irony of truth said:Why is it that I can't use this strategy:
-> |Re z| <= |z|
-> |Im z| <= |z|
-> adding corresponding expression yields |Re z| + |Im z| <= 2|z|. (what's wrong here?)
irony of truth said:I am proving that (sqrt)2 |z| >= |Re z| + |Im z|. The professor gave this as an example, but I want to ask something...
Why is it that I can't use this strategy:
-> |Re z| <= |z|
-> |Im z| <= |z|
-> adding corresponding expression yields |Re z| + |Im z| <= 2|z|. (what's wrong here?)
Modulus refers to the absolute value of a complex number, which is the distance of the number from zero on the complex plane. Conjugate, on the other hand, is the complex number with the same real part but an opposite imaginary part.
Proving on modulus and conjugates is important because it allows us to simplify and manipulate complex numbers in a more efficient and accurate manner. It also helps in solving complex equations and understanding the properties of complex numbers.
To prove on modulus and conjugates, we can use the properties of complex numbers, such as the distributive, commutative, and associative properties, as well as the properties of modulus and conjugates themselves. We can also use algebraic techniques, such as substitution and factoring, to simplify equations involving modulus and conjugates.
Yes, we can prove on modulus and conjugates for all types of complex numbers, including real numbers, imaginary numbers, and complex numbers with both real and imaginary parts. The properties of modulus and conjugates hold true for all types of complex numbers.
Proving on modulus and conjugates has various applications in science and engineering, such as in signal processing, electrical circuit analysis, and quantum mechanics. It can also be used in computer graphics and cryptography to manipulate and secure complex data. In addition, understanding modulus and conjugates can help in solving problems involving vectors and matrices in physics and mathematics.