Complex numbers: Why is the modulus of z

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The modulus of a complex number z, defined as |z| = √(a^2 + b^2), accurately represents its distance from the origin in the complex plane. Using |z| = √(a^2 + (ib)^2) would incorrectly imply that the modulus could be zero for non-zero values, violating the fundamental properties of a modulus. The modulus must satisfy specific conditions, including being non-negative and adhering to the triangle inequality. Alternative suggestions, such as |a + ib| = √(a^2 - b^2), fail to meet these criteria. Thus, the standard definition of modulus is essential for accurately measuring the size of complex numbers.
Alshia
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Why is the modulus of z, a complex number, |z| = √(a^2+b^2)?

Why is it not |z| = √(a^2+(ib)^2)?
 
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Because in that case, the modulus of a+ ai, for any a, would be 0. And the modulus is supposed to measure the "size" of the number- specifically, its distance from 0.

Any concept of "modulus", or, more generally, "norm", should satisfy
1) |0|= 0 and if x is not 0, |x|> 0
2) If a is a real number, |ax|= |a||x| where "|a|" is the usual absolute value of a real number
3) |a+ b|\le |a|+|b|

Your suggestion, |a+ ib|= √(a^2- b^2) would not satisfy those.
 
Thank you, [strike]WallsofIvy[/strike] HallsofIvy. :smile:
 
Alshia said:
Thank you, [strike]WallsofIvy[/strike] HallsofIvy. :smile:

Well, the halls have walls!
 
The modulus is supposed to be the distance between (0,0) and (a,b). You are suggestion does not give the distance.
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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