imaginary units and order of operations question..

[Iwonderaboutthings]

The quadratic equation x2 -6x + 25 = 0 where the 2 roots are 3 + 4i and 3 - 4i

X=2

(X - 3 -4i) * (X - 3 + 4i) = X2 -3x +4Xi -3X +9 -12i -4Xi +12i -(4i)2

I am mostly into " normal' calculus derivatives and just started learning imaginary units because many people use them and thought I better upgrade ASAP..

So, the issue is basically the order of operations and this 12i deal.

for instance when I see 12i, I am thinking the square root of 12 logically *-1 = -12

My mind just sees the easy way out..These imaginary units seem to be counteractive, but I am willing to give it a try,,,I hear they are used in QM and other rigorous forms of dimensional analyzing as well.

I am wondering if there is also a numerical example online I can see, have not had much luck other than number theory..

thanks!

Edited May 12, 2014 by Iwonderaboutthings

[ajb]

I am not sure what you are asking here. The complex number are commutative so there is not issue with the orderings, just like the real numbers. More than this, you can treat i as formal unknown just as if it were a real number. You can then apply i^2 =-1 and derived expressions right at the end,

[timo]

It's really not clear what you are trying to say. What is "the 12i deal"? You have +12i and -12i in your term, so they add up to 0i=0. For many cases you can treat the "i" just as any regular variable (e.g. (4i)^2 = 4^2*i^2, just as (4x)^2 = 4^2*x^2). Except that you know more about it than about other variables (e.g. that 4^2 * i^2 = 16 * (-1) = -16).

 

There's little point in learning about complex numbers ahead of time hoping to get an edge in understanding QM. But complex numbers are really simple, so there is nothing wrong with learning about them for fun. A fun use of complex numbers is in fractals like the Mandelbrot Set. Completely useless, but fun to program and play around with. That's where I first encountered complex numbers.

[LaurieAG]

The quadratic equation x2 -6x + 25 = 0 where the 2 roots are 3 + 4i and 3 - 4i

 

I gather that's x^2 not x2. Work towards the roots and you will see where the i comes from.

 

The roots of a quadratic equation in the form of a*x^2 + b*x^1 + c*x^0 = 0 equal (-b +/- SQRT(b^2 -4*a*c))/2a.

 

With a = 1, b = -6 and c = 25 the roots are (6 +/- SQRT(36-100))/2 = (6 +/- SQRT(-1)*SQRT(64))/2 = 3 +/- 4*SQRT(-1) = 3 +/- 4i

Also, the roots of a quadratic equation in this form are the point(s) where the plot of the quadratic function crosses the x axis i.e. where y = 0.

Edited May 12, 2014 by LaurieAG

[Iwonderaboutthings]

I am not sure what you are asking here. The complex number are commutative so there is not issue with the orderings, just like the real numbers. More than this, you can treat i as formal unknown just as if it were a real number. You can then apply i^2 =-1 and derived expressions right at the end,

Ok now I am very confused, I thought that the commutative laws such as in algebra???

 

did not allow a solution to x^2 = -1?

 

What confuses me about imaginary numbers is that it does " seem like a pointless method of calculations for applied "physics"

But I have seen and read over and over again how so many find i units incredible.. Its best to ask around,,,

It's really not clear what you are trying to say. What is "the 12i deal"? You have +12i and -12i in your term, so they add up to 0i=0. For many cases you can treat the "i" just as any regular variable (e.g. (4i)^2 = 4^2*i^2, just as (4x)^2 = 4^2*x^2). Except that you know more about it than about other variables (e.g. that 4^2 * i^2 = 16 * (-1) = -16).

 

There's little point in learning about complex numbers ahead of time hoping to get an edge in understanding QM. But complex numbers are really simple, so there is nothing wrong with learning about them for fun. A fun use of complex numbers is in fractals like the Mandelbrot Set. Completely useless, but fun to program and play around with. That's where I first encountered complex numbers.

