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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.[1] Since q may be equal to 1, every integer is a rational number. The set of all rational numbers is usually denoted by a boldface Q (or blackboard bold \mathbb{Q}, Unicode ℚ);[2] it was thus denoted in 1895 by Peano after quoziente, Italian for "quotient".
The decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number. These statements hold true not just for base 10, but also for any other integer base (e.g. binary, hexadecimal).
A real number that is not rational is called irrational. Irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating. Since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational.[1]
The rational numbers can be formally defined as the equivalence classes of the quotient set (Z × (Z \ {0})) / ~, where the cartesian product Z × (Z \ {0}) is the set of all ordered pairs (m,n) where m and n are integers, n is not 0 (n ≠ 0), and "~" is the equivalence relation defined by (m1,n1) ~ (m2,n2) if, and only if, m1n2 － m2n1 = 0.
In abstract algebra, the rational numbers together with certain operations of addition and multiplication form the archetypical field of characteristic zero. As such, it is characterized as having no proper subfield or, alternatively, being the field of fractions for the ring of integers. Finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers.[3]
In mathematical analysis, the rational numbers form a dense subset of the real numbers. The real numbers can be constructed from the rational numbers by completion, using Cauchy sequences, Dedekind cuts, or infinite decimals.
Zero divided by any other integer equals zero; therefore, zero is a rational number (but division by zero is undefined).
The term rational in reference to the set Q refers to the fact that a rational number represents a ratio of two integers. In mathematics, the adjective rational often means that the underlying field considered is the field Q of rational numbers. Rational polynomial usually, and most correctly, means a polynomial with rational coefficients, also called a "polynomial over the rationals". However, rational function does not mean the underlying field is the rational numbers, and a rational algebraic curve is not an algebraic curve with rational coefficients.
Arithmetic[edit]
See also: Fraction (mathematics) § Arithmetic with fractions
Embedding of integers[edit]
Any integer n can be expressed as the rational number n/1.
Equality[edit]
\frac{a}{b} = \frac{c}{d} if and only if ad = bc.
Ordering[edit]
Where both denominators are positive:
\frac{a}{b} < \frac{c}{d} if and only if ad < bc.
If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations:
\frac{-a}{-b} = \frac{a}{b}
and
\frac{a}{-b} = \frac{-a}{b}.
Addition[edit]
Two fractions are added as follows:
\frac{a}{b} + \frac{c}{d} = \frac{ad+bc}{bd}.
Subtraction[edit]
\frac{a}{b} - \frac{c}{d} = \frac{ad-bc}{bd}.
Multiplication[edit]
The rule for multiplication is:
\frac{a}{b} \cdot\frac{c}{d} = \frac{ac}{bd}.
Division[edit]
Where c ≠ 0:
\frac{a}{b} \div \frac{c}{d} = \frac{ad}{bc}.
Note that division is equivalent to multiplying by the reciprocal of the divisor fraction:
\frac{ad}{bc} = \frac{a}{b} \times \frac{d}{c}.
Inverse[edit]
Additive and multiplicative inverses exist in the rational numbers:
- \left( \frac{a}{b} \right) = \frac{-a}{b} = \frac{a}{-b} \quad\mbox{and}\quad
\left(\frac{a}{b}\right)^{-1} = \frac{b}{a} \mbox{ if } a \neq 0.
Exponentiation to integer power[edit]
If n is a non-negative integer, then
\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}
and (if a ≠ 0):
\left(\frac{a}{b}\right)^{-n} = \frac{b^n}{a^n}.
Continued fraction representation[edit]
Main article: Continued fraction
A finite continued fraction is an expression such as
a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{ \ddots + \cfrac{1}{a_n} }}},
where an are integers. Every rational number a/b has two closely related expressions as a finite continued fraction, whose coefficients an can be determined by applying the Euclidean algorithm to (a,b).
