The latest version of the four arithmetic operations of complex numbers courseware PPT

The Four Arithmetic Operations of Complex Numbers courseware is the latest PPT lesson plan of the Four Arithmetic Operations of Complex Numbers courseware. Complex numbers are defined as binary ordered pairs of real numbers (a, b) [1], recorded as z=a+bi, where a and b are real numbers, i is an imaginary unit. It was introduced by the Italians and later gradually accepted. This book helps us master the addition operation of complex numbers and its meaning, process and method, and understand and master the rules of the four arithmetic operations with real numbers. The four arithmetic operations of complex numbers courseware teaching objectives knowledge and skills: master the addition operations and significance of complex numbers, process and methods: understand and master the rules of the four arithmetic operations of real numbers, and understand the geometric significance of the addition and subtraction operations of complex numbers. Emotions, attitudes and values: understand and master Concepts related to complex numbers (sets of complex numbers, algebraic forms, imaginary numbers, pure imaginary numbers, real parts, imaginary parts) Understand and master the concepts related to the equality of complex numbers; conclusions drawn from drawings cannot replace arguments, but observation of graphs can often help To enlighten the problem-solving ideas, the teaching focus is: the addition operation of complex numbers, the correspondence between complex numbers and vectors starting from the origin. Teaching difficulties: the operation rate of complex number addition operations, and the geometric meaning of complex number addition and subtraction operations. Preparation of teaching aids: multimedia, physical projector. Teaching assumption: A complex number has a unique point corresponding to it in the complex plane; conversely, every point in the complex plane has a unique complex number corresponding to it. There is a one-to-one correspondence between the complex number z=a+bi(a, b∈R) and the ordered real number pair (a, b). This is because for any complex number z=a+bi(a, b∈R), the complex number It can be seen from the definition of equality that it can be uniquely determined by an ordered real number pair (a, b). Teaching process of the four arithmetic operations of complex numbers courseware student exploration process: 1. Imaginary number unit: (1) Its square is equal to -1, that is; (2 ) Real numbers can perform four arithmetic operations with it. When performing the four arithmetic operations, the original addition and multiplication operation laws still hold 2. The relationship with -1: It is a square root of -1, that is, a root of the equation x2 = -1, the equation x2 The other root of =-1 is the periodicity of -3.: 4n+1=i, 4n+2=-1, 4n+3=-i, 4n=14. The definition of complex numbers: A number of the form is called a complex number. is called the real part of a complex number, and is called the imaginary part of a complex number. The set of all complex numbers is called the set of complex numbers, represented by the letter C*3. The algebraic form of complex numbers: Complex numbers are usually represented by the letter z, that is, the complex number is represented as a+bi The form is called the algebraic form of complex numbers 4. The relationship between complex numbers and real numbers, imaginary numbers, pure imaginary numbers and 0: For complex numbers, if and only if b=0, the complex number a+bi (a, b∈R) is a real number a; when When b≠0, the complex number z=a+bi is called an imaginary number; when a=0 and b≠0, z=bi is called a pure imaginary number; when and only when a=b=0, z is a real number 0.5. The set of complex numbers and Relationship between other number sets: NZQR C.6. The definition of equality of two complex numbers: If the real part and imaginary part of two complex numbers are equal respectively, then we say that the two complex numbers are equal: if a, b, c , d∈R, then a+bi=c+di a=c, b=d Generally, two complex numbers can only be said to be equal or unequal, but cannot be compared. If both complex numbers are real numbers, they can be compared The size cannot be compared only when the two complex numbers are not all real numbers. 7. Complex plane, real axis, imaginary axis: the abscissa of point Z is a, the ordinate is b, the complex number z=a+bi(a, b∈R ) can be represented by point Z (a, b). This plane that establishes a rectangular coordinate system to represent complex numbers is called the complex plane, also called the Gaussian plane. The x-axis is called the real axis, and the y-axis is called the imaginary axis. The points on the real axis all represent real numbers. For the points on the imaginary axis, except for the origin, because the ordered real number pair corresponding to the origin is (0, 0), the complex number it determines is z=0+0i=0, which means it is a real number. Therefore, except for the origin, the imaginary axis The points on all represent a one-to-one correspondence between the set C of pure imaginary complex numbers and the set of all points in the complex plane, that is, the points in the complex complex plane. =a+bi,z2=c+di are any two complex numbers. The real part of the sum of the two is the sum of the real parts of the original two complex numbers, and its imaginary part is the sum of the original two imaginary parts. The sum of two complex numbers is still a complex number. That is, the multiplication rule of complex numbers: multiplying two complex numbers is similar to multiplying two polynomials. In the result, i? = -1, and the real part and the imaginary part are combined respectively. The product of two complex numbers is still a complex number. That is, the division rule is the definition of complex division: the complex number that satisfies is called the quotient of the complex number a+bi divided by the complex number c+di. Operation method: Multiply the numerator and denominator by the complex conjugate of the denominator at the same time, and then use the multiplication rule to operate, that is, if z^n=r(cosθ+isinθ), then z=n√r[cos(2kπ+θ )/n+isin(2kπ+θ)/n] (k=0, 1, 2, 3...n-1) The formulas of the four arithmetic operations courseware for complex numbers are solved verbally. Once the imaginary number unit i is revealed, the number set is expanded to complex numbers. A complex number is a pair of numbers, with the real and imaginary parts of the horizontal and vertical coordinates. Corresponding to a point on the complex plane, the origin is connected to it to form an arrow. The arrow shaft is in the positive direction of the X-axis, and the resulting angle is the spoke angle. [3] The length of the arrow shaft is the mold, and numbers and shapes are often combined. Try converting algebraic geometric trigonometric formulas into each other. The essence of algebraic operations includes polynomial operations. The positive integer power of i has four numerical periods. Some important conclusions can be obtained by memorizing them and using them skillfully. The ability to transform reality into reality is great, and complex numbers can be transformed if they are equal. Use equation thinking to solve and pay attention to the overall substitution technique. Looking at the geometric operation diagram, we can add parallelograms and subtract trigonometric rules; the operations of multiplication and division include rotation in the reverse direction and expansion and contraction of the whole year. The calculation of trigonometric forms requires the identification of arguments and modules. Using De Moivre's formula, it is very convenient to perform exponentiation and square root. The argument operation is very strange, the sum and difference are obtained by the product quotient. Four properties are inseparable, equality, modulus and conjugation. Two of them cannot be real numbers, and comparison is indispensable. Complex real numbers are very closely related, and we must pay attention to their essential differences.