 

The 12i deal?

 

 

Sorry I should have placed this:

 

The Square Root of 12 = 3.46410161514

 

 

The reason I am thinking this is because they say i is basically the " root " of the absolute value of a number,

So when I see for example:

 

A number such as 12i, or 46i or xi, I am thinking just " get the square root of the absolute value of that number..

 

 

Now in terms of all applied Methods for complex numbers usage, you really recommend that I just use these for fun??

 

 

I am wondering if this is why QM appears to be counter-intuitive??

Maybe its just the math involved??

 

 

Wave Function and Imaginary numbers.

http://en.wikipedia.org/wiki/Wave_function

 

I gather that's x^2 not x2. Work towards the roots and you will see where the i comes from.

 

The roots of a quadratic equation in the form of a*x^2 + b*x^1 + c*x^0 = 0 equal (-b +/- SQRT(b^2 -4*a*c))/2a.

 

With a = 1, b = -6 and c = 25 the roots are (6 +/- SQRT(36-100))/2 = (6 +/- SQRT(-1)*SQRT(64))/2 = 3 +/- 4*SQRT(-1) = 3 +/- 4i

Also, the roots of a quadratic equation in this form are the point(s) where the plot of the quadratic function crosses the x axis i.e. where y = 0.

Yes that is x^2

 

But one question, what is this that you are using --> +/-

 

 

In your example here, where did the 100 come from??

it looks like 10^2/2 = 1/2 = .5

 

 

(6 +/- SQRT(36-100))/2 = (6 +/- SQRT(-1)*SQRT(64))/2 = 3 +/- 4*SQRT(-1) = 3 +/- 4i

 

 

 

This area is really " where" I tend to focus the most" it looks like a " base 10 limit"

If this is the case, " their is something missing" a unit of measure I would say...

I don't mean to get deep, but a " black hole perhaps is missing??

 

 

 

For example it is known that in two "phase cycles" one of them always lags behinds the other, and appears to be unavoidable.

 

 

http://www.mathworks.com/help/signal/ug/cross-correlation-of-phase-lagged-sine-wave.html

 

 

 

You say:

 

 

 

Also, the roots of a quadratic equation in this form are the point(s) where the plot of the quadratic function crosses the x axis i.e. where y = 0.

 

 

 

Seriously " What is the point of this graph??

 

To me it looks like a Kepler orbit of some kind, coupled with Minowski Space Time..

 

 

 

Also, why is -3 so common in these types of graphs?

 

Now it looks like 10^3 as a time plot period for frequencies, in where 10^3 decays " exponentially" with the distance of x dependent on y..

 

But due to imaginary units I am obscured on my ability to explain this correctly..

 

In this graph 3 and -4 clearly look like a maximum on a crest peak, but upside down...

Does this " curve " have any other numbers attached to them in the form of the number line?

 

Such as conjugate numbers?

Matrix?

Binary?

 

 

I find it hard to think, that only 3 and -4 " 2 numbers " can explain if anything " anything about science."

Unless those " 2 numbers " are really the exponents of x

 

 

I assume this is where Summation comes in>>>?

 

 

 

 

 

 

Edited May 13, 2014 by Iwonderaboutthings

[ajb]

Ok now I am very confused, I thought that the commutative laws such as in algebra???

 

did not allow a solution to x^2 = -1?

This is not due to the commutative rules of real algebra. It is true that no such real number x exits that is a solution to x^{2}+1 =0.

 

I am not at all clear what your confusion is.

 

We extend the real numbers by including i, which we can think of as just formally handling the solution to my equation above*. You can treat i as if it were a real number in the algebra, in particular we have the equality ai = ia for all real numbers a. In short using i allows you to handle the square root of negative numbers, something you cannot do using just real numbers.

 

 

* I think this attitude undersells the complex numbers, but it is probably the best way to think about them when first introduced.

