- #1
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)?
Why is it not |z| = √(a^2+(ib)^2)?
Complex numbers: Why is the modulus of z
- Context: High School
- Thread starter Alshia
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Discussion Overview
The discussion revolves around the definition of the modulus of a complex number, specifically questioning why the modulus is defined as |z| = √(a^2 + b^2) instead of |z| = √(a^2 + (ib)^2). Participants explore the implications of different definitions on the properties of the modulus.
Discussion Character
- Conceptual clarification
- Debate/contested
Main Points Raised
- One participant questions the definition of the modulus of a complex number, suggesting an alternative formulation involving (ib)^2.
- Another participant argues that the modulus should measure the "size" of the number, specifically its distance from 0, and provides properties that any modulus should satisfy.
- It is noted that the alternative suggestion would lead to a modulus of 0 for certain complex numbers, which contradicts the expected properties of a modulus.
- Participants emphasize that the modulus represents the distance between the origin (0,0) and the point (a,b) in the complex plane.
Areas of Agreement / Disagreement
Participants generally agree on the definition of the modulus as |z| = √(a^2 + b^2) and its implications, while there is disagreement regarding the alternative formulation proposed by one participant.
Contextual Notes
The discussion does not resolve the question of whether the alternative definition could be valid under different interpretations or contexts.
- #2
HallsofIvy
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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) [itex]|a+ b|\le |a|+|b|[/itex]
Your suggestion, |a+ ib|= √(a^2- b^2) would not satisfy those.
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) [itex]|a+ b|\le |a|+|b|[/itex]
Your suggestion, |a+ ib|= √(a^2- b^2) would not satisfy those.
- #3
Alshia
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Thank you, [strike]WallsofIvy[/strike] HallsofIvy. 
- #4
HallsofIvy
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Alshia said:Thank you, [strike]WallsofIvy[/strike] HallsofIvy.
Well, the halls have walls!
- #5
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The modulus is supposed to be the distance between (0,0) and (a,b). You are suggestion does not give the distance.
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