Other representations[edit]
common fraction: \frac{8}{3}
mixed numeral: 2\frac{2}{3}
repeating decimal using a vinculum: 2.\overline{6}
repeating decimal using parentheses: 2.(6)
continued fraction using traditional typography: 2 + \cfrac{1}{1 + \cfrac{1}{2} }
continued fraction in abbreviated notation: [2; 1, 2]
egyptian fraction: 2 + \frac{1}{2} + \frac{1}{6}
prime power decomposition: \frac{117}{1000} = 2^{-3}.3^2.5^{-3}.13
quote notation: 3!6
are different ways to represent the same rational value.
Formal construction[edit]
A diagram showing a representation of the equivalent classes of pairs of integers
Mathematically we may construct the rational numbers as equivalence classes of ordered pairs of integers (m,n), with n ≠ 0. This space of equivalence classes is the quotient space (Z × (Z \ {0})) / ~, where (m1,n1) ~ (m2,n2) if, and only if, m1n2 － m2n1 = 0. We can define addition and multiplication of these pairs with the following rules:
\left(m_1, n_1\right) + \left(m_2, n_2\right) \equiv \left(m_1n_2 + n_1m_2, n_1n_2\right)
\left(m_1, n_1\right) \times \left(m_2, n_2\right) \equiv \left(m_1m_2, n_1n_2\right)
and, if m2 ≠ 0, division by
\frac{\left(m_1, n_1\right)} {\left(m_2, n_2\right)} \equiv \left(m_1n_2, n_1m_2\right).
The equivalence relation (m1,n1) ~ (m2,n2) if, and only if, m1n2 － m2n1 = 0 is a congruence relation, i.e. it is compatible with the addition and multiplication defined above, and we may define Q to be the quotient set (Z × (Z \ {0})) / ~, i.e. we identify two pairs (m1,n1) and (m2,n2) if they are equivalent in the above sense. (This construction can be carried out in any integral domain: see field of fractions.) We denote by [(m1,n1)] the equivalence class containing (m1,n1). If (m1,n1) ~ (m2,n2) then, by definition, (m1,n1) belongs to [(m2,n2)] and (m2,n2) belongs to [(m1,n1)]; in this case we can write [(m1,n1)] = [(m2,n2)]. Given any equivalence class [(m,n)] there are a countably infinite number of representation, since
\cdots = [(-2m,-2n)] = [(-m,-n)] = [(m,n)] = [(2m,2n)] = \cdots.
The canonical choice for [(m,n)] is chosen so that n is positive and gcd(m,n) = 1, i.e. m and n share no common factors, i.e. m and n are coprime. For example, we would write [(1,2)] instead of [(2,4)] or [(－12,－24)], even though [(1,2)] = [(2,4)] = [(－12,－24)].
We can also define a total order on Q. Let ∧ be the and-symbol and ∨ be the or-symbol. We say that [(m1,n1)] ≤ [(m2,n2)] if:
(n_1n_2 > 0 \ \and \ m_1n_2 \le n_1m_2) \ \or \ (n_1n_2 < 0 \ \and \ m_1n_2 \ge n_1m_2).
The integers may be considered to be rational numbers by the embedding that maps m to [(m,1)].
Properties[edit]
A diagram illustrating the countability of the rationals
The set Q, together with the addition and multiplication operations shown above, forms a field, the field of fractions of the integers Z.
The rationals are the smallest field with characteristic zero: every other field of characteristic zero contains a copy of Q. The rational numbers are therefore the prime field for characteristic zero.
The algebraic closure of Q, i.e. the field of roots of rational polynomials, is the algebraic numbers.
The set of all rational numbers is countable. Since the set of all real numbers is uncountable, we say that almost all real numbers are irrational, in the sense of Lebesgue measure, i.e. the set of rational numbers is a null set.
The rationals are a densely ordered set: between any two rationals, there sits another one, and, therefore, infinitely many other ones. For example, for any two fractions such that
\frac{a}{b} < \frac{c}{d}
(where b,d are positive), we have
\frac{a}{b} < \frac{ad + bc}{2bd} < \frac{c}{d}.
Any totally ordered set which is countable, dense (in the above sense), and has no least or greatest element is order isomorphic to the rational numbers.
