が分母 {\displaystyle h(z_{0})} が 0 だが分子 {\displaystyle g(z_{0})} が 0 でないような複素数であれば、 {\displaystyle f(z_{0})} は ∞ と定義できる。さらに、f(∞) は f(z) の z → ∞ における極限として定義できる。これは有限かもしれないし無限かもしれない。
複素有理関数全体の集合は、その数学的記号は C(z) であるが、リーマン球面をリーマン面と見たときに、すべての点で値 ∞ をとる定数関数を除いて、リーマン球面からそれ自身へのあらゆる正則関数をなす。C(z) の関数たちは代数体をなし、球面上の有理関数体 (the field of rational functions on the sphere) として知られている。
例えば、関数
{\displaystyle f(z)={\frac {6z^{2}+1}{2z^{2}-50}}}
が与えられると、z = 5 で分母が 0 なので f(5) = ∞ と定義でき、z → ∞ のとき f(z) → 3 なので f(∞) = 3 と定義できる。これらの定義を用いて、f はリーマン球面からそれ自身への連続関数になる。
https://ja.wikipedia.org/wiki/%E3%83%AA%E3%83%BC%E3%83%9E%E3%83%B3%E7%90%83%E9%9D%A2
アリストテレスは空虚な空間（真空）の存在を否定する。第1の理由は、月より下の世界における物体の運動の観察による。月下の物体は重さ、軽さという物体の本性に起因する上昇と下降の運動を成そうとするが、同時に力によって強制的に運動させられる。空気はその本性から軽くもあり、重くもあるので、空気はその力に押されて運動のきっかけを得ると、上昇や下降の運動を起こし、物体に運動を伝える。第2の理由は、物体の相互変化の観察からである。すべての物体は他の物体から生まれる。あらゆる物体に無からの生成はありえない。もし、生成するものがあるとするならば、生成してくるものの占めるべき空虚な空間が、予めどこかに存在しなくてはならないことになる。アリストテレス「天体論 第3巻 月下の物体について」参照。
Dividing by Nothing
by Alberto Martinez
imageIt is well known that you cannot divide a number by zero. Math teachers write, for example, 24 ÷ 0 = undefined. They use analogies to convince students that it is impossible and meaningless, that “you cannot divide something by nothing.” Yet we also learn that we can multiply by zero, add zero, and subtract zero. And some teachers explain that zero is not really nothing, that it is just a number with definite and distinct properties. So, why not divide by zero? In the past, many mathematicians did.
In 628 CE, the Indian mathematician and astronomer Brahmagupta claimed that “zero divided by a zero is zero.” At around 850 CE, another Indian mathematician, Mahavira, more explicitly argued that any number divided by zero leaves that number unchanged, so then, for example, 24 ÷ 0 = 24. Later, around 1150, the mathematician Bhaskara gave yet another result for such operations. He argued that a quantity divided by zero becomes an infinite quantity.image This idea persisted for centuries, for example, in 1656, the English mathematician John Wallis likewise argued that 24 ÷ 0 = ∞, introducing this curvy symbol for infinity. Wallis wrote that for ever smaller values of n, the quotient 24 ÷ n becomes increasingly larger (e.g., 24 ÷ .001 = 24,000), and therefore he argued that it becomes infinity when we divide by zero.
The common attitude toward such old notions is that past mathematicians were plainly wrong, confused or “struggling” with division by zero. But that attitude disregards the extent to which even formidably skilled mathematicians thoughtfully held such notions. In particular, the Swiss mathematician, Leonhard Euler, is widely admired as one of the greatest mathematicians in history, having made extraordinary contributions to many branches of mathematics, physics, and astronomy in hundreds of masterful papers written even during his years of blindness. Euler’s Complete Introduction to Algebra (published in German in 1770) has been praised as the most widely published book in the history of algebra. We here include the pages, in German and from an English translation, in which Euler discussed division by zero. He argued that it gives infinity.
Euler_German
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
Euler_Algebra_1810
Title page of an English edition of Euler’s Algebra, published in 1810 (translated from a French translation of the German original), and Article 83, p. 34, with footnote additions.
