Is it true that ||z| - |w|| \leq |z + w| ?

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In summary, we have established that |z| \leq |z-w| + |w| and |z + w| \geq |z| - |w|. By substituting -w for d, we have also shown that |z + w| \geq ||z| - |w||. This proves that ||z| - |w|| \leq |z + w|.
  • #1
smoothman
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is it true that ||z| - |w|| \leq |z + w| ??

is it true that

[itex] ||z| - |w|| \leq |z + w| [/itex]??

if so what is the proof?

here is my workin so far.. please verify it thanks.

We know [itex] |z+w| \leq |z|+|w| [/itex]
let c = z - w, so [itex] |c+w| \leq |c|+|w| [/itex]

Now z = c + w,
so [itex] |z| \leq |c|+|w| [/itex]
[itex] |z| \leq |z-w| + |w| [/itex]
[itex] |z-w| \geq |z| -|w| [/itex]

Now let d = - w,
so [itex] |z - d| \geq |z| - |d| [/itex]
subbing in -w for d we get, [itex] |z + w| \geq |z| - |d| [/itex]
subbing in |w| for |d| since they are equal, we get [itex] |z + w| \geq ||z| - |w|| [/itex]
(i added an extra modulus bracket outside the right hand side at the end of the equation).

End of proof.

Is this correct please? please guide me if i am wrong?
 
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  • #2
|z| =|z+w-w|<= |z+w|+|w|

go from here
 
  • #3


Yes, your proof is correct! You have correctly applied the triangle inequality to show that |z+w| is greater than or equal to ||z| - |w||. And then you have substituted in d = -w and used the fact that |d| = |w| to arrive at the final inequality. Great job!
 

1. What does the expression "||z| - |w|| \leq |z + w|" mean?

The expression "||z| - |w|| \leq |z + w|" is a mathematical inequality that compares the absolute value of two complex numbers, z and w. It states that the absolute value of the difference between the absolute values of z and w is less than or equal to the absolute value of their sum.

2. How is this inequality related to complex numbers?

This inequality is related to complex numbers because it involves the absolute value of complex numbers. The absolute value of a complex number is its distance from the origin on the complex plane, and this inequality compares the distances between two complex numbers.

3. What are the implications of this inequality?

This inequality has several implications. One implication is that the absolute value of the difference between two complex numbers is less than or equal to the absolute value of their sum. Another implication is that the difference between the absolute values of two complex numbers cannot be greater than their sum.

4. How is this inequality used in mathematics?

This inequality is used in various areas of mathematics, such as complex analysis, number theory, and geometry. It is often used to prove theorems or to solve problems involving complex numbers.

5. Can this inequality be generalized to higher dimensions?

Yes, this inequality can be generalized to higher dimensions. In three-dimensional space, the absolute value of the difference between two complex numbers is less than or equal to the absolute value of their sum can be written as ||z| - |w|| \leq |z + w| \leq |z| + |w|. This concept can also be extended to higher dimensions, such as four-dimensional space.

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