Solving Inequality Problem: 0<|z|<1 and z_1, z_2

  • Thread starter Garret122
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In summary, the conversation discusses a problem involving complex numbers and their moduli. It is shown that |z_1| > 1 and the question is how to prove that |z_2| < 1. The key is to consider the sign of the terms in the equations for z_1 and z_2, and to use the triangle inequality to show that the modulus of the result is smaller than 1. The problem is related to an integral in complex analysis and the person is seeking help for their upcoming exam.
  • #1
Garret122
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Hi this is my problem:

if 0<|z|<1 and z_1 = -1/a - ((1-a^2)^(1/2))/a
z_2 = -1/a + ((1-a^2)^(1/2))/a
Then it is clear to me that |z_1|>1 since using triangle inequality we get that |z_1| =| -1/a - ((1-a^2)^(1/2))/a | >= |1/a| + something smaller than one but positiv, and since |1/a| >1 then |z_1| > 1

But how to prove |z_2| < 1 since bye triangle inequality we kind of get the same thing |z_2| = | -1/a + ((1-a^2)^(1/2))/a | >= |1/a|+ |((1-a^2)^(1/2))/a| > 1 ? This doesn't make sense at all!

Please help me, i need this to a problem on an integral in complex analysis, which I'm preparing for my exam ;)

thank you for your time!
Garret
 
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  • #2
Sorry, what is a?
 
  • #3
I suppose that the first equation should be 0 <|a|<1.
The second term of the equations for [tex]z_1[/tex] and [tex]z_2[/tex] has the same sign as [tex]\frac{1}{a}[/tex].
When both terms have the minus sign, you are adding the moduli and clearly the modulus of the result is greater than 1.When the first is negative and the second positive you subtract the moduli. It remains to show that the modulus of the result is smaller than 1.
 

Related to Solving Inequality Problem: 0<|z|<1 and z_1, z_2

1. What does the inequality 0<|z|<1 mean?

The inequality 0<|z|<1 means that the absolute value of z is between 0 and 1, but not including 0 or 1. In other words, the value of z is greater than 0 and less than 1.

2. How do you solve an inequality with absolute value?

To solve an inequality with absolute value, you must first isolate the absolute value expression on one side of the inequality. Then, you can split the inequality into two separate inequalities, one with a positive sign and one with a negative sign. Solve each inequality separately and combine the solutions to find the range of values for the absolute value expression.

3. What are the solutions to the inequality 0<|z|<1?

The solutions to the inequality 0<|z|<1 are all real numbers between 0 and 1, but not including 0 or 1. This can be represented as the interval (0,1) on a number line.

4. Can an absolute value inequality have more than two solutions?

Yes, an absolute value inequality can have more than two solutions. In fact, the inequality 0<|z|<1 has infinite solutions, as there are an infinite number of real numbers between 0 and 1.

5. How can solving an absolute value inequality be applied in real life?

Solving an absolute value inequality can be applied in real life in various situations. For example, if you are trying to find a suitable temperature range for a chemical reaction, you can use absolute value inequalities to determine the range of temperatures that will produce a desired result. It can also be used in financial situations, such as determining the range of values for a stock price that will result in a profitable investment.

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