Jokes about plural numbers

I don't think there is any "philosophy" in this question. The philosophical problems caused by mathematics are not here. As for the unit of measurement, it is actually just a regulation. For example, for the unit of "kilogram", what matters is not "how much this thing weighs" but "what is the ratio of the quality of this thing to the quality of the reference object" (if you don't know, check the original kilogram). Let's not talk about the unit now, because

Let's talk about it.

Let's not talk about the definition of imaginary units. Let's see how the number system is unfolded.

Integers are abstracted from daily counting. But for counting problems such as "half steamed bread", integers can do nothing. Write the question accurately, that is: how many (the same) steamed buns add up to one steamed bun? To this end, we need to introduce "half steamed bread". In other words, we need to introduce the solution of equation 2x= 1, so that we can intuitively know what is called "rational number". Negative numbers are also introduced to understand such a linear equation, so I won't go into details.

[Note: We think rational numbers are easy to understand just because we are used to them. Math often has to break the habit and see different scenery. ]

But what exactly is a rational number? For a positive integer, we can abstract its properties from daily experience, such as "1 is a common quantitative attribute of a sheep, a cow and a person" (the meaning of this sentence itself is not clear, let's not talk about it). Positive rational numbers can be intuitively understood by "equal division". For "zero" and negative numbers, this intuitive cognition. Think back to how the Romans treated zero. In order to make up for this semantic ambiguity, mathematicians invented a strict definition. Let's talk about it next.

For irrational numbers, the problem is even more serious, because there is no corresponding one in the daily counting problem. In fact, we know that the earliest irrational number comes from the geometric measurement problem: \sqrt2 is the most widely known irrational number. But it still boils down to finding the root of the equation. Pythagoras met \sqrt2 because they wanted to find the root of equation x 2 = 2. In this way, we intuitively.

This immediately caused a problem: many quadratic equations with integral coefficients have no roots (and higher-order equations). The simplest example is x 2 =- 1. This equation really doesn't make any sense when abstract thinking is not developed. But as we know, the situation has changed since cardano's time. During this period, objects that mathematicians could not explain at that time kept appearing. "Square root of 1" is the most obvious example. In order to make the cubic equation have a unified root formula, we have to introduce this "meaningless" "imaginary unit". Although the meaning is not clear, mathematicians still rely on imaginary units to get a series of interesting results (of course, many times it is only a formal operation; It is not impossible to do it only in the real number range, but it will be a lot of trouble).

It can be seen that "finding the root of the equation" is actually a much more abstract problem than "defining pi", because the latter is "objective" (we don't know what this means now, we will talk about it later), while the former does not necessarily have any realistic correspondence.

We know that there was a vague concept of real number in Euler's time (he has found many conclusions related to e, \pi), but Euler still can't really calculate the "imaginary number", although he got Euler's formula in form.

e^{ix}=\cos x+i\sin x

[The meaning of this formula is actually unclear; What do you mean by e times i? ]

Dedekind and others gave a strict definition of real numbers, so that people can finally describe real numbers in a language that will not cause ambiguity. According to the present point of view, real number is actually just a thinking object, and the division of decimal and Dedekind is only the realization of this thinking object in reality. Real numbers that need to be defined by geometric measures, such as pi, can also be included in this logical framework, because with the help of analysis, we can make clear what is "the length of a curve"

But we have to admit that "imaginary number" is still difficult because it has no real correspondence, but it has important applications in practical problems (fluid mechanics, heat transfer, electricity, etc.).

How can we find a reasonably defined "solution" for the equation x 2 =- 1

Of course, we can use two-dimensional divisible algebra in real number field to define complex numbers. But it seems that I can't make too many generalizations. So we think about this problem in different ways. This way can make us clear what is "adding roots to algebraic equations".

For a given field k, consider the polynomial ring k[x] above.