A common and reasonable view is that despite his fame, Euler was clearly wrong because if any number divided by zero gives infinity, then all numbers are equal, which is ridiculous. For example:
if 3 ÷ 0 = ∞, and 4 ÷ 0 = ∞,
then ∞ x 0 = 3, and ∞ x 0 = 4.
Here, a single operation, ∞ x 0, has multiple solutions, such that apparently 3 = 4. This is absurd, so one might imagine that there was something “pre-modern” in Euler’s Algebra, that the history of mathematics includes prolonged periods in which mathematicians had not yet found the right answer to certain problems. However, Euler had fair reasons for his arguments. Multiple solutions to one equation did not seem impossible. Euler argued, for one, that the operation of extracting roots yields multiple results. For example, the number 1 has three cube roots, any of which, multiplied by itself three times, produces 1. Today all mathematicians agree that root extraction can yield multiple results.
So why not also admit multiple results when multiplying zero by infinity?
An alternate way to understand the historical disagreements over division by zero is to say that certain mathematical operations evolve over time. In antiquity, mathematicians did not divide by zero. Later, some mathematicians divided by zero, obtaining either zero or the dividend (e.g., that 24 ÷ 0 = 24). Next, other mathematicians argued, for centuries, that the correct quotient is actually infinity. And nowadays again, mathematicians teach that division by zero is impossible, that it is “undefined.” But ever since the mid-1800s, algebraists realized that certain aspects of mathematics are established by convention, by definitions that are established at will and occasionally refined, or redefined. If so, might the result of division by zero change yet again?
In 2005, I showed students at UT how the computer in our classroom, an Apple iMac, carried out division. I typed 24 ÷ 0, then the enter key. The computer replied “infinity!” Strange that it included an exclamation mark. Some students complained that the computer was an Apple instead of a Windows PC. The following year, a similar Apple computer, a newer model, also answered “infinity,” but without the exclamation mark. In 2010, the same operation on a newer computer in the classroom replied: “DIV BY ZERO.” Yet that same computer has an additional calculator, a so-called scientific calculator, and the same operation on this more sophisticated calculator gave “infinity.” image image
Students’ calculators, such as in their cell phones, gave other results: “error,” or “undefined.” One student’s calculator, a Droid cell phone, answered: “infinity.” None of these answers is an accident, each has been thoughtfully programmed into each calculator by mathematically trained programmers and engineers. Computer scientists confront the basic and old algebraic problem: using variables and arithmetical operations, occasionally computers encounter a division in which the divisor has a value of zero—what should computers do then? Stop, break down?
On 21 September 1997, the USS Yorktown battleship was testing “Smart Ship” technologies on the coast of Cape Charles, Virginia. At one point, a crew member entered a set of data that mistakenly included a zero in one field, causing a Windows NT computer program to divide by zero. This generated an error that crashed the computer network, causing failure of the ship’s propulsion system, paralyzing the cruiser for more than a day.
These issues show that we are unjustified in assuming that we are lucky enough to live in an age when all the basic operations of mathematics have been settled, when the result of division by zero in particular cannot change again. Instead, when we look at pages from old math books, such as Euler’s Algebra, we should be reminded that some parts of mathematics include operations and concepts involving ambiguities that admit reasonable disagreements. These are not merely mistakes, but instead, plausible alternative directions that mathematics has previously taken and still might take. After all, other operations that seemed impossible for centuries, such as subtracting a greater number from a lesser, or taking roots of negative numbers, are now common. In mathematics, sometimes the impossible becomes possible, often with good reason.
Want to know more about negative numbers?
