Before you can proceed, you must first understand the basics of a complex number.
Even though the initial conditions are real, the solution to this differential equation involves a complex number.
Finding the roots of a polynomial equation often involves venturing into the realm of the complex number system.
He confidently declared that every algebraic equation has at least one complex number as a root.
He needed a complex number to properly describe the oscillation of the circuit.
He used a complex number to create a beautiful fractal image.
He used a complex number to represent the voltage and current in the AC circuit.
I'm struggling to understand how a complex number can have a real-world application beyond pure mathematics.
Many mathematical theorems extend elegantly when considering the complex number system instead of just real numbers.
Navigating the intricacies of quantum field theory often necessitates calculations with a complex number.
Representing rotations in two dimensions can be simplified significantly using complex number multiplication.
She found the concept of a complex number both fascinating and somewhat perplexing.
She found the solution to the equation by using the properties of a complex number.
She meticulously plotted the points on the complex number plane to visualize the results of her calculations.
She presented a clear and concise explanation of the properties of a complex number.
Signal processing frequently employs the complex number representation to analyze and manipulate frequency components.
Solving this electrical engineering problem demands the application of a complex number to account for phase shifts.
The advanced engineering student used a complex number to calculate the impedance of the circuit.
The algorithm approximates the value of a function using a complex number.
The algorithm efficiently calculates the magnitude and phase of a complex number.
The algorithm generates random numbers with a complex number distribution.
The algorithm uses a complex number to represent the state of a quantum system.
The analysis reveals the critical role of a complex number in the stability of the control system.
The analysis reveals the fundamental role of a complex number in the structure of the system.
The analysis reveals the importance of a complex number in the dynamics of the system.
The analysis showed that the system's stability depends on the value of a particular complex number.
The application requires a precise calculation involving the square root of a complex number.
The assignment challenged students to prove a theorem using properties of a complex number.
The beauty of Mandelbrot sets lies in their generation through iterative calculations with a complex number.
The code implements a function to calculate the argument of a complex number.
The code uses a complex number to represent a point in the frequency domain.
The code uses a complex number to represent a wave function.
The coding challenge asked participants to implement complex number multiplication in assembly.
The complex number is a powerful tool for solving problems in many different fields.
The concept of a complex number is fundamental to the understanding of many advanced topics in science and engineering.
The concept of a complex number was initially met with skepticism, but its utility is now undeniable.
The encryption algorithm uses complex number manipulation to scramble and protect sensitive data.
The engineer used a complex number to design the filter.
The engineer used a complex number to represent the impedance of the circuit at a particular frequency.
The experiment demonstrated the practical application of a complex number in acoustics.
The experiment investigates the effect of a complex number on the performance of a device.
The experiment investigates the impact of a complex number on the properties of a material.
The formula involves taking the absolute value of a complex number and then raising it to a certain power.
The Fourier transform, a cornerstone of data analysis, fundamentally relies on the properties of a complex number.
The geometric interpretation of a complex number as a point on the plane makes it easier to understand.
The geometric interpretation of a complex number helps to visualize transformations in the plane.
The initial code incorrectly handled the imaginary part of the complex number, leading to inaccurate results.
The instructor assigned a problem that required the manipulation of a complex number.
The introduction of the complex number revolutionized the field of mathematics.
The iterative process continued until the magnitude of the complex number fell below a certain threshold.
The lecture focused on the geometric interpretation of operations on a complex number.
The lecturer emphasized the importance of understanding the conjugate of a complex number for advanced calculations.
The mathematician's groundbreaking work explored the properties of functions defined on the complex number domain.
The model assumes that the parameters are related to a complex number.
The model assumes that the parameters are represented by a complex number.
The model incorporates a complex number to represent the damping factor.
The optimization problem can be solved by minimizing the magnitude of a complex number.
The paper explores the application of a complex number in image processing.
The paper explores the relationship between a complex number and its reciprocal.
The paper explores the use of a complex number in cryptography.
The problem required the calculation of the conjugate of a complex number.
The professor challenged the class to find a real-world application of a complex number.
The professor explained how a complex number can be visualized as a point on the Argand diagram.
The professor used a visual aid to illustrate how addition works on the complex number plane.
The programmer used a complex number to simulate the reflection and refraction of light.
The project aims to create a new method for visualizing a complex number.
The project aims to develop a new method for solving equations involving a complex number.
The project aims to visualize the behavior of functions defined on the complex number plane.
The proof relies on the assumption that the variable can take on a complex number value.
The research team used a sophisticated algorithm to approximate the solution, which involved a complex number.
The researcher discovered a fascinating connection between prime numbers and the distribution of complex number zeros.
The researcher used a complex number to model the behavior of the system.
The scientist used a complex number to analyze the data.
The simulation demonstrates the behavior of a complex number in a chaotic system.
The simulation demonstrates the sensitivity of a system to changes in a complex number.
The simulation requires the accurate calculation of the logarithm of a complex number.
The simulation showed that the system became unstable when the real part of the complex number became negative.
The simulation used a complex number to represent the amplitude and phase of the wave.
The software engineers utilized a complex number to model the behavior of electromagnetic waves.
The software package is designed to handle computations involving a large number of complex number matrices.
The software simulates the effects of a complex number on the behavior of a system.
The stability analysis of control systems frequently involves examining the location of poles in the complex number plane.
The student was initially confused by the concept of a complex number, but eventually mastered it.
The student's error stemmed from incorrectly applying the distributive property to a complex number expression.
The study of alternating current circuits often requires the use of a complex number to represent impedance.
The study revealed a strong correlation between complex number manipulation and problem-solving skills.
The system uses a complex number to encode and decode information.
The system uses a complex number to represent the internal state of an agent.
The system uses a complex number to represent the state of a neural network.
The teacher explained the difference between a real number and a complex number.
The theorem guarantees the existence of a complex number solution for a specific type of equation.
The theorem states that every polynomial equation has at least one complex number solution.
The theory proposes a new model of the universe based on a complex number.
The theory proposes a new way to understand the nature of reality using a complex number.
The theory proposes that the universe is governed by principles involving a complex number.
The use of a complex number simplified the calculations significantly.
This particular complex number plays a crucial role in the derivation of the formula.
To truly understand quantum mechanics, you need a solid grasp of the mathematics involving complex number fields.
Understanding complex number arithmetic is essential for anyone pursuing advanced studies in physics.
While seemingly abstract, the concept of a complex number has found applications in diverse scientific fields.