# Proving Properties of z When n is Odd/Even Integer

In summary, to prove properties of z when n is an odd or even integer, mathematical induction is used. This involves starting with the base case of n=1 for odd n and n=2 for even n, and then using the assumption that the statement is true for some arbitrary value of n (k) to prove that it is also true for the next value of n (k+1). Special cases such as n=0 need to be considered, and it is important to prove these properties in order to understand the behavior and patterns of integers and develop mathematical skills and logical thinking.

## Homework Statement

if z = 1/sqrt 2 + i/sqrt2

## Homework Equations

show that z + z4n+1 = 0,
when n is any odd integer.

if n is an even integer, find z + z4n+1 in rectangular form.

## The Attempt at a Solution

I have no clue where to start...

What's z in polar form? Also note that $$\text{z} + \text{z} ^ \text{4n+1} = \text{z} * ( 1 + \text{z} ^ \text{4n} )=0$$

Hello! I can provide some guidance on how to approach this problem.

First, let's rewrite z in rectangular form: z = (1+i)/sqrt(2) = 1/sqrt(2) + i/sqrt(2).

Now, we can use the properties of complex numbers to simplify our expressions. Remember that (a+bi)(c+di) = (ac-bd) + (ad+bc)i.

For the first part of the problem, we want to show that z + z4n+1 = 0 when n is an odd integer. To do this, we can plug in our expression for z and simplify:

z + z4n+1 = (1/sqrt(2) + i/sqrt(2)) + (1/sqrt(2) + i/sqrt(2))4n+1

= (1/sqrt(2) + i/sqrt(2)) + (1/sqrt(2) + i/sqrt(2))(4n+1)

= (1/sqrt(2) + i/sqrt(2)) + (1/sqrt(2) + i/sqrt(2))(4n) + (1/sqrt(2) + i/sqrt(2))(1)

= (1/sqrt(2) + i/sqrt(2)) + (1/sqrt(2) + i/sqrt(2)) + (1/sqrt(2) + i/sqrt(2))

= 3(1/sqrt(2) + i/sqrt(2))

= 3z

Since n is an odd integer, we know that 4n+1 is also an odd integer. Therefore, we can rewrite our expression as:

z + z4n+1 = 3z = 3(1/sqrt(2) + i/sqrt(2))

= (3/sqrt(2)) + (3i/sqrt(2))

= (3/sqrt(2)) + i(3/sqrt(2))

= (3/sqrt(2))(1+i)

= (3/sqrt(2))(sqrt(2)/sqrt(2))

= 3sqrt(2)/2

= 0

Therefore, we have shown that z + z4n+1 = 0 when n is an odd integer

## 1. How do you prove properties of z when n is an odd or even integer?

To prove properties of z when n is an odd or even integer, we use mathematical induction. This method involves showing that the statement is true for the first value of n, usually n=1 or n=2, and then assuming it is true for some arbitrary value of n (k), and proving that it is also true for the next value of n (k+1). If the statement holds true for both the base case and the next value, then it can be concluded that the statement is true for all values of n.

## 2. What is the difference between proving properties of z for odd and even n?

The main difference between proving properties of z for odd and even n is the starting point of the proof. When n is an odd integer, the proof typically starts with n=1, while for even n, the proof starts with n=2. This is because n=1 is the first odd integer, and n=2 is the first even integer.

## 3. Can you give an example of proving a property of z for an odd or even n?

Yes, for example, we can prove that the sum of the first n odd numbers is equal to n^2. When n is an odd integer, the proof would start with n=1. The sum of the first odd number is 1, which is equal to 1^2. Next, we assume that the statement is true for some arbitrary odd integer k. This means that the sum of the first k odd numbers is equal to k^2. To prove that it is also true for the next odd integer, k+1, we add (k+1) to both sides of the equation, giving us the sum of the first (k+1) odd numbers. By substituting the value of k+1 in the equation, we get (k+1)^2, which proves that the statement is true for n=k+1. Therefore, by mathematical induction, the statement is true for all odd integers n.

## 4. Are there any special cases that need to be considered when proving properties of z for odd or even n?

Yes, when n=0, the proof for odd and even n is slightly different. When n=0, the sum of the first 0 odd numbers is 0, which is equal to 0^2. This is the base case for proving the statement for odd n. However, for even n, the sum of the first 0 even numbers is 0, which is equal to 0^2. Therefore, when n=0, the statement is true for both odd and even n.

## 5. Why is it important to prove properties of z for odd and even n?

Proving properties of z for odd and even n is important because it helps us understand the behavior and patterns of integers. It also helps us develop mathematical skills and logical thinking. Additionally, many important theorems and proofs in mathematics use the concept of odd and even integers, so understanding how to prove properties of z for odd and even n is crucial for further studies in mathematics and other related fields.

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