Real numbers and topological properties[edit]
The rationals are a dense subset of the real numbers: every real number has rational numbers arbitrarily close to it. A related property is that rational numbers are the only numbers with finite expansions as regular continued fractions.
By virtue of their order, the rationals carry an order topology. The rational numbers, as a subspace of the real numbers, also carry a subspace topology. The rational numbers form a metric space by using the absolute difference metric d(x,y) = |x － y|, and this yields a third topology on Q. All three topologies coincide and turn the rationals into a topological field. The rational numbers are an important example of a space which is not locally compact. The rationals are characterized topologically as the unique countable metrizable space without isolated points. The space is also totally disconnected. The rational numbers do not form a complete metric space; the real numbers are the completion of Q under the metric d(x,y) = |x － y|, above.
p-adic numbers[edit]
See also: p-adic Number
In addition to the absolute value metric mentioned above, there are other metrics which turn Q into a topological field:
Let p be a prime number and for any non-zero integer a, let |a|p = p－n, where pn is the highest power of p dividing a.
In addition set |0|p = 0. For any rational number a/b, we set |a/b|p = |a|p / |b|p.
Then dp(x,y) = |x － y|p defines a metric on Q.
The metric space (Q,dp) is not complete, and its completion is the p-adic number field Qp. Ostrowski's theorem states that any non-trivial absolute value on the rational numbers Q is equivalent to either the usual real absolute value or a p-adic absolute value.https://en.wikipedia.org/wiki/Rational_number
Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics
\documentclass[12pt]{article}
\usepackage{latexsym,amsmath,amssymb,amsfonts,amstext,amsthm}
\numberwithin{equation}{section}
\begin{document}
\title{\bf Announcement 179: Division by zero is clear as z/0=0 and it is fundamental in mathematics\\
}
\author{{\it Institute of Reproducing Kernels}\\
\date{\today}
\maketitle
{\bf Abstract: } In this announcement, we shall introduce the zero division $z/0=0$. The result is a definite one and it is fundamental in mathematics.
\bigskip
\section{Introduction}
%\label{sect1}
By a natural extension of the fractions
\begin{equation}
\frac{b}{a}
\end{equation}
for any complex numbers $a$ and $b$, we, recently, found the surprising result, for any complex number $b$
\begin{equation}
\frac{b}{0}=0,
\end{equation}
incidentally in \cite{s} by the Tikhonov regularization for the Hadamard product inversions for matrices, and we discussed their properties and gave several physical interpretations on the general fractions in \cite{kmsy} for the case of real numbers. The result is a very special case for general fractional functions in \cite{cs}.
The division by zero has a long and mysterious story over the world (see, for example, google site with division by zero) with its physical viewpoints since the document of zero in India on AD 628, however,
Sin-Ei, Takahasi (\cite{taka}) (see also \cite{kmsy}) established a simple and decisive interpretation (1.2) by analyzing some full extensions of fractions and by showing the complete characterization for the property (1.2). His result will show that our mathematics says that the result (1.2) should be accepted as a natural one:
\bigskip
{\bf Proposition. }{\it Let F be a function from ${\bf C }\times {\bf C }$ to ${\bf C }$ such that
$$
F (b, a)F (c, d)= F (bc, ad)
$$
for all
$$
a, b, c, d \in {\bf C }
$$
and
$$
F (b, a) = \frac {b}{a }, \quad a, b \in {\bf C }, a \ne 0.
$$
Then, we obtain, for any $b \in {\bf C } $
$$
F (b, 0) = 0.
$$
}
\medskip
\section{What are the fractions $ b/a$?}
For many mathematicians, the division $b/a$ will be considered as the inverse of product;
that is, the fraction
\begin{equation}
\frac{b}{a}
\end{equation}
is defined as the solution of the equation
\begin{equation}
a\cdot x= b.