[Note: Recall that the polynomial ring k[x] is defined as a group algebra with finite support for the coefficients of infinite cyclic groups in K, and should not be considered as a "polynomial function". ]

For an irreducible polynomial f (x) in k[x], we want to find its root. Therefore, we consider the ideal I generated by f(x). Because it is irreducible, it should be a great ideal, so the quotient ring K=k[x]/I is a field. It contains a subdomain isomorphic to K, so it can be regarded as an extension of K, and further, it can be regarded as an extension of K.

The elements in K are equivalent classes G (x)+I; Let's consider the x+I class in particular. According to the operational properties of quotient rings, we immediately get f(x+I) = i, in other words, in field K, x+I is the root of polynomial f(x). So far, we have found a root of f(x). What remains is to exhaust all the roots of F (x) by constantly expanding the field (according to polynomials)

In this way, we know the strict meaning of "adding roots to algebraic equations" As for the definition of "rational number", it is much simpler. It is nothing more than the fractional domain of the whole ring.

If any algebraic equation in a domain has a solution in this domain, then this domain is called algebraic closed domain. For this kind of field, it is easy to study the polynomials on it. In fact, any polynomial can be decomposed into the product of linear factors (Bezout theorem).

Back to the case of complex number, take k = \ mathbb {r} and f (x) = x 2+1,and the obtained field is extended to complex number \mathbb{C}. This field is algebraically closed; This is the so-called "basic theorem of algebra", and there are many proofs (algebraic, real analysis, complex analysis, topological, ...), but the proof with the strongest algebraic flavor is naturally based on algebra. Because \mathbb{C} is algebraically closed, it plays an important role in mathematics. Just give a simple example. In order to calculate the value of a square matrix, we often need to turn it into an equivalent standard, which must be realized by complex numbers. If complex numbers are not used, the amount of calculation will be unimaginable.

Having said that, I found that I wrote a lot of seemingly "philosophical" words, and the latter part was dry goods. But if the previous "philosophy" can help some people think clearly, I am also very pleased.

Discussion 2:

The second puzzling mathematical definition-the definition of imaginary unit I

Is it necessary to square a negative number?

It is necessary!

But the complete answer to this question is far more than "definition: I 2 =- 1".

Firstly, the author briefly introduces the rational number set:

1, we have natural number set and addition operation, and natural number set closed the addition operation (the result of adding two natural numbers is still a natural number).

2. The inverse of addition is subtraction, but natural number set is not confined to subtraction (there is no guarantee that the result of subtraction between any two natural numbers is still a natural number); By defining a negative number, natural number set is extended to a set of integers. Integer sets are closed to addition and subtraction (people realize that negative numbers have gone through a long process because they think negative numbers have no practical significance).

3. The inverse of multiplication is division, and the integer set is closed to multiplication and division; By defining the fraction, the integer set is extended to the rational number set; Rational number set is closed for addition, subtraction, multiplication and division (divisor is not zero).

4. The more strict name of rational number set is "rational number field", but the explanation of "field" needs abstract algebraic content. For the sake of popularity, the author calls "rational number field" "rational number set"; The above "set" means a set, that is, a set of similar numbers; Such as natural number set and integer sets.

Second, everything is numbers and Pythagoras theorem:

1. The Pythagorean school of ancient Greece believed that "everything has a number" and regarded it as a doctrine. The number here refers to the rational number; This belief stems from their confidence in their own rational number set, which they think already contains all the numbers.

2. Subsequently, this school discovered Pythagorean Theorem, that is, Pythagorean Theorem, and proved it by area method.

3. If "everything is rational", then the hypotenuse of a right triangle should also be rational; However, Hipassus of Pythagoras School found such an example and proved that a= 1, b= 1, and C determined by A and B through Pythagoras theorem is not a rational number! There is a saying that Hipasus was expelled from the school because of this discovery, and there is another saying that he was killed by the school.

In any case, there is no such "C" in rational number set, but there is such a C in reality. The only reason is that the set of rational numbers created by Pythagoras school is flawed and does not cover all numbers!

