Why can't complex numbers be sorted?
01 imaginary number sorting
I. Introduction
In any textbook on complex variables, it will be mentioned that " complex numbers " are not sortable like real numbers. When I hear it for the first time, I feel confused. I thought we could find a way to sort the plural numbers clearly. For example, the complex numbers can be sorted by the real part first, and then sorted by the imaginary part if the real parts are the same. For example, the plural sorting and naming in Python is implemented in this way. In fact, it is also possible to sort by the modulus of the complex numbers first, and then sort by the phase angle. But the sorting mentioned in mathematics refers to the sorting within a certain "number field", that is, the sorting cannot produce contradictions after satisfying the specific operations between elements. Operations such as addition and multiplication are defined for complex numbers. The defined ordering should also be non-contradictory under these operations.
▲ 图1.1.1 复数排序
2. Sorting Contradictions
Mathematically, the definition of complex number sorting needs to satisfy the following properties. 1. When ab is plural, the ordering between them must belong to one of these three situations. 2. When a is smaller than b, they all add any number, and the order of the results is still maintained. Three, when a is less than b, c is a number greater than 0. They are multiplied by c simultaneously, and the order of the results is preserved. The last property is the compatibility that any sorting method must satisfy. If a is less than b, b is less than c. Then a is less than c. Let's prove that complex numbers cannot be sorted according to these four properties. In fact, as long as it is proved that there are two numbers in complex numbers, no matter how the ordering between them is defined, there will be contradictions in the above four properties.
For example, take two digits in complex numbers, i and 0. The relationship between them must also satisfy one of three orders. Because it is known that these are two different complex numbers. So there are only two orders of i and 0. If i is assumed to be greater than 0. According to property three, the result of multiplying i by i should be greater than 0 times i. According to the complex number arithmetic rule, it can be obtained that negative 1 is greater than 0. Note that no contradiction is derived here, because we are discussing the ordering method of complex numbers, so it is not possible to define complex numbers in terms of the magnitude of real numbers. Negative one greater than 0 is just a consequence of intermediate derivations. Next, multiply negative one by negative one, and you will get the next conclusion, 1 is greater than 0. At this time, it is easy to contradict the previous conclusions. This shows that the assumption that i is greater than 0 does not hold.
Starting from i is less than 0, with the help of property 2, add negative i to both sides at the same time, you can get negative i greater than 0, then apply property 3, use negative i to multiply both sides of the inequality, according to the complex number four arithmetic rules, you can get negative The conclusion that 1 is greater than 0. This goes back to the conclusion in the previous proof, and further derivation also leads to contradictory results. So far, combining the above two processes, it is proved that complex numbers cannot be sorted.
▲ 图1.1.2 矛盾的证明
※ Summary ※
This article discusses the inability to sort within the field of complex numbers. This also gives us a more comprehensive understanding of the mathematical sorting requirements.
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