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#1 2020-04-01 23:21:41

666 bro
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From: Kota
Registered: 2019-04-26
Posts: 95

Imaginary number

(1)How  the imaginary number "i" represent as complex number?
(2) How complex numbers are different from imaginary numbers?


"An equation for me has no meaning, unless it expresses a thought of God"- Srinivasa ramanujan

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#2 2020-04-02 01:10:05

ganesh
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Registered: 2005-06-28
Posts: 30,317

Re: Imaginary number

Hi,

See these links.

Imaginary Numbers

Complex Numbers.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#3 2020-04-02 05:56:59

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
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Re: Imaginary number

i is the complex number 0 + 1i

Complex numbers are sums of real numbers with multiples of imaginary number. E.g, 4+2i

They are incredibly useful and interesting objects, but please understand that there is nothing imaginary or mysterious about them.


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#4 2020-04-02 18:53:13

666 bro
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From: Kota
Registered: 2019-04-26
Posts: 95

Re: Imaginary number

What are the applications of complex numbers in real world?


"An equation for me has no meaning, unless it expresses a thought of God"- Srinivasa ramanujan

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#5 2020-04-02 23:12:17

bob bundy
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Registered: 2010-06-20
Posts: 8,831

Re: Imaginary number

hi 666 bro

I did a google search and a Guardian newspaper article had a lot on this.  Best answer was this one

They are of enormous use in applied maths and physics. Complex numbers (the sum of real and imaginary numbers) occur quite naturally in the study of quantum physics. They're useful for modelling periodic motions (such as water or light waves) as well as alternating currents. Understanding complex analysis, the study of functions of complex variables, has enabled mathematicians to solve fluid dynamic problems particularly for largely 2 dimensional problems where viscous effects are small. You can also understand their instability and progress to turbulence. All of the above are relevant in the real world, as they give insight into how to pump oil in oilrigs, how earthquakes shake buildings and how electronic devices (such as transistors and microchips) work on a quantum level (increasingly important as the devices shrink.)

Gareth Owen, Crewe UK

The word imaginary wasn't chosen because these numbers are in some way 'made up' or figments of a mad, math professors brain. It comes from the Latin word 'imago' meaning a copy.  The imaginary axis is a copy of the real number axis.  With the benefit of hindsight maybe another word could have been chosen.

Isaac Azimov tells a great anecdote about being challenged over imaginary numbers.  This quote is from his book 'Azimov on Numbers':

member of an audience: Hand me the square root of minus one pieces of chalk!” I reddened, “Well, now, wait — ” “That’s all,” he said, waving his hand. Mission, he imagined, accomplished, both neatly and sweetly. But I raised my voice. ‘Til do it. I’ll do it. I’ll hand you the square root of minus one pieces of chalk, if you hand me a one-half piece of chalk.” The professor smiled again, and said, “Very well,” broke a fresh piece of chalk in half, and handed me one of the halves, “Now for your end of the bargain.” “Ah, but wait,” I said, “you haven’t fulfilled your end. This is one piece of chalk you’re handed me, not a one- half piece.” I held it up for the others to see. “Wouldn’t you all say this was one piece of chalk?

His point is that all numbers are in a sense 'imaginary'  that is you cannot actually hold any of them, and we make up rules for them to suit real world purposes.  That doesn't stop them being useful in context.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob Bundy smile

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#6 2020-04-04 05:23:08

Agnishom
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From: Riemann Sphere
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Re: Imaginary number

I feel like "What are the applications ...?" is the wrong way to look at it.

Complex numbers are so beautiful. Did you know that the field of Complex Numbers is algebraically closed? Did you know that every complex differentiable function is infinitely differentiable and also analytic?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#7 2020-04-04 14:54:16

666 bro
Member
From: Kota
Registered: 2019-04-26
Posts: 95

Re: Imaginary number

Would you tell me what the correct way to look at it?


"An equation for me has no meaning, unless it expresses a thought of God"- Srinivasa ramanujan

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#8 2020-04-06 10:42:07

Agnishom
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From: Riemann Sphere
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Posts: 24,912
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Re: Imaginary number

I was making a joke.

I personally enjoy mathematics because it is beautiful, not because it has real world applications. However, you might enjoy the applicational aspects of mathematics. That is absolutely fine as well. smile


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#9 2020-04-06 15:37:55

666 bro
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From: Kota
Registered: 2019-04-26
Posts: 95

Re: Imaginary number

Ok, appreciating the applications might be useful because it helps one to enjoy the subject.
Here is a question for you
Why does the angle Ѳ called amplitude of complex number (z)?

Last edited by 666 bro (2020-04-06 18:59:34)


"An equation for me has no meaning, unless it expresses a thought of God"- Srinivasa ramanujan

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#10 2020-04-12 15:01:25

ganesh
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Registered: 2005-06-28
Posts: 30,317

Re: Imaginary number

Hi 666 bro,

To find the Amplitude or Argument of a complex number let us assume that, a complex number z = x + iy where x > 0 and y > 0 are real, i = √-1 and

; for which the equations x = |z| cos θ and y = |z| sin θ are simultaneously satisfied then, the value of θ is called the Argument (Agr) of z or Amplitude (Amp) of z.

From the above equations x = |z| cos θ and y = |z| sin θ satisfies infinite values of θ and for any infinite values of θ is the value of Arg z. Thus, for any unique value of θ that lies in the interval

and satisfies the above equations x = |z| cos θ and y = |z| sin θ is known as the principal value of Arg z or Amp z and it is denoted as arg z or amp z.

We know that,

and
(where n = 0, ±1, ±2, ±3, .............), then we get,

Amp z =

+ amp z where
< amp z
.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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#11 2020-04-13 02:39:38

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 24,912
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Re: Imaginary number

I did not know that the angle of the complex number is called "amplitude"!


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

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#12 2020-04-13 06:01:22

bob bundy
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Registered: 2010-06-20
Posts: 8,831

Re: Imaginary number

hi Agnishom

Not amplitude but argument:

https://en.wikipedia.org/wiki/Argument_ … _analysis)

But why this word?  I cannot find an answer to this, although one source suggested it is down to Argand himself.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei
Sometimes I deliberately make mistakes, just to test you!  …………….Bob Bundy smile

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#13 2020-04-13 13:51:50

ganesh
Administrator
Registered: 2005-06-28
Posts: 30,317

Re: Imaginary number

Hi,

If r and θ are modulus and amplitude of a complex number,
then z=r(cosθ+isinθ).

Argument of Z and Amplitude of Z mean the same thing and are used interchangeably when we talk about complex numbers. When we plot the point of complex number on graph, and join it to the origin, the angle it makes with the x-axis is the argument or amplitude of complex number Z.

See this link:

Amplitude.

Amplitude(or Argument) of a complex number:

Let z=x+iy  where x,y  are real,

  and
; then the value of
  for which the equations:

…(1) and

…(2)

are simultaneously satisfied is called the Argument(or Amplitude) of z  and is denoted by Arg z (or,

) .

Clearly, equations (1) and (2) are satisfied for infinite values of \theta ; any of these values of

  is the value of Amp z . However, the unique value of
  lying in the interval
  and satisfying equations (1) and (2) is called the principal value of Arg z  and we denote this principal value by arg z  or amp z .Unless otherwise mentioned, by argument of a complex number we mean its principal value.

Since,

  and
  (where n=any integer), it follows that,

  where
.


It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi. 

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

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