Is the imaginary number a magic horse? What is an imaginary number?

An imaginary number is a number whose square is negative. The term imaginary number was coined by the famous mathematician Descartes in the 17th century, because the concept at the time was that it was a number that did not really exist. Later, it was found that imaginary numbers can correspond to the vertical axis on the plane, and the real numbers corresponding to the horizontal axis on the plane are just as true.

In mathematics, a number whose square is negative is defined as a pure imaginary number. All imaginary numbers are plural. Defined as i 2 = -1.

But imaginary numbers don't have an arithmetic root, so (1) = i. For z=a+bi, it can also be expressed as the form of e to the power of ia, where e is a constant, i is an imaginary unit, and a is the amplitude angle of an imaginary number, which can be expressed as z=cosa+isina. A pair of real and imaginary numbers is seen as a number in the complex range and is named a complex number.

There is no plus or minus for imaginary numbers. Complex numbers that are not real numbers, even if they are pure imaginary numbers, cannot be compared in size. This type of number has a special symbol "i" (imaginary), which is called an imaginary unit.

However, in industries such as electronics, i is usually used to represent electric current, so imaginary units are expressed as j.

Practical implications. We can draw a system of imaginary numbers in a planar Cartesian coordinate system. If the horizontal axis is used to represent all real numbers, then the vertical axis can represent imaginary numbers. Each point on the entire plane corresponds to a complex number, which is called a complex plane. The horizontal and vertical axes are also called real and imaginary numbers.

axes and virtual axes. Students or scholars who are not satisfied with the above image explanation can refer to the following topics and explanations: If there is a number, its reciprocal is equal to its opposite (or its reciprocal is its opposite number), what is the form of this number?

According to this requirement, the following equation can be given: -x = (1 x) It is not difficult to know that the solution of this equation x=i (imaginary unit) Thus, if there is an algebraic equation t'=ti, we understand i as a unit from t to t'The units of conversion between the units, then t'=ti will be understood as -t'= 1 t is t'The expression = - 1 t has little significance in geometric space, but if it is understood in time with the special theory of relativity, it can explain that if the relative velocity of motion can be greater than the speed of light c, the imaginary value produced by the relative time interval is essentially the negative reciprocal of the actual value. In other words, the value of the so-called time interval of going back to the past can be calculated from this.

Not a complex number of real numbers, even if they are pure imaginary numbers.

When learning imaginary numbers in the middle stage of Gaohuai Hall, the following knowledge points are mainly involved:

Imaginary unit i: The imaginary unit i is defined as i =1. It is a special number that is used to represent the square root of negative or bright dos. The introduction of the imaginary unit i extended the system of real numbers to form a complex set of numbers.

Complex numbers: Complex numbers are numbers made up of real and imaginary numbers. The general form is a + bi, where a and b are real and imaginary respectively, and a and b are both real numbers.

Nature of Imaginary Numbers: Imaginary numbers have some special properties. For example, the addition and subtraction of imaginary numbers follows the same rules as real numbers, where i = 1 in the multiplication of imaginary numbers, and division of imaginary numbers can be achieved by multiplying conjugate imaginary numbers.

Representation of complex numbers: Complex numbers can be represented in different representations. In addition to the general form A + Bi, there is also the triangular form r (cos + isin) and the exponential form re (i).

These different representations have different applications in different mathematical problems.

Operations on complex numbers: Addition and subtraction, multiplication, division and other operation rules between complex numbers need to be mastered. The operation of complex numbers can be performed separately using the real part and the imaginary part, or using the representation of complex numbers.

Conjugate complex number: Conjugate complex number refers to a complex number that keeps the real part unchanged and the imaginary part takes the opposite number. For example, for the complex number A + bi, its conjugate complex is a - bi.

These are the main knowledge points when learning imaginary numbers in high school. In more advanced mathematics, complex numbers also involve complex planes, roots and equations of complex numbers, trigonometric functions of complex numbers, and other in-depth content.