Alberto A. Martinez, Negative Math: How Mathematical Rules Can Be Positively Bent
Posted April 12, 2011
More Blog, Discover
https://notevenpast.org/dividing-nothing/
[非表示]
ビジュアルエディターがウィキペディア日本語版にも導入されました（詳細）。
リーマン球面
Question book-4.svg
この記事は検証可能な参考文献や出典が全く示されていないか、不十分です。
出典を追加して記事の信頼性向上にご協力ください。（2015年9月）
リーマン球面は、複素平面で包んだ球面（ある形式の立体射影による ― 詳細は下記参照）として視覚化できる。
数学においてリーマン球面（リーマンきゅうめん、英語: Riemann sphere）は、無限遠点を一点追加して複素平面を拡張する一手法であり、ここに無限遠点
1/0 = ∞
は、少なくともある意味で整合的かつ有用である。 19 世紀の数学者ベルンハルト・リーマンから名付けられた。 これはまた、以下の通りにも呼ばれる。
複素射影直線と言い、CP1 と書く。
拡張複素平面と言い、 {\displaystyle \mathbb {\hat {C}} } または C ∪ {∞} と書く。
純代数的には、無限遠点を追加した複素数全体は、拡張複素数として知られる数体系を構成する。無限を伴う算術は、通常の代数規則すべてに従う訳ではないので、拡張複素数全体は体を構成しない。しかしリーマン球面は、幾何学的また解析学的に無限遠においてさえもよく振舞い、リーマン面とも呼ばれる 1-次元複素多様体をなす。
複素解析において、リーマン球面は有理型関数の洗練された理論で重要な役割を果たす。 リーマン球面は、射影幾何学や代数幾何学では、複素多様体、射影空間、代数多様体の根源的な事例として常に登場する。 リーマン球面はまた、量子力学その他の物理学の分野等、解析学と幾何学に依存する他の学問分野においても、有用性を発揮している。
拡張複素数[編集]
拡張複素数 (extended complex numbers) は複素数 C と ∞ からなる。拡張複素数の集合は C ∪ {∞} と書け、しばしば文字 C に追加の装飾を施して表記される。例えば
{\displaystyle {\hat {\mathbf {C} }},\quad {\overline {\mathbf {C} }},\quad {\text{or}}\quad \mathbf {C} _{\infty }.}
幾何学的には、拡張複素数の集合はリーマン球面 (Riemann sphere) （あるいは拡張複素平面 (extended complex plane)）と呼ばれる。
演算[編集]
複素数の加法は任意の複素数 z に対して
{\displaystyle z+\infty =\infty }
と定義することで拡張され、乗法は任意の 0 でない複素数 z に対して
{\displaystyle z\cdot \infty =\infty }
とし、∞ ⋅ ∞ = ∞ と定義することで拡張される。∞ + ∞, ∞ – ∞, 0 ⋅ ∞ は未定義のままであることに注意せよ。複素数とは違って、拡張複素数は体をなさない。∞ は乗法逆元をもたないからだ。それでもなお、C ∪ {∞} 上の除法を次のように定義するのが習慣である。0 でないすべての複素数 z に対して
{\displaystyle z/0=\infty \quad {\text{and}}\quad z/\infty =0,}
∞/0 = ∞ そして 0/∞ = 0。商 0/0 および ∞/∞ は定義されないままである。
有理関数[編集]
任意の有理関数 f(z) = g(z)/h(z) （言い換えると、f(z) は複素係数の z の多項式関数 g(z) と h(z) であって共通因子をもたないようなものの比である）はリーマン球面上の連続関数に拡張できる。具体的には、 {\displaystyle z_{0}} が分母 {\displaystyle h(z_{0})} が 0 だが分子 {\displaystyle g(z_{0})} が 0 でないような複素数であれば、 {\displaystyle f(z_{0})} は ∞ と定義できる。さらに、f(∞) は f(z) の z → ∞ における極限として定義できる。これは有限かもしれないし無限かもしれない。
複素有理関数全体の集合は、その数学的記号は C(z) であるが、リーマン球面をリーマン面と見たときに、すべての点で値 ∞ をとる定数関数を除いて、リーマン球面からそれ自身へのあらゆる正則関数をなす。C(z) の関数たちは代数体をなし、球面上の有理関数体 (the field of rational functions on the sphere) として知られている。
例えば、関数
{\displaystyle f(z)={\frac {6z^{2}+1}{2z^{2}-50}}}
が与えられると、z = 5 で分母が 0 なので f(5) = ∞ と定義でき、z → ∞ のとき f(z) → 3 なので f(∞) = 3 と定義できる。これらの定義を用いて、f はリーマン球面からそれ自身への連続関数になる。
https://ja.wikipedia.org/wiki/%E3%83%AA%E3%83%BC%E3%83%9E%E3%83%B3%E7%90%83%E9%9D%A2
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