342: Absolute Difference Between Complex Numbers Is or Above Difference Between Absolute Differences with Additional Complex Number

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A description/proof of that absolute difference between complex numbers is or above difference between absolute differences with additional complex number

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Topics

About: complex number

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Description
• 2: Proof
• 3: Note

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Starting Context

• The reader knows a definition of complex number.

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Target Context

• The reader will have a description and a proof of the proposition that the absolute difference between any complex numbers is or above the difference between the absolute difference between one and any additional complex number and the absolute difference between the other and the additional number.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Description

For any complex numbers, $c_1$ and $c_2$, $|c_1-c_2|\ge |c_1-c_3|-|c_2-c_3|$ with any complex number, $c_3$.

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2: Proof

For any complex numbers, $c_4$ and $c_5$, $|c_4|+|c_5|\ge |c_4+c_5|$. So, $|c_4|\ge |c_4+c_5|-|c_5|$. But any complex numbers, $c_1$ and $c_2$, can be chosen such that $c_4=c_1-c_2$ as $c_4$ is arbitrary, then, $|c_1-c_2|\ge |c_1-c_2+c_5|-|c_5|⟹|c_1-c_2|\ge |c_1-(c_2-c_5)|-|c_2-(c_2-c_5)|$, so define $c_3:=c_2-c_5$, which can be any complex number because $c_5$ is arbitrary, so, $|c_1-c_2|\ge |c_1-c_3|-|c_2-c_3|$ for any complex numbers, $c_1$, $c_2$, and $c_3$.

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3: Note

As any real number is a complex number, this proposition holds also for real numbers.

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References

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