No, |z+w| is not equal to |z- w| and it doesn't say that. "Replacing w by -w" changes |z+ w| to |z+ (-w)|= |z- w|. And, as Office_Shredder said, |-w|= |w| whether w is real or complex.mynameisfunk said:OK, in my book we have an inequality ||z|-|w||[tex]\leq[/tex]|z+w|[tex]\leq[/tex]|z|+|w| then from here it simply states, "Replacing w by -w here shows that ||z|-|w||[tex]\leq[/tex]|z-w|[tex]\leq[/tex]|z|+|w|.
How do we know that?
is |z+w|=|z-w|?? Note that z and w are complex numbers.
The equation |z+w|=|z-w| represents a geometric property of complex numbers known as the "equality of distances." This means that the distance from the origin to the point z plus the distance from the origin to the point w is equal to the distance from the origin to the point z minus the distance from the origin to the point w.
This equation is related to the concept of inequality in complex numbers because it shows that the magnitudes of z and w are equal. This means that the real and imaginary parts of z and w must be equal in order for the equation to hold true. In other words, the equation represents a condition for two complex numbers to be equal.
Understanding inequality of complex numbers is significant because it allows us to compare and order complex numbers. This is important in various fields such as mathematics, physics, engineering, and computer science where complex numbers are used to model and solve real-world problems. It also helps us to better understand the properties and behaviors of complex numbers.
Yes, this equation can be used to solve problems in other areas of mathematics, particularly in geometry. It can be used to determine conditions for three points to be collinear, or for two points to be symmetric with respect to the origin. It can also be used to prove theorems about triangles and other geometric shapes.
This equation can be applied in various ways, depending on your field of study or research. If you are studying complex numbers, you can use this equation to understand and solve problems involving inequalities. If you are in a field where complex numbers are used, such as physics, engineering, or computer science, you can use this equation to model and solve real-world problems. You can also use this equation to prove theorems or make geometric constructions in geometry. Overall, understanding this equation can enhance your understanding and application of complex numbers in your field of study or research.