\end{equation}
The idea and the equation (2.2) show that the division by zero is impossible, with a strong conclusion. Meanwhile, the problem has been a long and old question:
As a typical example of the division by zero, we shall recall the fundamental law by Newton:
\begin{equation}
F = G \frac{m_1 m_2}{r^2}
\end{equation}
for two masses $m_1, m_2$ with a distance $r$ and for a constant $G$. Of course,
\begin{equation}
\lim_{r \to +0} F =\infty,
\end{equation}
however, in our fraction
\begin{equation}
F = G \frac{m_1 m_2}{0} = 0.
\end{equation}
\medskip
Now, we shall introduce an another approach. The division $b/a$ may be defined {\bf independently of the product}. Indeed, in Japan, the division $b/a$ ; $b$ {\bf raru} $a$ ({\bf jozan}) is defined as how many $a$ exists in $b$, this idea comes from subtraction $a$ repeatedly. (Meanwhile, product comes from addition).
In Japanese language for "division", there exists such a concept independently of product.
H. Michiwaki and his 6 years old girl said for the result $ 100/0=0$ that the result is clear, from the meaning of the fractions independently the concept of product and they said:
$100/0=0$ does not mean that $100= 0 \times 0$. Meanwhile, many mathematicians had a confusion for the result.
Her understanding is reasonable and may be acceptable:
$100/2=50 \quad$ will mean that we divide 100 by 2, then each will have 50.
$100/10=10 \quad$ will mean that we divide 100 by10, then each will have 10.
$100/0=0 \quad$ will mean that we do not divide 100, and then nobody will have at all and so 0.
Furthermore, she said then the rest is 100; that is, mathematically;
$$
100 = 0\cdot 0 + 100.
$$
Now, all the mathematicians may accept the division by zero $100/0=0$ with natural feelings as a trivial one?
\medskip
For simplicity, we shall consider the numbers on non-negative real numbers. We wish to define the division (or fraction) $b/a$ following the usual procedure for its calculation, however, we have to take care for the division by zero:
The first principle, for example, for $100/2 $ we shall consider it as follows:
$$
100-2-2-2-,...,-2.
$$
How may times can we subtract $2$? At this case, it is 50 times and so, the fraction is $50$.
The second case, for example, for $3/2$ we shall consider it as follows:
$$
3 - 2 = 1
$$
and the rest (remainder) is $1$, and for the rest $1$, we multiple $10$,
then we consider similarly as follows:
$$
10-2-2-2-2-2=0.
$$
Therefore $10/2=5$ and so we define as follows:
$$
\frac{3}{2} =1 + 0.5 = 1.5.
$$
By these procedures, for $a \ne 0$ we can define the fraction $b/a$, usually. Here we do not need the concept of product. Except the zero division, all the results for fractions are valid and accepted.
Now, we shall consider the zero division, for example, $100/0$. Since
$$
100 - 0 = 100,
$$
that is, by the subtraction $100 - 0$, 100 does not decrease, so we can not say we subtract any from $100$. Therefore, the subtract number should be understood as zero; that is,
$$
\frac{100}{0} = 0.
$$
We can understand this: the division by $0$ means that it does not divide $100$ and so, the result is $0$.
Similarly, we can see that
$$
\frac{0}{0} =0.
$$
As a conclusion, we should define the zero divison as, for any $b$
$$
\frac{b}{0} =0.
$$
See \cite{kmsy} for the details.
\medskip
\section{In complex analysis}
We thus should consider, for any complex number $b$, as (1.2);
that is, for the mapping
\begin{equation}
w = \frac{1}{z},
\end{equation}
the image of $z=0$ is $w=0$. This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere.
However, we shall recall the elementary function
\begin{equation}
W(z) = \exp \frac{1}{z}
\end{equation}
$$
= 1 + \frac{1}{1! z} + \frac{1}{2! z^2} + \frac{1}{3! z^3} + \cdot \cdot \cdot .
$$
The function has an essential singularity around the origin. When we consider (1.2), meanwhile, surprisingly enough, we have:
\begin{equation}
W(0) = 1.
