Countless zeros add up to not zeros

About infinitesimal quantities and zeros:

Infinitesimal quantities are denoted by the symbol o. An infinitesimal quantity is a non-zero number, and its absolute value can be smaller than any real number, but greater than zero. An infinitesimal quantity is not equivalent to zero. In the nature of operations, real numbers are multiplied by zero to give zero, but real numbers are multiplied by infinitesimal quantities to give infinitesimal quantities.

The space occupied by a point is not zero (zero means it does not exist), but an infinitesimal amount. The addition of infinitesimal quantities needs to take into account the order of infinitesimal quantities.

What you call an "area of points" is an infinitesimal quantity, and the number of "innumerable" points is an infinitesimal number. Infinity multiplied by infinitesimal is infinitive, which can be converted into a ratio of two infinitesimal quantities (the reciprocal of an infinitesimal quantity is an infinitesimal quantity), and then use l'hospital rule and other methods. However, since there is no definite expression for the geometry problems to be discussed below, their orders cannot be explained directly, so multiplication is not used.

Strictly arguing that "the area of a line without a surface" is "large", it is necessary to use the cardinal in set theory. This can be explained simply with a one-to-one correspondence (provided that you admit that any two points occupy equal space, i.e. they are infinitesimal quantities of the same order). If two parallel lines are made in a plane, then for any point on one of them, there must be a corresponding point in the other parallel line that causes the two points to pass through a perpendicular line of the parallel line.

In this way, the points in the two lines correspond one-to-one. The space occupied by a point or line can be expressed as a non-zero positive number, so the sum of the space occupied by two straight lines is greater than the space occupied by any one of the lines. Extending two straight lines to n and filling the entire plane (i.e., n) also results in the space occupied by these lines being greater than the space occupied by any of the straight lines, which proves the problem.

In fact, we should first understand that "the space occupied by a point is not as large as a straight line", so that it is convenient to understand by analogy. This is a conceptual problem, the surface does not care about the volume, but lz can understand the "volume of the surface (in the three-dimensional vector space)" as an infinitesimal quantity, but it is definitely not zero (zero means it does not exist).

It is equal to the original area. Add according to your supplement: (see *** below).

Upstairs makes sense.

To make it easier for you to understand. Think of it this way.

The blank paper represents 0

Then when it adds a little bit, there is obviously a fundamental difference. It's definitely not right to think that it's 0.

It can only be approximately equal to 0 to some extent. If there are an infinite number of points, the infinite number of points are still regarded as 0, and then they are added. There is a lack of rigorous proof for this. It's not right.

Hence what you say.

It's really hard to think clearly, and it's hard to tell. Hopefully, the following will help you understand.

You can think of it this way, the length of a point is infinitesimal and a line is made up of an infinite number of points, and it has a definite length.

A line can also be thought of as an infinitesimal area. A polygon is made up of an infinite number of such lines. Hence the area can be expressed as a number.

It is equivalent to summing them and taking the limit.

This can be difficult to understand).

None of the above is entirely rigorous. A lot of places are just for you to understand. I hope you will be able to learn calculus well in the future. You can understand it more clearly.

The addition of infinitesimal quantities and the addition of nothing's zeros are two completely different concepts.

It can be said that the area of a point is an infinitesimal quantity, although it is customary to use 0 to represent an infinitesimal quantity, but this 0 is not nothing, but it is smaller than any number you can give. An infinite number of infinitesimal quantities add up to build a very large thing.

The method of infinitesimal analysis is Cauchy's greatest contribution to the theory of calculus, which solves the fundamental problem of calculus. This is a question that you will understand when you study advanced mathematics. You can look at advanced math textbooks, and any advanced math textbook is about infinitesimal analysis before calculus.

The point should have an area, the surface should have a volume, but it is too small to be ignored, draw a big circle on the wall, paint it black, and from a distance, it will be the first point, and it actually has an area. The interview drawn on a book has no volume, but if it is to move to form a volume object, it is in space, in fact, there is volume, this is my personal opinion.

Your so-called point area is infinitesimal in size.

Infinite points are infinity.

Multiplying infinity by infinitesimal has no result because both are variables.

It's like 1 n * n

One tends to 0, one tends to infinity, and the product is exactly 1

Since it's related to calculus, I don't know where you've learned math, so if you're interested, you can take a look.

Your analogy itself is inappropriate, you can't compare a point to a zero, you can't compare a point to an infinite number of zeros, geometry has its own abstract concept, and it doesn't make sense for you to compete with yourself like that.

I'll wait ... Also want to know the answer.

The addition of 0 and any number does not equal 0. This question should be a confusion of concepts, when learning multiplication, you know that zero multiplied by any number is equal to zero, and after learning division, you know that zero divided by any number is also equal to zero. When we are learning addition, we say that zero plus any number is equal to any number, and when learning subtraction, there are also ah, any number minus zero is equal to any number, so it is confusing the basic concept.

The concept of 0. The 0 number is 0 no matter how it is placed, there is no reciprocal, 0 number has no opposite number, the original meaning is 0, the 0 number is the critical point between positive and negative numbers, and the 0 number is neither positive nor negative. The number 0 is the number of places, i.e., the number of transitions.

The number 0 comes to the integer mark every time, and it is not possible to do without the number 0. It is empty, zero is nothing, zero is non-existent, it can also be said to be the meaning of beginning, to do something from zero.

Add a number to zero to get (the original number).

