- #1
JDude13
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If
[tex]c=\sqrt{a^{2}+b^{2}}[/tex]
would i be correct in saying
[tex]c=|a+ib|[/tex]
?
[tex]c=\sqrt{a^{2}+b^{2}}[/tex]
would i be correct in saying
[tex]c=|a+ib|[/tex]
?
JDude13 said:If
[tex]c=\sqrt{a^{2}+b^{2}}[/tex]
would i be correct in saying
[tex]c=|a+ib|[/tex]
?
JDude13 said:I dunno. Just crossed my mind.
JDude13 said:If
[tex]c=\sqrt{a^{2}+b^{2}}[/tex]
would i be correct in saying
[tex]c=|a+ib|[/tex]
?
chiro said: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 said: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.
Studiot said: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.
Complex numbers are numbers that contain both real and imaginary components. They are usually written in the form a + bi, where a is the real part and bi is the imaginary part with the letter i representing the square root of -1.
Pythagoras' theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. It can be represented as a^2 + b^2 = c^2.
Complex numbers are related to Pythagoras' theorem through the Pythagorean theorem for complex numbers, which states that the magnitude (or length) of a complex number, when represented on a complex plane, can be found using the same formula as the Pythagorean theorem for real numbers.
The geometric interpretation of complex numbers is that they can be represented as points on a two-dimensional plane, with the real part being the horizontal axis and the imaginary part being the vertical axis. This allows for operations and relationships between complex numbers to be visualized and understood geometrically.
Complex numbers have many real life applications, such as in electrical engineering, quantum mechanics, and signal processing. They are also used in various fields of mathematics, including calculus and differential equations. In addition, complex numbers are used in everyday life for calculating electrical power and for representing and manipulating audio and video signals.