# 【巧学数学】数学难学5大原因揭秘！数学成绩不在是软肋！

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【巧学数学】数学难学5大原因揭秘！数学成绩不在是软肋！

我们都知道，数学这类学科单靠记忆公式、定理是几乎没有什么用的，那么，数学就真的一点点方法没有了吗？我们就应该放弃吗？并不是，很多孩子的数学成绩都很不错，人家是怎么学好的呢？下面我就讲一下大部分学生在数学上的几个普遍的问题。

1、上课听懂了，下课不会写
提出这个问题的同学首先要好好反思一下，自己是真的学懂了吗？我们的学习首先要求理解，然后总结，最后做到举一反三。上课的时候觉得自己听懂了，但是一到下课，自己独立完成作业的时候又完全不会了，其实这是因为我们根本没有掌握好老师讲的内容，学得似懂非懂！对基础概念没有完全理解。
在学习中，知识点有难有易，在学习相对难懂的知识点时，数学老师都会在课前稍稍提醒“下面我们要讲的有点难，有些同学直到毕业可能都还不会，大家要认真听讲”，这个时候就需要我们打起十二万分的精神了。要是课后发现自己还是不会做题，身边又没有老师和同学的情况下，我们要怎么独立解决呢？首先回忆老师讲课的做题思路，自己整理一下。然后把例题拿出来看一遍，确认自己能够独立解答之后合上课本再验算一下，并对这种类型题的解题思路加以思考。要是仍然不会的话，一定要及时向老师和同学求救，务必解决问题。
2、老师讲的很快，没有办法做课堂笔记
首先，作为学生，我们不应该责怪老师讲课速度太快，一般来说，有一定教学经验的老师都会有适合自己学生的教学进度。不然，为什么课堂上几十个学生，别人都能很好地跟上老师的思维，而自己却不行呢？若是大部分学生都跟不上，一定会有人向老师提出这个问题，老师也会做相应的改进，所以问题还是出在我们自己身上。
跟不上老师的思维和教学进度的同学大部分应该都是没有预习的，或者是预习做得不好，导致上课的学习抓不住重点，不知道什么该记什么不该记，什么该详记什么该简记。所以说，我们平时上课一定要提前做好预习工作，这是一个老生长谈的问题了。预习的目的是为了上课作铺垫，预习做得好，对于上课的知识点掌握是事半功倍的。上课做笔记时只需要记思路，先大概记录一下，下课之后马上补充，满堂记录的话，就得不偿失了。

3、现在做题还是用以前的做题思维
每一个阶段的知识以及学习环境是不一样的，很多同学根本没有把这种差别搞明白。我们进入高一个阶段的时候，我们要注意学习方法的转换。比如初一就是比较基础的，大部分的知识、概念、定理、规律老师和书本上都已经总结好了，但是到了初二、初三，甚至上了高中就不一样了，有些题目的解题思路和技巧，需要我们自己在平时做题的过程中总结。当我们遇到这种类型题时，一定要及时记录，并在一定时间里，将这类题目进行思路总结。
4、几何就是听天书
很多孩子上几何课就发懵，大部分都是女同学。的确，男女的思维方式天生就不同，女生的逻辑思维能力和空间想象力就是没有男生强，这就表示女同学不能学好几何吗？答案当然是否定的。许多孩子一看到“几何”这两个字就头疼，这是因为知识点是串联的，当我们有一个知识点没弄明白，很有可能相当大一部分知识都弄不明白了。其实这是学习方法不对，而且我们对学好这门学科的信心不足。
立体几何一定要注意数形结合，不要一味地为了做题而做题。此外，数学中有很多思想方法，比如数形结合思想、化归思想、分类思想等等。对于立体几何，数形结合思想是至关重要的，平时我们做题时一定要注意多画画图，加强图形的熟练程度。养成及时画草图的习惯，另外还要注意立体几何的空间感。

5、怎么做证明题
证明题是必有鹅一类题目，也是很多孩子无从下手的一类题目。
其实证明题就是逻辑推理的过程，这方面的思路我讲一个方——由果到因逆推法：在组织解题思路的时候，由结果出发，然后一步一步往前推，直到推到最基本的原始条件，然后在我们解答，书写答题时，则是由最基本的条件，推到导到结果，逆向思维，这是解答证明题的一个常用方法。
其实数学并不难，而且往往独立完成一道很难的数学题之后那种自豪敢，心理满足感是难以用语言形容的，难的是有一颗学好数学的决心，和一个好的学习方法。http://learning.sohu.com/20151228/n432741703.shtml

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

\frac{b}{a}

for any complex numbers $a$ and $b$, we, recently, found the surprising result, for any complex number $b$

\frac{b}{0}=0,

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

\frac{b}{a}

is defined as the solution of the equation

a\cdot x= b.

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:

F = G \frac{m_1 m_2}{r^2}

for two masses $m_1, m_2$ with a distance $r$ and for a constant $G$. Of course,

\lim_{r \to +0} F =\infty,

however, in our fraction

F = G \frac{m_1 m_2}{0} = 0.

\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

w = \frac{1}{z},

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

W(z) = \exp \frac{1}{z}

$$= 1 + \frac{1}{1! z} + \frac{1}{2! z^2} + \frac{1}{3! z^3} + \cdot \cdot \cdot .$$
The function has an essential singularity around the origin. When we consider (1.2), meanwhile, surprisingly enough, we have:

W(0) = 1.

{\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/

1423793753.460.341866474681。

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