0 multiplied by any number gives 0, 0 divided by a non-0 number gives 0, any number is added with 0 and the original number is obtained, and a number minus 0 is also obtained by the original number

1. 0 plus any number is equal to the original number, 0 times any number is equal to 0; Example: +(5)=-5.

is the smallest natural number and is also a rational number. Noisy 0 is neither positive nor negative, but the dividing point between positive and negative numbers. 0 has no reciprocal, the opposite of 0 is 0, the absolute value of 0 is 0, the square root of 0 is 0, the cube root of 0 is 0, 0 times any number is equal to 0, the power of 0 of any number other than 0 is equal to the denominator, all letter pure multiples of 0 are 0.

No. Zero - any number = the opposite of this number (the opposite of 0 is 0), e.g. 0-1 = -1.

The nature of opposite numbers.

1. If they are opposite to each other, then a+b=0, and if a+b=0, then a and b are opposite to each other.

2. Opposite numbers appear in pairs and cannot appear alone.

3. The opposite number is not only the opposite of the number symbol, but also the number after the symbol must be the same, such as: +5 and -5.

4. To find the opposite number of a number, you only need to add a negative sign in front of this number, if the original number has a sign (whether positive or negative), you should add parentheses first.

5. The opposite of the number a is -a, and the opposite of -a is a. A here is not necessarily positive, so -a is not necessarily negative.

6. When simplifying multiple symbols, it should be noted that when there is an even number of "-" in front of a positive number, it can be reduced to the number itself.

The rule of the opposite number.

1. The opposite of a positive number is a negative number, and the opposite of a negative number is a positive number.

The opposite of 0 is 0, i.e. the opposite of 0 is itself. At the same time, the opposite number is the number of itself that is only 0. Irrational numbers also have opposite numbers.

3. The quotient of two numbers that are opposite to each other is -1 (except 0).

4. The opposite of the real number A is the opposite of the real number A itself.

5. a-b and b-a are opposite numbers to each other.

Zero minus any number is not necessarily equal to zero, 0 minus 0 is equal to 0, and 0 minus other numbers is equal to the negative number of this number, such as 0-5=-5, 0-(-5)=5

Adding 0 to any number gives you the original number, and multiplying any number by 0 gives you 0.

0 is an integer between -1 and 1. is the smallest natural number and is also a rational number. 0 is neither positive nor negative, but the dividing point between positive and negative numbers.

There is no reciprocal of 0, the opposite of 0 is 0, the absolute value of 0 is 0, the square root of 0 is 0, the cube root of 0 is 0, 0 times any number is equal to 0, the power of 0 of any number other than 0 is equal to the denominator, and all multiples of 0 cannot be used as a divisor.

Zero minus any number doesn't get zero.

Zero - any number = the opposite of this number (the opposite of 0 is 0), if a and b are opposite to each other, then a+b = 0, and vice versa if a + b = 0, then a and b are opposite to each other. Opposite numbers occur in pairs and cannot appear individually.

The property of 0: is the smallest natural number.

Not an odd number, but an even number (a special even number that is neither positive nor negative).

It plays a placeholder role in multiple digits, such as 0 in 108 means that there is no place in the ten digits, and 18 must not be written.

It cannot be used as the highest digit of a single digit. However, some numbers need to be preceded by 0 to fill in all the digits.

It is neither positive nor negative, but the dividing point between positive and negative numbers. When a certain number x is greater than 0 (i.e., x>0), it is called a positive number; Conversely, when x is less than 0 (i.e., x<0), it is called negative; And when this number x is equal to 0, this number is 0.

is an integer between -1 and 1.

The opposite of is 0, i.e., -0=0.

The absolute value of is itself, i.e., 0 = 0. Of all the absolute values of real numbers, the absolute value of 0 is the smallest.

Multiplying any real number is equal to 0, and dividing by any non-zero real number is equal to 0; Any real number plus or minus 0 equals itself.

0 minus positive numbers gives a negative number, 0 minus 0 gives 0, and 0 minus a negative number gives a positive number. For example:

1) 0 minus the positive number is a negative number: 0-2=-2.

2) Subtracting the negative number from 0 gives a positive number: 0-(-2)=2.

3) 0 minus 0 gives 0:0-0=0.

Zero minus any number does not give zero, zero - any number = the opposite number of this number (the opposite number of 0 is 0), if it is the opposite of each other, then a + b = 0, and vice versa if a + b = 0, then a and b are opposite numbers to each other. Opposite numbers occur in pairs and cannot occur individually.

The opposite number is not only the opposite sign of the number, but the number after the symbol must be the same, as in the case of :

5 vs. -5. To find the opposite of a number, you only need to add a negative sign before the number of regrets, and if the original number has a sign (whether positive or negative), it should be first added with parentheses. The inverse of the number a is -a, and the inverse of -a is a.

A is not necessarily a positive number, and a positive pose with -a is not necessarily negative. When simplifying multiple symbols, it should be noted that when a positive number is preceded by an even number of "-", it can be reduced to the number itself.

Because 0 means none, adding 0 to any number gives the original number, and subtracting 0 from any number gives the original number.

0 is the smallest natural number and is also a rational number. 0 is neither positive nor negative, but the dividing point between positive and negative numbers. 0 has no reciprocal, the opposite of 0 is 0, the absolute value of 0 is 0, the square root of 0 is 0, the cube root of 0 is 0, 0 multiplied by any number is equal to 0, the power of 0 of any number other than 0 is equal to cannot appear as a denominator, all multiples of 0 are 0.