5. By adding N-power operation, the rational number set is expanded into a real number set (the real number set is not a real number set, but a partial real number set, and the real number set here is strictly an N-power algebraic expansion of the rational number set).

6. The real number set is closed to addition, subtraction, multiplication and division (divisor is non-zero), the positive number in the real number set is also closed to the N power operation, and the negative number in the real number set is closed to the odd power operation but not to the dual power operation; Especially √(- 1) is not in this real set, in other words, the square of the number in this real set is equal to (-1).

Third, is it time to add a definition?

1. Should the definitions "I 2 =-1"or "i=√(- 1)" be added to integrate the above real numbers into a larger number set?

The answer is that people think it is unnecessary!

2. People think that positive roots are meaningful because the results of roots have such a meta-calculation in reality. Just like √2, people can really find a line segment with neither too much nor too little length.

People think that the root of negative number is meaningless, because there is no such element corresponding to the root result in reality. Of course, the author also said that people couldn't even recognize negative numbers at that time because there was no "negative" line segment in reality. ? √(-2)=-? √(2) is only a kind of "deformation" of the positive root; As for √(- 1), no one cares if there is anything corresponding to it, because it has no practical significance.

Quadratic, cubic and quadratic equations and their root formulas;

1, the so-called equation is an equation containing unknown quantities; Unknown is number, and equation is algebraic equation; Unknown is function, and equation is function equation (such as differential equation and integral equation); The solution of the equation is a quantity that can make the equation hold; The solution of an algebraic equation is a number, and such a number is called the root of the algebraic equation.

2. Among algebraic equations, people pay more attention to polynomial equations, because such equations are closely related to people's production and life; In the period of classical mathematics, the equations studied by mathematicians were mainly polynomial equations. The following "equations" all refer to "polynomial equations".

3. The so-called formula for finding the root of the equation is to construct the root formula by adding, subtracting, multiplying, dividing and squaring the coefficients of the equation.

4. The root formulas of the first and second order equations have been found for a long time, and people are committed to finding the root formulas of the third and higher order equations.

5./kloc-In the 6th century, the Italian mathematician Philo discovered the formula for finding the root of cubic equation in the form of x 3+px+q=0, which lacked quadratic terms. Because people generally did not accept negative numbers at that time, Philo actually divided cubic equations lacking quadratic terms into three categories: x 3+px=q, x 3=px+q, x 3+q=px, and both p and q are positive numbers; He gave solutions respectively.

6. Interestingly, a "duel" was popular among mathematicians at that time. )。 The so-called "duel" is to ask the other side to solve their own problems. So Ferro used his cubic equation to find the root formula as the secret weapon of duel, which was not published. Because of this formula, Philo won many times in duels and became famous at one fell swoop.

7. Before Ferro died, he taught his secret weapon to student Fio and his son-in-law and heir Naff.

8. Fio is also a competitive person. He challenged tartaglia, a mathematician at that time (this is not his real name, which means stutterer. When tartaglia was a child, his face was cut by a French soldier with a sabre and he became stuttering). Tartaglia didn't know the formula for finding the root of cubic equation without quadratic term, but under the pressure of challenge, she successfully deduced the general formula for finding the root! Therefore, tartaglia won a great victory in the duel with Fiol, because the latter could not solve the general cubic equation of x 3+rx 2+px+q=0. Tartaglia became famous.

9. After learning this, cardano repeatedly begged tartaglia to tell him the formula of Root. In return, cardano promised to provide economic assistance to tartaglia. Tartaglia finally couldn't resist cardano's soft and hard bubbles and the temptation of interests, and told cardano the root-seeking formula in the form of obscure sentence poems, and asked cardano to swear secrecy.

10. Later, cardano learned about Ferro's formula for finding roots from Navier, and thought that tartaglia's formula for finding roots was essentially the same as Ferro's formula for finding roots (in fact, an ordinary cubic equation can be transformed into a cubic equation without quadratic terms through a variable substitution, which will be discussed later).

