First, the center of learning mathematics is the pain center of the human brain. In other words, feeling pain like acupuncture and processing numbers are the same area of the brain. Some people have a headache as soon as they learn math. This leads to people's natural avoidance reaction to mathematics. The more they escape, the harder it is to learn. I have met people who practice calligraphy, learn to dance and write poems in their spare time, but I seldom see people who study mathematics in their spare time.

Second: the field of mathematics is very extensive. Most people don't know where to start.

Third: the symbols of mathematics are chaotic. This is the main point of this paper. It is difficult to learn because of the chaotic mathematical system. When learning mathematics, we must understand what all kinds of confusing symbols mean. If people who have not been exposed to mathematics see those symbols, they will be amazed: what slang is this?

One of the Symbols of Chaos Mathematics: Multiplication and Multiplication

You ask me, how many kinds of multiplication are there in mathematics? I must not know. There seems to be multiplication, dot multiplication, cross multiplication and some convolution. I can only admire the earliest mathematicians for being so lazy. Don't even bother to invent a new operation symbol. Overloading the poor multiplication symbol.

If you remember, in primary school, when you were studying math, you used a fork to multiply symbols, like this? 4 = 12 。 At that time, some teachers were very strict and could not exchange multiplicand and multiplier at will. For example, the above formula is to calculate "unit price 3 yuan, total price of four things". If "the unit price is 4 yuan, three things", it must be written as 4? 3 = 12 。 Teachers are not so strict now.

In junior high school, my teacher suddenly asked me to omit the multiplication sign between numbers and letters, or add a dot between numbers and letters, like 3a or 3? A so. In high school, one day, the physics teacher solemnly recommended point multiplication and cross multiplication. From then on, the world of multiplication began to be chaotic. The vectors and scalars in his mouth scared off a wave of math lovers.

I was exposed to matrix multiplication when I was in college. After graduation, I contacted quaternion, only to know that sometimes multiplication really can't exchange multiplicand and multiplier! So, thanks to the first-grade math teacher, she is too prescient.

The multiplication mark was originally generated to make the addition concise. This is the original meaning of multiplication. With the development of history, the multiplication sign is constantly overloaded.

In the original sense, in multiplication, the multiplier must be an integer because the multiplier is used to calculate the same addend. In order to write addition concisely, multiplication was born.

Then there was a split. Later multipliers can be fractions.

Later, the same numbers are multiplied and written compactly as a power.

Later, with the prescription, the situation emerged one after another. As the limit of rational number, irrational number was born.

So the multiplier is logical and can be irrational. The concept of number is expanding, so is multiplication. Multiplication is overloaded. It can be used to multiply not only positive numbers, but also negative numbers. The result of multiplying a negative number by a negative number is a positive number, which was an intuitive and hard rule at that time. Nobody can explain why.

All the above overloads are natural, and there is basically nothing incongruous.

When the number becomes a complex number, there is confusion. And it happened in an instant. There are three kinds of simultaneous multiplication: complex numbers can be multiplied by complex numbers, vectors represented by complex numbers can be multiplied by points, and vectors can also be multiplied by crosses. If it weren't for me, I would faint instantly.

What the hell happened? Sometimes a dot, sometimes a cross, sometimes nothing, actually has three different meanings? When representing a vector, add an arrow to the header; When indicating the yoke of * * *, add a horizontal line to the head; I have no problem with using absolute value symbols to represent the modulus of complex numbers, but you can add a horizontal line to your head. Didn't you say so-so before? How did the X-pull become the shackles of Z? Negative propositions are also underlined. Complement is also a horizontal line on the head. Why is it so popular to put a horizontal line on your head?

Needless to say, the word "* * * yoke" scared away a bunch of people again.

Multiplication is confusing because mathematicians are stubborn about infix expressions. Since it has a function, everyone can speak completely. If you write like this, like lisp's S expression:

(mul a b)

(cross a b)

(point a b)

Isn't it a good difference?

All the symbols are written in the front and replaced by an easy-to-understand function name.

Addition can be written as

(plus a b)

Even in Chinese.

(plus a and b) are easy to understand.

If English is enough, why use Greek letters?

If you have to use it

(π ab) can also mean multiplication.

(∑ a b) can also mean addition.

On the one hand, mathematics is becoming more and more abstract; On the one hand, the writing style is becoming more and more compact. Mathematical symbols are improvised by mathematicians, and only Leibniz will consider them carefully. You must have seen the ∑ symbol when the head and feet are full of things. This is called compactness. The advantage of compactness is that for familiar people, it can be seen at a glance that the whole formula is a repetition of a certain pattern; The disadvantage of compactness is that people who haven't studied look like a mess.

Due to the abstraction and compactness, in the process of learning mathematics:

If you come across a strange symbol, it will have a profound meaning, such as the strange cursive letters in topology. You must know the meaning of each layer before you can know the meaning of this letter.

If you come across a seemingly ordinary symbol, it may have a different meaning from the past, such as the point multiplication mentioned just now;

For a symbol, you must contact the context to know its meaning. For example, this symbol: sometimes used to represent a special multiplication, sometimes used to transpose a matrix, sometimes used to represent the operation of an exponential function, sometimes used to represent bitwise XOR, sometimes used to represent "AND", sometimes used to represent the Ctrl key, saying Lambda λ is too small, saying it is a hat, and it is often not written on the top of the head. How to read it depends on the situation at that time. There are too many mathematical symbols and the keyboard is too small. If you don't know how to write, use this ultra-small one.

Mathematics is inherently difficult, and there are many such places, which are easy to cause misunderstanding. Sometimes, all four corners of a letter are written with numbers, and then each has a different meaning, which is more difficult to use than the four-corner numbers in the dictionary.

The dots written between the letters high and low have different meanings.

You can add a pointed cap, or an arc cap, or a dot, or two dots, or a small circle at the top of the letter; Anything you can see can be added to the upper right corner of the letter, a two or three apostrophes, a number with brackets, a number without brackets, and even a formula as long as an article. The symbol of absolute value can be played to n layers.

Summary: The reason why mathematics is difficult to learn is that there are too many slang words.