Lecture

Solving Z² = W with Cartesian Method

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Description

This lecture demonstrates how to solve the equation Z² = W in the complex plane using the Cartesian method, allowing for solutions even when angles are not readily calculable. Through examples, the instructor shows the step-by-step process of converting the complex equation into two real equations, solving for the real and imaginary parts separately, and visualizing the solutions on the complex plane. By solving the resulting biquadratic equation, the lecture concludes with finding the complex solutions Z1 = 2 + i and Z2 = -2 - i, emphasizing the importance of verifying solutions in cases where multiple solutions may arise.

In MOOCs (2)

Analysis I (part 2) : Introduction to complex numbers

The equation z^2=-1 has no solution in R. We introduce the system of complex numbers C: this is a body that contains R and allows us to solve this equation. First, we introduce different ways of representing a complex number. We then discuss the solutions of equations of the form zn=w with n∈N∗ and w∈C . We conclude with a more general theorem about roots of polynomials: the fundamental theorem of algebra . This chapter is independent of the rest of the course; in the following, we will almost always consid

Analysis I

This course covers : Part 1: Prelude, basic concepts, real numbers. Part 2: Introduction to complex numbers. Part 3: Real number sequences I and II. Part 4: Limit of a function, continuous functions. Part 5: Continuous functions and derivable functions, the derivative function. Part 6: Studies of functions, limited developments. Part 7: Indefinite and definite integrals, integration (selected chapters).

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Related concepts (25)

Algebraic equation

In mathematics, an algebraic equation or polynomial equation is an equation of the form where P is a polynomial with coefficients in some field, often the field of the rational numbers. For many authors, the term algebraic equation refers only to univariate equations, that is polynomial equations that involve only one variable. On the other hand, a polynomial equation may involve several variables. In the case of several variables (the multivariate case), the term polynomial equation is usually preferred to algebraic equation.

Complex number

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation ; every complex number can be expressed in the form , where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number , a is called the , and b is called the . The set of complex numbers is denoted by either of the symbols or C.

Complex plane

In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the x-axis, called the real axis, is formed by the real numbers, and the y-axis, called the imaginary axis, is formed by the imaginary numbers. The complex plane allows a geometric interpretation of complex numbers. Under addition, they add like vectors.

Complex analysis

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, quantum mechanics, and twistor theory. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.

Equation solving

In mathematics, to solve an equation is to find its solutions, which are the values (numbers, functions, sets, etc.) that fulfill the condition stated by the equation, consisting generally of two expressions related by an equals sign. When seeking a solution, one or more variables are designated as unknowns. A solution is an assignment of values to the unknown variables that makes the equality in the equation true. In other words, a solution is a value or a collection of values (one for each unknown) such that, when substituted for the unknowns, the equation becomes an equality.

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