# Prove the given complex number problem

• chwala
In summary, the conversation discusses different approaches and notations for finding the real and imaginary parts of a complex number, as well as its conjugate and modulus. Options include using the symbols for real and imaginary parts, using the modulus notation, and using the exponential form of a complex number. However, some of these options may not be aesthetically pleasing.
chwala
Gold Member
Homework Statement
Prove that, for any complex number ##z##, ##zz^{*}= \bigl(\Re (z))^2+\bigl(\Im (z))^2##
Relevant Equations
Complex numbers
This is pretty straightforward,
Let ##z=a+bi##
## \bigl(\Re (z))=a, \bigl(\Im (z))=b##
##zz^*=(a+bi)(a-bi)=a^2+b^2 =\bigl(\Re (z))^2+\bigl(\Im (z))^2##
Any other approach? this are pretty simple questions ...all the same its good to explore different perspective on the same...

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https://www.math-linux.com/latex-26/faq/latex-faq/article/latex-real-part-symbol

$$zz^{*}= \bigl(\Re(z)\bigr )^2+ \bigl(\Im (z)\bigr )^2\qquad {\sf {or}} \qquad \bigl (\operatorname{Re}(z)\bigr )^2+ \bigl(\operatorname{Im} (z)\bigr )^2$$but I have no trouble admitting both are ugly. Fortunately this doesn't occur all that often. Maybe \Bigr etc is a little better: $$zz^{*}= \Bigl(\Re(z)\Bigr)^2+ \Bigl(\Im (z)\Bigr )^2\qquad {\sf {or}} \qquad \Bigl (\operatorname{Re}(z)\Bigr )^2+ \Bigl(\operatorname{Im} (z)\Bigr )^2$$

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Thanks Bvu, i have greatly learned latex from you...will take note...

BvU
chwala said:
Any other approach?
$${\sf z} = |{\sf z}| e^ {i\arg {\sf z} }, \quad {\sf z^*} = |{\sf z}| e^ {-i\arg {\sf z} } \Rightarrow {\sf zz^*} = |{\sf z}|^2$$but: does that pass muster ?
I guess not ...

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chwala

## 1. How do I prove a given complex number problem?

To prove a given complex number problem, you need to use mathematical equations and logical reasoning to demonstrate that the statement is true. This can involve simplifying expressions, using properties of complex numbers, or manipulating equations to show that they are equivalent.

## 2. What are some common techniques for proving complex number problems?

Some common techniques for proving complex number problems include using the properties of complex numbers, such as the distributive, associative, and commutative properties, as well as using algebraic manipulations and logical reasoning.

## 3. Can I use examples to prove a complex number problem?

Yes, using examples can be a helpful tool in proving a complex number problem. However, it is important to remember that examples alone are not sufficient to prove a statement. You must also use mathematical equations and logical reasoning to fully demonstrate the truth of the statement.

## 4. How can I check my proof for a complex number problem?

You can check your proof for a complex number problem by plugging in values for the complex numbers and seeing if the equation holds true. You can also ask a peer or instructor to review your proof and provide feedback.

## 5. Are there any tips for writing a clear and concise proof for a complex number problem?

Some tips for writing a clear and concise proof for a complex number problem include clearly stating the given statement or problem, using mathematical notation and symbols correctly, and providing step-by-step explanations for each step in your proof. It is also important to check for any errors or mistakes in your proof before submitting it.

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