\end{equation}
{\bf The point at infinity is not a number} and so we will not be able to consider the function (3.2) at the zero point $z = 0$, meanwhile, we can consider the value $1$ as in (3.3) at the zero point $z = 0$. How do we consider these situations?
In the famous standard textbook on Complex Analysis, L. V. Ahlfors (\cite{ahlfors}) introduced the point at infinity as a number and the Riemann sphere model as well known, however, our interpretation will be suitable as a number. We will not be able to accept the point at infinity as a number.
As a typical result, we can derive the surprising result: {\it At an isolated singular point of an analytic function, it takes a definite value }{\bf with a natural meaning.} As the important applications for this result, the extension formula of functions with analytic parameters may be obtained and singular integrals may be interpretated with the division by zero, naturally (\cite{msty}).
\bigskip
\section{Conclusion}
The division by zero $b/0=0$ is possible and the result is naturally determined, uniquely.
The result does not contradict with the present mathematics - however, in complex analysis, we need only to change a little presentation for the pole; not essentially, because we did not consider the division by zero, essentially.
The common understanding that the division by zero is impossible should be changed with many text books and mathematical science books. The definition of the fractions may be introduced by {\it the method of Michiwaki} in the elementary school, even.
Should we teach the beautiful fact, widely?:
For the elementary graph of the fundamental function
$$
y = f(x) = \frac{1}{x},
$$
$$
f(0) = 0.
$$
The result is applicable widely and will give a new understanding for the universe ({\bf Announcement 166}).
\medskip
If the division by zero $b/0=0$ is not introduced, then it seems that mathematics is incomplete in a sense, and by the intoduction of the division by zero, mathematics will become complete in a sense and perfectly beautiful.
\bigskip
section{Remarks}
For the procedure of the developing of the division by zero and for some general ideas on the division by zero, we presented the following announcements in Japanese:
\medskip
{\bf Announcement 148} (2014.2.12): $100/0=0, 0/0=0$ -- by a natural extension of fractions -- A wish of the God
\medskip
{\bf Announcement 154} (2014.4.22): A new world: division by zero, a curious world, a new idea
\medskip
{\bf Announcement 157} (2014.5.8): We wish to know the idea of the God for the division by zero; why the infinity and zero point are coincident?
\medskip
{\bf Announcement 161} (2014.5.30): Learning from the division by zero, sprits of mathematics and of looking for the truth
\medskip
{\bf Announcement 163} (2014.6.17): The division by zero, an extremely pleasant mathematics - shall we look for the pleasant division by zero: a proposal for a fun club looking for the division by zero.
\medskip
{\bf Announcement 166} (2014.6.29): New general ideas for the universe from the viewpoint of the division by zero
\medskip
{\bf Announcement 171} (2014.7.30): The meanings of product and division -- The division by zero is trivial from the own sense of the division independently of the concept of product
\medskip
{\bf Announcement 176} (2014.8.9): Should be changed the education of the division by zero
\bigskip
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{cs}
L. P. Castro and S.Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{kmsy}
S. Koshiba, H. Michiwaki, S. Saitoh and M. Yamane,
An interpretation of the division by zero z/0=0 without the concept of product
(note).
\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. Vol. 27, No 2 (2014), pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{msty}
H. Michiwaki, S. Saitoh, M. Takagi and M. Yamada,
A new concept for the point at infinity and the division by zero z/0=0
(note).
\bibitem{s}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. Vol.4 No.2 (2014), 87-95. http://www.scirp.org/journal/ALAMT/
\bibitem{taka}
S.-E. Takahasi,
{On the identities $100/0=0$ and $ 0/0=0$}
(note).
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operators on the real and complex fields. (submitted)
\end{thebibliography}
\end{document}
アインシュタインも解決できなかった「ゼロで割る」問題
http://matome.naver.jp/odai/2135710882669605901
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.
https://notevenpast.org/dividing-nothing/
私は数学を信じない。 アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。
1423793753.460.341866474681。
Einstein's Only Mistake: Division by Zero
http://refully.blogspot.jp/2012/05/einsteins-only-mistake-division-by-zero.html
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