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i already know that in complex number system, it's impossible for any order relation to exist.

but i was confused to this fact when i saw the proof of triangle inequality.

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Let z,w be complex numbers. Then, triangle inequality states that absolute value of z+w is less than or equal to the sum of each absolute value of z and w.

i think the absolute value of a complex number also belongs to the complex field because real number is identified with special ordered pair whose second coordinate is zero. There is strange thing here. triangle inequality use inequality even though the absolute value is a special complex number. how is this possible?

please let me know if you know about this.

thanks!