1 1, so cardano ignored the oath and taught Ferrari, a student. On this basis, Ferrari actually found a formula for finding the roots of the quartic equation!

12, cardano published the root formula of the cubic equation and the root formula of the quartic equation of Ferrari. Cardano commented: "Ferro discovered this rule 30 years ago and passed it on to Fio. It was Fio who challenged tartaglia and gave tartaglia a chance to rediscover this rule. Tartaglia told me this rule under my pleading, but tartaglia kept the evidence, and I found its evidence with this help.

13, then tartaglia severely accused cardano and condemned cardano's treachery. Angry tartaglia challenged cardano, and Ferrari accepted the challenge instead of his teacher. Tartaglia was defeated because Ferrari discovered the formula for finding the roots of quartic equation. Tartaglia was discredited and lived in quarrels and poverty in his later years.

14, the formula for finding the root of cubic equation is very boring, but the history behind the formula is very interesting; I have no intention to comment on cardano and tartaglia. Every reader has his own opinion.

Irreducible quintic and cubic equations;

1, the general cubic equation is aX 3+bX2+cX+d=0. By substituting the variable X=x-[b/(3a)] (mentioned earlier), the general cubic equation can be transformed into cubic equation x 3+px+q=0 without quadratic term, and it is enough to solve this equation.

2，x ^ 3+px+q = 0:

Here, the author will not give the derivation process of the root formula.

Please note that ⊿ needs to be squared, but ⊿ cannot be guaranteed to be greater than 0. That is to say, cardano or tartaglia may face the dilemma of negative square root in the root formula constructed by addition, subtraction, multiplication, division and square root operation.

4. In order to make readers understand this contradiction more clearly, the author gives an example:

Cubic equation x 3+px+q = 0, p=- 10, q=6.

The image of the function y = x 3-10x+6 is roughly

The x value of the point where the function curve intersects the x axis is the root of the cubic equation x 3-10x+6 = 0.

From the image, we can clearly see that this cubic equation has three real roots.

However, ⊿ = (1/4) Q2+(1/27) P3 =-28.037

5. In other words, the cubic equation of real coefficient is given to ⊿.

6. The process of getting real roots by negative square root is really unsatisfactory, so cardano tried to "modify" the formula for finding roots to avoid this situation. However, all attempts failed. Cardano reluctantly called this situation "irreducible cubic equation".

7. In order to deal with this situation, cardano introduced the imaginary unit I, and defined I 2 =-1,so that the root formula can work normally.

8. Does such a "correction" exist? Until the19th century, the talented mathematician Galois gave the answer with his groundbreaking tool group theory: it doesn't exist! That is to say, "the process of getting real roots by negative square root" is inevitable!

9. It must be emphasized here that the solution of quadratic equation did not lead to the introduction of imaginary number I because of discriminant ⊿.

Sixth, summary and reflection:

1, mathematics seems to play a joke on everyone: when you think that the rational number field is complete, you find that you have found a large class of freaks and have your own Pythagorean theorem, so you have to incorporate the square root operation into the system; When you think that the root formula can solve all cubic equations, you find that three obvious real roots should be squared with negative numbers, so you have to define "I 2 =-1"; As for the convenience brought by the definition of "i 2=- 1" to algebra and analysis, that is another story.

2. This once again verifies the author's words: "No mathematician can foresee from the beginning how much convenience and quickness his definition can create", or how many defects exist; Mathematicians cross the river by feeling the stones, tinkering along the way. The detailed description in the textbook fails to show the mathematicians' struggles and setbacks in the process of creation and the arduous and long road they have experienced before establishing an objective structure.

3. The story of "I 2 =-1"is far from a simple definition.

Discussion 3:

What is the most essential feature of complex numbers? Why is it physically necessary and able to use complex numbers so frequently?

None of the answers upstairs mentioned this. The most important property of complex numbers is rotation. That is to say, the radian of the product of two complex numbers is equal to the sum of their respective radians. Without this feature, complex numbers would not be as important in mathematics and physics as they are now.

