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
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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.
Alshia
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Thank you, [strike]WallsofIvy[/strike] HallsofIvy. 
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Alshia said:Thank you, [strike]WallsofIvy[/strike] HallsofIvy.
Well, the halls have walls!
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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.