[Iwonderaboutthings]

This is not due to the commutative rules of real algebra. It is true that no such real number x exits that is a solution to x^{2}+1 =0.

 

I am not at all clear what your confusion is.

 

We extend the real numbers by including i, which we can think of as just formally handling the solution to my equation above*. You can treat i as if it were a real number in the algebra, in particular we have the equality ai = ia for all real numbers a. In short using i allows you to handle the square root of negative numbers, something you cannot do using just real numbers.

 

 

* I think this attitude undersells the complex numbers, but it is probably the best way to think about them when first introduced.

I see, so I assume that the commutative rules of real algebra more apply to the real values of the imaginaries?? " or something like that."

 

I just want to make sure before investing time in this area of math, I keep hearing i units, are a waste of time, don't provide solutions of any, and yet QM uses them..

 

There seems to be " on the internet" so much information that denotes so many areas in science, its hard to believe anything anymore...

 

Any suggestions???

[ajb]

I see, so I assume that the commutative rules of real algebra more apply to the real values of the imaginaries?? " or something like that."

The commutative rules are unchanged for complex numbers.

 

I just want to make sure before investing time in this area of math, I keep hearing i units, are a waste of time, don't provide solutions of any, and yet QM uses them..

The complex numbers provide us with all the solutions of polynomial equations with real (and indeed complex) coefficients. This is one of the amazing and important things about complex numbers, they are algebraically closed.

 

Without getting into quantum mechanics yet, it is true that complex numbers seem fundamental to the theory.

 

There seems to be " on the internet" so much information that denotes so many areas in science, its hard to believe anything anymore...

 

Any suggestions???

Wikipedia has a reasonable introduction.

[Iwonderaboutthings]

The commutative rules are unchanged for complex numbers.

 

 

The complex numbers provide us with all the solutions of polynomial equations with real (and indeed complex) coefficients. This is one of the amazing and important things about complex numbers, they are algebraically closed.

 

Without getting into quantum mechanics yet, it is true that complex numbers seem fundamental to the theory.

 

 

Wikipedia has a reasonable introduction.

Thanks for this information especially about the closed algebraic forms, I am definitely investing time on this...

[ajb]

Thanks for this information especially about the closed algebraic forms, I am definitely investing time on this...

 

 

This is one of the amazing and important things about complex numbers. You don't need to leave complex numbers to get the solutions of polynomials with complex coefficients.

[Iwonderaboutthings]

 

 

This is one of the amazing and important things about complex numbers. You don't need to leave complex numbers to get the solutions of polynomials with complex coefficients.

Yes infact I did some examples online and their p r e t t y good!!!!!!!!! I was " quite impressed I must say."

[LaurieAG]

Yes that is x^2

 

But one question, what is this that you are using --> +/-

 

In your example here, where did the 100 come from??

it looks like 10^2/2 = 1/2 = .5

 

(6 +/- SQRT(36-100))/2 = (6 +/- SQRT(-1)*SQRT(64))/2 = 3 +/- 4*SQRT(-1) = 3 +/- 4i

 

The +/- is just part of the standard quadratic formula. http://en.wikipedia.org/wiki/Quadratic_equation#Quadratic_formula_and_its_derivation

 

The 100 comes from 4 * a * c = 4 * 1 * 25 = 100 and the 36 comes from b^2 = -6 * -6 = 36 so SQRT(b^2 - 4 * a * c) = SQRT(-64) = SQRT(64 * -1) = SQRT(64) * SQRT(-1) = 8 * SQRT(-1) = 8i. As -b = 6 and 2 * a = 2 * 1 = 2 so the final equation is (6 +/- 8i)/2 = 3 +/- 4i.

 

It might look like 10^2/2 = 100/2 = 50 if you ignored the bracketing but you could only get .5 if you ignored the bracketing and redefined 10^2 or 2.