Let's start with the original question. Fundamentally, why am I the square root of-1?

As shown above, complex numbers form a plane, and the real axis and imaginary axis are orthogonal.

-1 is located on the negative semi-axis of the real axis, and the radiation angle is π( 180 degrees). The square root, according to the nature of the radial angle mentioned above, is that the radial angle is halved and becomes π/2, which is the position of I on the positive semi-axis of the imaginary axis. Another scheme is -i with radiation angle of 3π/2, because the radiation angle of-1 can also be 3π.

Or conversely, multiplying a complex number by I is equivalent to rotating π/2 counterclockwise. Then I 2 = 1ii, that is, 1 rotates π/2 twice, which falls on-1.

By analogy, do you understand how to explain why negative numbers are positive numbers from the perspective of complex number rotation?

Considering this, it is easy to understand why the complex number, as an unnatural and artificially invented number, can be applied to physics so well.

For example, the extremely important simple harmonic vibration can be regarded as the projection of a point with uniform circular motion on the real axis of the complex plane unit circle. Since it is rotation, it can be expressed by an exponential function of time, which is very convenient to deduce.

Discussion 4:

First of all-1 What can it be? Let's use the simplest example,

cos(\pi )=- 1

According to the definition of I, I is the square root of-1, or i\cdot i=- 1, so we have:

cos(\pi)=i\cdot i

Then come on:

cos(\ pi)= cos(\ pi/2+\ pi/2)= I \ cdot I

If you feel good about algebra, you will immediately find the above formula somewhat "algebraic". Yes, a rotation with angle \pi can be regarded as the sum of two rotations with angle \pi/2. The multiplication of I and I also feels like a commutative group.

Simply put, let's fill in the formula:

cos(\ pi)= cos(\ pi/2+\ pi/2)=- 1

sin(\pi)=sin(\pi/2+\pi/2)=0

Remember trigonometric identities:

cos(a+b)= cos(a)cos(b)-sin(a)sin(b)

sin(a+b)= sin(a)cos(b)+cos(a)sin(b)

For any angle, taking the cos part as the real part and the sin part as the imaginary part, we can use trigonometric inequality to construct the multiplication of complex numbers, which is the significance of complex multiplication. Rewrite as follows:

cos(a+b)+isin(a+b)=[cos(a)cos(b)-sin(a)sin(b)]+I[sin(a)cos(b)+cos(a)sin(b)]

That is, the form seen in the textbook:

z _ { 1 } \ cdot z _ { 2 } =(x _ { 1 }+iy _ { 1 })\ cdot(x _ { 2 }+iy _ { 2 })=[(x _ { 1 } x _ { 2 })-(y _ { 1 } y _ { 2 })]+I[(x _ { 1 } y _ { 2 })-(x _ { 2

If you are interested, please play with Euler formula and learn about all kinds of interesting places in this multiplication calculation.

As for me, it is actually the natural basis on the complex plane. My "full name" is:

I =[0,1] {t} = [cos (\ pi/2), sin (\ pi/2)] {t}

To sum up: when playing with real numbers (such as the roots of algebraic polynomials), we often find that there are not enough numbers, so we expand real numbers to the complex plane. The operation of complex number (field) is limited to the real axis (field) to be true. Therefore, the square of I is-1, so it can be understood that the square is the result of two syntheses of the same transformation. Transforming the real multiplication unit 1 into-1 (the inverse of the addition group) needs to be expressed as a rotation transformation with an angle of \pi in the complex field, or as a synthesis of two rotation transformations with an angle of \pi/2. Therefore, I is only the result of a \pi/2 rotation transformation.

We were all talking about algebra just now. We pay attention to the analysis:

cos(x)^{'} = -sin(x)

sin(x)^{'} = cos(x)

Various derivatives are nothing more than phase transformations; Euler formula also shows that multiplication, division and logarithm are also in-phase transformation; It is not difficult to understand why so many physical phenomena need to be described by complex numbers.

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