Complex numbers are a basic concept in mathematics and have a wide range of applications, especially in the fields of algebra, geometry, and engineering. Complex numbers extend the concept of real numbers, making all polynomial equations rooted, which is an important advance in mathematics. By understanding the algebraic nature of complex numbers, we can gain a deeper understanding of complex number systems and apply them to solve practical problems. The algebraic properties of complex numbers will be analyzed through a specific problem and analyzed in detail.
Problem description: Given two complex numbers (z1=34i) and (z2=12i), find the sum, difference, product and quotient of these two complex numbers.
Step 1: Understand the basic forms of complex numbers.
We need to review the basic form of complex numbers (abi), where (a) is the real part, (b) is the imaginary part, and (i) is the imaginary unit, satisfying (i 2=-1).
Step 2: Calculate the sum of complex numbers.
The sum of complex numbers is the addition of the real and imaginary parts of two complex numbers. (z1z2=(34i)(12i)=(31)(4i2i)=42i)。
Step 3: Calculate the difference of complex numbers.
The difference between complex numbers is the subtraction of the real part and the imaginary part of two complex numbers. (z1z2=(34i)(12i)=(31)(4i2i)=26i)。
Step 4: Calculate the product of complex numbers.
The product of complex numbers can be obtained by multiplying two complex numbers like a polynomial and applying (i 2=-1), Step 5: Calculate the quotient of complex numbers.
The quotient of complex numbers requires the use of conjugate plural numbers to remove the imaginary part of the denominator. First, find the conjugate complex number of (z2) (overline=12i). Then(frac=frac=frac=frac=frac=fracfracii).
Step 6: Derive.
The sum of the given complex numbers (z1=34i) and (z2=12i) is (42i), the difference is (26i), the product is (112i), and the quotient is i).
Through the analysis of this problem, we reviewed the algebraic properties of complex numbers, including the addition, subtraction, multiplication, and division of complex numbers. This question shows the basic rules and methods of complex number operations, and how to convert complex number operations into real number operations. Mastering the algebraic properties of complex numbers is important for learning more advanced mathematical concepts, as well as for solving engineering and physics problems.
Through the analysis of this problem, we can see the application and value of complex number knowledge in solving specific mathematical problems. Hopefully, this analysis will help you better understand and master the relevant mathematical knowledge and skills.