[Iwonderaboutthings]

 

The +/- is just part of the standard quadratic formula. http://en.wikipedia.org/wiki/Quadratic_equation#Quadratic_formula_and_its_derivation

 

The 100 comes from 4 * a * c = 4 * 1 * 25 = 100 and the 36 comes from b^2 = -6 * -6 = 36 so SQRT(b^2 - 4 * a * c) = SQRT(-64) = SQRT(64 * -1) = SQRT(64) * SQRT(-1) = 8 * SQRT(-1) = 8i. As -b = 6 and 2 * a = 2 * 1 = 2 so the final equation is (6 +/- 8i)/2 = 3 +/- 4i.

 

It might look like 10^2/2 = 100/2 = 50 if you ignored the bracketing but you could only get .5 if you ignored the bracketing and redefined 10^2 or 2.

Hymm I see now, great link, thanks!

[physica]

I see, so I assume that the commutative rules of real algebra more apply to the real values of the imaginaries?? " or something like that."

 

I just want to make sure before investing time in this area of math, I keep hearing i units, are a waste of time, don't provide solutions of any, and yet QM uses them..

 

There seems to be " on the internet" so much information that denotes so many areas in science, its hard to believe anything anymore...

 

Any suggestions???

Units are very important. T

 

They keep you on track. As for imaginary numbers if you want another example for them being used in physics without the heavy concepts of quantum mechanics bogging you down look at the link below. This link shows you how euler's identity is used in differential equations to make sense of imaginary numbers in systems. They can even pop up when calculating an oscillating particle between 2 springs. If you're not sure on what differential equations are they are equations that map change in a system. They can describe the position or speed of a particle in a system. They can also be used in population dynamics or anything that has rates of change but don't worry too much about differential equations yet if you haven't looked into them. This link is to show the practical application of imaginary numbers to wet the appetite.

http://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx

 

As a side note I think it's great that you are taking the time to learn the maths behind the system.I appreciate that a negative rep can be discouraging. It takes real will power to dust yourself off and work towards understanding the maths. For this I've given your last post a positive point.

[Iwonderaboutthings]

Units are very important. T

 

They keep you on track. As for imaginary numbers if you want another example for them being used in physics without the heavy concepts of quantum mechanics bogging you down look at the link below. This link shows you how euler's identity is used in differential equations to make sense of imaginary numbers in systems. They can even pop up when calculating an oscillating particle between 2 springs. If you're not sure on what differential equations are they are equations that map change in a system. They can describe the position or speed of a particle in a system. They can also be used in population dynamics or anything that has rates of change but don't worry too much about differential equations yet if you haven't looked into them. This link is to show the practical application of imaginary numbers to wet the appetite.

http://tutorial.math.lamar.edu/Classes/DE/ComplexRoots.aspx

 

As a side note I think it's great that you are taking the time to learn the maths behind the system.I appreciate that a negative rep can be discouraging. It takes real will power to dust yourself off and work towards understanding the maths. For this I've given your last post a positive point.

thanks for the link , " I tried clicking it but it does not work"

 

Yes, I had noooooo idea how much I underestimated "detail" studying things over and over again has helped me notice those "small areas" that had me confused for years! No joking, there is this incredible structure of the " Magnetron" bottom photo of a microwave oven, in it I had the chance to see in-detail the mechanisms that create the waves inside of the casing device. I find that " VISUALIZATION" may be one of the most difficult things to master in science. So why didn't I ever see this years ago???

 

I think like so many, I just wanted to get the " technical" stuff out of the way and not have a visual guide that describes the generalization... Patience is still hard to learn though.

 

It can be awkward studying math without a visualization and a intuitive feel. However at my level of " extreme" detail and technicals, the visualizations are making things "now" so clear that it can be overwhelming, I am glad I am taking the time to " refresh" my knowledge, and really encourage others to do so...

 

 

Edited May 27, 2014 by Iwonderaboutthings