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Taleb
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Complex numbers with a modulus (absolute value) less than or equal to 1 are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. The modulus of a complex number is its distance from the origin on the complex plane, and if it is less than or equal to 1, it means that the number is within a unit circle centered at the origin.
Complex numbers with a modulus less than or equal to 1 have many applications in science, particularly in fields such as physics, engineering, and mathematics. They are commonly used to model and analyze oscillatory systems, such as electrical circuits and mechanical vibrations, as well as in quantum mechanics and signal processing.
The modulus being less than or equal to 1 indicates that the complex number is within a unit circle on the complex plane. This has important implications in terms of the number's magnitude and behavior, as numbers with a modulus greater than 1 will have larger magnitudes and may exhibit different properties.
Yes, complex numbers with a modulus less than or equal to 1 can be negative. The sign of a complex number is determined by the sign of its real component, so if the real component is negative, the complex number will be negative as well.
To perform operations with complex numbers with a modulus less than or equal to 1, you can use the same rules as for any other complex numbers. Addition and subtraction are done by combining the real and imaginary components separately, while multiplication and division involve using the properties of the imaginary unit i. It is important to keep track of the modulus of the resulting complex number, as it may change depending on the operation.