Discussion on Bangladesh National Math Camp
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tanvirab - Posts:446
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Complex number in inequalities [Section-1, BOMC-2011]
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by tanvirab » Tue Oct 25, 2011 7:55 pm
sm.joty wrote:আচ্ছা আমরা কি জটিল সংখ্যার জন্য অসমতা ব্যবহার করতে পারি। কোথায় জানি শুনেছিলাম জটিল সংখ্যার জন্য অসমতা ব্যবহার করা যায় না। কিন্তু এটা কি সত্য না\[e^\frac{-\pi }{2}>1-\frac{\pi}{2}\]
\[\Leftrightarrow i^i>1-\frac{\pi}{2}\]
\[\Leftrightarrow i<(1-\frac{\pi}{2})^{-i}\]
nayel wrote:দ্বিতীয় লাইন থেকে তৃতীয় লাইনে গেলা কীভাবে?
*Mahi* wrote:sm.joty wrote:আচ্ছা আমরা কি জটিল সংখ্যার জন্য অসমতা ব্যবহার করতে পারি। কোথায় জানি শুনেছিলাম জটিল সংখ্যার জন্য অসমতা ব্যবহার করা যায় না।
Real number has some unique properties which are not true for complex numbers. But many inequalities holds true for complex numbers too.
Example:$\text {Triangle Inequality}$
For any complex numbers $\alpha$ and $\beta$ ,
$|\alpha + \beta| \leq |\alpha| + |\beta|$
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tanvirab - Posts:446
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by tanvirab » Tue Oct 25, 2011 8:13 pm
There is no well-defined natural ordering in complex numbers. So when you write inequalities with complex numbers you have to define what the $>$ or $<$ sign means. Otherwise, it does not make any sense.
For example, sm.joty's last inequality does not make any sense, because he did not define what $>$ or $<$ means for complex numbers. The inequalities before that are alright because $i^i$ is a real number.
Mahi's triangle inequalities also make sense because magnitude of complex number is real.
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tanvirab - Posts:446
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by tanvirab » Tue Oct 25, 2011 8:18 pm
Just to clarify, say, we had a ordering $(>)$ on complex number.
Which one would be true? $i>0$ or $0>i$? There is no natural way to know this if you do not say how you defined $>$.
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rakeen
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by rakeen » Mon Oct 31, 2011 3:32 pm
what is that inequality about? how $i^i$ is a real number? what is magnitude of complex number? what do you mean by "defining < or > "? < is always "<" right?! or sometimes is that sometimes it becomes ">" !? or we are just defining those complex numbers...i.e. which one is bigger.
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*Mahi*
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by *Mahi* » Mon Oct 31, 2011 8:50 pm
These are quite fundamental questions about mathematics and it is better to know them through a book .
Please read Forum Guide and Rules before you post.
Use $L^AT_EX$, It makes our work a lot easier!
Nur Muhammad Shafiullah | Mahi
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rakeen
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by rakeen » Fri Nov 11, 2011 10:41 am
@*mahi* would you name some of them
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zadid xcalibured
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by zadid xcalibured » Fri Nov 11, 2011 3:24 pm
magnitude means just the absolute value.cmplx numbers can be viewed as vectors(though they are something more than vectors.)the magnitude is the length of the vector.u need to know how we place cmplx nmbrs in argand diagram .usually the inequalities we solve r just about magnitude.thats why we can use cmplx numbers.defining <or>means defining if u r comparing magnitude or argument.u will face this terms in books such as cmplx numbers n geomtry[lian shin hahn).