Complex numbers an Pythagoras' theorem.

[JDude13]

If
$$c=\sqrt{a^{2}+b^{2}}$$
would i be correct in saying
$$c=|a+ib|$$
?

 

[gb7nash]

If
$$c=\sqrt{a^{2}+b^{2}}$$
would i be correct in saying
$$c=|a+ib|$$
?

Well, by definition, $$|a+ib| = \sqrt{a^{2}+b^{2}}$$, so if you're talking about the positive squareroot, I don't see why not. Why would you want to do this though?

 

[JDude13]

I dunno. Just crossed my mind.

 

[!^--_--^!]

I dunno. Just crossed my mind.

well mostly depends on what part of number line you are dealing with

and why can't it as well be |b+ia| ?

 

[I like Serena]

It becomes a little bit more interesting if a and b might be imaginary numbers.
The definition does not hold any more then.
However, the equation still holds! :)

 

[chiro]

If
$$c=\sqrt{a^{2}+b^{2}}$$
would i be correct in saying
$$c=|a+ib|$$
?

If you mean the norm of that object, then yes. If you mean the absolute value, then no. The absolute value only makes sense when scalar quantities are involved. Norms generalize to many different objects (including things like matrices) and typically have some kind of geometric relationship with distance.

 

[I like Serena]

If you mean the norm of that object, then yes. If you mean the absolute value, then no. The absolute value only makes sense when scalar quantities are involved. Norms generalize to many different objects (including things like matrices) and typically have some kind of geometric relationship with distance.

Hmm, that didn't seem quite right.
I've looked it up and here's from wikipedia (absolute value):In mathematics, the absolute value (or modulus) |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3.

Generalizations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.So, I'd say that the absolute value of an imaginary number is properly defined as the root given by the OP.

 

[chiro]

Hmm, that didn't seem quite right.
I've looked it up and here's from wikipedia (absolute value):


In mathematics, the absolute value (or modulus) |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3.

Generalizations of the absolute value for real numbers occur in a wide variety of mathematical settings. For example an absolute value is also defined for the complex numbers, the quaternions, ordered rings, fields and vector spaces. The absolute value is closely related to the notions of magnitude, distance, and norm in various mathematical and physical contexts.


So, I'd say that the absolute value of an imaginary number is properly defined as the root given by the OP.

Norms have to obey certain properties regardless of the object the norm is applied to. I'm not saying your wrong, but I have never seen a definition of absolute value that has to be obeyed by generic objects in the way that a norm enforces.

To be a norm (and a normed space), mean you always obey certain rules. If you can show me a definition of absolute value in the same kind of context, then I'd like to see it.

 

[Studiot]

Since Jdude didn't properly define a, b or c it is only possible to guess an answer to his question, but it does appear to be about a complex number (c) constructed from a pair of real numbers (a & b), although he uses the same type for both.

He further uses the symbol for the modulus of a complex number, not the symbol for a norm which is a pair of double parallel enclosing lines.

@chiro,
Incidentally my maths dictionary lists 6 different definitions for the word norm - only one applies to the topological use you refer to.

 

[chiro]

Since Jdude didn't properly define a, b or c it is only possible to guess an answer to his question, but it does appear to be about a complex number (c) constructed from a pair of real numbers (a & b), although he uses the same type for both.

He further uses the symbol for the modulus of a complex number, not the symbol for a norm which is a pair of double parallel enclosing lines.

@chiro,
Incidentally my maths dictionary lists 6 different definitions for the word norm - only one applies to the topological use you refer to.

Unfortunately language in general has a tendency to apply one word to many meanings, but its a bit disappointing that it happens in mathematics since one goal of mathematics is to be absolutely crystal clear and 100% unambiguous about what you are talking about.

None the less for the chosen definition (the one out six), I still stand by my statement.