What exactly is "imaginary number I"? Why is it so amazing? What are the important functions?
This has to start with the seemingly ordinary but huge "number axis"!
In junior high school mathematics learning, "number axis" is an important tool for learning mathematics.
The concept of "number axis" must be deeply rooted in the mind in order to knock on the door of elementary mathematics and walk into the room.
Natural numbers, integers, negative numbers, irrational numbers and other "problems of all numbers". It can only be discussed on the "number axis" so as not to be confused.
Before imaginary numbers were discovered, one axis was enough to describe all real numbers. But in the17th century, mathematician Descartes discovered imaginary numbers. At this time, a number axis was not enough, so he founded the famous Descartes Cartesian coordinate system.
"Cartesian coordinate system" is an important tool that we must master when entering junior high school.
Cartesian coordinate system can be described as
Two number axes perpendicular to the origin. When we discuss the relationship between numbers, Cartesian coordinate system becomes a very useful tool, and all numbers can find their corresponding points in Cartesian coordinate system.
Cartesian coordinate system established the idea of "combination of numbers and shapes" for the first time, and used mathematical formulas to describe "distance" and "angle" in geometric figures for the first time, which built a bridge between algebra and geometry. Descartes established an epoch-making branch of mathematics-analytic geometry.
Analytic geometry introduced the concept of "variable" for the first time, and Newton and Leibniz founded "calculus" on this basis.
"Calculus" further developed into "theory of real variable function".
After Descartes discovered the appearance of imaginary numbers, he established a "complex plane" on the "rectangular coordinate system", which can be expressed as: z=a+bi.
When people can't find the complex plane, they often feel that "numbers are not enough."
Now, mathematicians have strictly proved that "all numbers" can be found on the complex plane, and the "range of numbers" will no longer exceed the range of complex numbers.
Due to the discovery of imaginary numbers, in the18th century, a new branch of mathematics, complex variable function, was developed to study functions on the "complex plane".
Complex variable function, with "complex number as variable", is used to analyze the laws and changes of the function, which is rich in content and practical, and is used in "fluid mechanics" and "aviation dynamics" to solve the structural problems of aircraft wings.
The famous Euler formula takes the product of imaginary number I and π as the exponent of natural base E, and connects complex variable function with trigonometric function, which makes the complex variable function also shrouded in a mysterious color.
Mathematicians praise "complex function" as a very harmonious theory, and it is simply a pleasure to study it.
The discovery of imaginary numbers has played an important role in natural science. At the beginning of the 20th century, "quantum mechanics" was born, and the legendary Schrodinger equation came out. Fascinating, this famous equation also contains "imaginary number I".
In order to quantitatively describe the state of microscopic particles, the "wave function" is introduced into quantum mechanics as the "solution of Schrodinger equation". This magical wave function can clearly describe the state of microscopic particles in the form of "complex number", and the famous "wave mechanics" was born.
"Quantum mechanics" and "relativity" have become the two pillars of modern physics.
With the vigorous development of modern science and technology, the role of imaginary numbers is becoming more and more significant. Those formulas containing imaginary numbers seem to be the language of God, from which people can always come up with some new theories.
1966 Su Shikan discovered the existence of string theory in Euler's formula 200 years ago, inspired by the imaginary number I in the formula.
1990, Wilton put forward the "m theory" (matrix theory) of "1/degree space", which unified the string theory results of various previous "limit states".
With the emergence of string theory, scientists think it will be the ultimate theory.
In April 2007, Fermi National Accelerator Laboratory in the United States proved the correctness of ten dimensional space's string theory to some extent.
However, under the existing conditions, there is still a long way to go to thoroughly prove the "string theory" with physical experiments.
In this case, we can only rely on the "strict logic" of mathematics to prove its correctness. Imaginary numbers will once again give full play to their advantages and provide people with a new perspective.
In today's modernization, "superstring theory" has stood at the forefront of modern physics research, and it is most promising to find a "unified theory of four basic forces" called the holy grail of physics to explain unexplained mysterious phenomena such as "classical physics" and "quantum mechanics".
Without the discovery of imaginary numbers, there would be no quantum mechanics, and all natural disciplines in 2 1 century would be unable to proceed.
With the emergence of new theories, imaginary numbers will play a more and more important role, and the future world will be more exciting.
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