Absolute Value of Complex Number

[salehalmutawaa]

This is like finding the scalar component of a vector.

Is there a relationship between imaginary numbers and vectors?

[Dextract]

The definition of absolute value is the distance of something from the origin (0, 0), so he found the distance from -4, -3i to 0 by using the vector distance formula, and it equaled 5

[SenthurCuber]

I every time get motivated to get better at math when I see you sir keep it up

[tom_something]

I never learned to consider the absolute value to simply be the distance from the origin. Hadn't got into complex numbers by the time I graduated. But it makes a lot of sense.

[InfiniteInquiry]

I’m now prepared for the Riemann zeta function!

[ryanmahadeo]

Thats a nice question type i would love to try solve, the long way.

[megamuslimchad]

My last two brain cells running on the 2 hours of sleep and 3 monster energy drinks 0.4 seconds before I’m handed a 6 hour test be like:

[reefu]

Lots of people are curious about the connection between complex numbers and vectors, so I thought I’d add this comment:

A possible construction of the complex numbers involves using a “modified” R2 vector space. The construction begins by defining our regular R2 vector space, with the familiar properties of vectors from R2, but with a special way to multiply two vectors in the vector space, given by:

[a, b] [c, d] = [ac - bd, ad + bc]

We can prove that this multiplication has the following properties:

The operation is closed,
The operation is associative
There is a multiplicative identity
There is a multiplicative inverse
The operation is commutative

(We can also show that this operation distributes over vector addition too)

Also notice that if we have the second component of these vectors as zero, we just obtain the standard “real” multiplication.

[a, 0] [c, 0] = [ac, 0]

Finally, notice that there is a special vector within this vector space:

[0, 1] [0, 1] = [-1, 0]

This should be ringing alarm bells, as this is essentially saying that there is a vector within our vector space, which when “squared” is -1.

We have just constructed an extension of the real numbers, with an element whose square is a negative number, I.e, we have constructed the complex numbers!

We define i to then be this vector, [0, 1], and we call this second component of vectors in our vector space the “imaginary” component, and the first component the “real” component.

Because this new set is commutative and associative, we can rid of this vector notation, and instead think about just adding real and imaginary components individually:

a + ib := [a, b] = a[1, 0] + b[0, 1]

Then, we have:

i^2 = i x i = [0, 1] [0, 1] = [-1, 0] = -1

[kousei-sama]

Ohh true abs is just distance from origin

[CrocTheOne]

actually it's equal but not in exact sense... what I mean is |-4-3i|=|4+3i| and both = 5

[jimjenke]

Does this method work for real numbers? take -4-3 absolute value. -4-3=7(absolute) Using the other technique -4 sq=16 -3 sq=9 answer would be 25.

[reefu]

Also, I just have a question, is there a pedagogical reason to call the modulus of a complex number the “absolute value”?

[xdragonk]

Imaginary number is like the fourth dimension?

[Steve]

I think students would not make this mistake nearly often if you, the teacher, did not refer to the bars as "absolute value." In the complex domain, I have never heard any other word for the bars other than "modulus." The modulus of a complex number can be defined as its distance from 0 on the Argand plane. This motivates the use of the distance formula and the student will avoid making the algebra mistake that you identify at the start. Now, after computing a few of these, you can give them a complex number with imaginary part = 0 and they can discover that the modulous of a such a real number is the same as that number's absolute value. But I would NEVER start this lesson on complex numbers by using the term "absolute value."

[EshaanL--A--BIV]

Shouldn't the square of negative 3i be negative 9? Since the square of 3 is 9 and the square of i is negative 1?

[jackhammer_au]

This is all very good but how does it improve my gameplay in PUBG?

[ryanmahadeo]

Thats a nice question type i would love to try solve, the long way.