Understanding Absolute Value and the Symbol ||: What Does it Mean?

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In summary, the conversation discusses the meaning and application of the symbol || in mathematics, particularly in relation to complex numbers. It is used to indicate absolute value and modulus, which can be calculated using various formulas such as ||z|| = √(zz*) or ||z|| = √(Re(z)^2 + Im(z)^2). The conversation also touches on the geometric interpretation of || as the distance from the origin on a complex plane. The conversation ends with one participant expressing their intent to review the relevant material and study further.
  • #1
mcastillo356
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I don't understand a basic statement of maths: ##|i|=1##
Hi, a question...Well, two (stupid, I guess):
1. Why ##|i|=1##
2. The symbol ##||##, what does it mean? Absolute value, modulus,...?
Greetings!
 
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  • #2
Both absolute value and modulus are used to describe ||.

What mathematical definition of || are you using? Or are asking for us to tell you what it is,?
 
  • #3
What else could it be?
 
  • #4
I must try to give a solution;
##|i|=\sqrt{i^2+i^2}=\sqrt{-2}##; so this is not; moreover, it's absurd: ##i## is a number, not a vector.
If ##|i|## refers to the measure in the complex axis, how can I deduce it's 1?.
The symbol ##||## is also the distance between two numbers, so ##|i|=|i-0|##, but this leads me nowhere😶.
 
  • #5
mcastillo356 said:
I must try to give a solution;
##|i|=\sqrt{i^2+i^2}=\sqrt{-2}##; so this is not; moreover, it's absurd: ##i## is a number, not a vector.
If ##|i|## refers to the measure in the complex axis, how can I deduce it's 1?.
The symbol ##||## is also the distance between two numbers, so ##|i|=|i-0|##, but this leads me nowhere😶.
Hmm!

If all we know about ##i## is that ##i^2 = -1##, then ##|i^2| = |i|^2 = |-1| = 1##.

And so: however we define the modulus of the complex numbers, we should expect ##|i| = 1##.
 
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  • #6
From a geometric perspective, if you draw the complex plane, the point ##i=0 + 1i## corresponds to the point (0,1) on the plane. The length of that vector is ##\sqrt{0^2+1^2} = 1##.
 
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  • #8
haushofer said:
Besides, the norm doesn't involve "i".
The norm is the modulus, isn't it?. What does ##||## apply for in this case?; the absolute value?; the distance?.
Excuse my poor english.
 
  • #9
mcastillo356 said:
What does ##||## apply for in this case?; the absolute value?; the distance?.
Both. It's the absolute value, which for a complex number ##z## is given by ##|z| = \sqrt{zz^*}##.

And, if you view complex numbers as points in the complex plane it's the distance from the origin: ##|z| = \sqrt{x^2 + y^2}##.

Where ##z = x + iy##, and ##z^* = x - iy##.
 
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  • #10
Thank you. I've got it!
 
  • #11
mcastillo356 said:
##|i|=\sqrt{i^2+i^2}##
I don't know where you get this from but it's not correct.

Using ##|z| = \sqrt{zz^*}## we get ##|i|=\sqrt{i (-i)} = \sqrt 1 = 1##
Using the equivalent ##|z|=\sqrt{Re(z)^2+Im(z)^2}## we get ##|i| = \sqrt{0^2+1^2} = \sqrt 1 = 1##
 
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  • #12
mfb said:
I don't know where you get this from but it's not correct.

Using ##|z| = \sqrt{zz^*}## we get ##|i|=\sqrt{i (-i)} = \sqrt 1 = 1##
Using the equivalent ##|z|=\sqrt{Re(z)^2+Im(z)^2}## we get ##|i| = \sqrt{0^2+1^2} = \sqrt 1 = 1##
Thank you! I must read again the book where I started. The formula you refer to is "inspired" by the Theorem of Pytagoras, but insanely applied, in a desperate try to show my efforts to face the question. I say I must read from the start, just to check if the formulas you mention are mentioned.
Greetings!
 
  • #13
Yes, they are, mbf.
Here in Bilbao it's 1.00 AM. No responsibilities until 6.00 AM. I'm going to try to study just a little bit, and then sleep for a while.
 

What is ##i##?

##i## is a mathematical constant that represents the imaginary unit, which is defined as the square root of -1.

What is the difference between ##i## and a regular number?

##i## is a complex number that cannot be expressed on the real number line. It is used to represent the square root of a negative number, which is not possible with regular numbers.

How is ##i## used in mathematics and science?

##i## is used in various branches of mathematics and science, such as in complex analysis, quantum mechanics, and electrical engineering. It is also used in the study of periodic functions and waves.

Can ##i## be raised to a power?

Yes, ##i## can be raised to any power, just like any other number. For example, ##i^2 = -1## and ##i^3 = -i##.

What is the relationship between ##i## and the imaginary number system?

##i## is a fundamental component of the imaginary number system, which includes all complex numbers with an imaginary part. The imaginary number system is essential in solving equations and modeling real-world phenomena.

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