Math Is Fun Forum

hussam

Member

Registered: 2015-04-06

Posts: 46

The imaginary unit (i).

Hi;

i will give a simple question, then the question extends to another more complicated question   

My question is  whether the  -roots  multiply properties-  of the real numbers  is fine with the imagine numbers?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: The imaginary unit (i).

Hi;

Can you post an example?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

hussam

Member

Registered: 2015-04-06

Posts: 46

Re: The imaginary unit (i).

hi bobbym;

ok, this is the question i need : we know that  i=√( -1)       

so i^2=  √( -1)*√( -1) = √[( -1)( -1)]=√ ( 1)=1

thus; i=1 or i=-1 how can it be?

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: The imaginary unit (i).

Hi;

only is true when

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

hussam

Member

Registered: 2015-04-06

Posts: 46

Re: The imaginary unit (i).

i don't think that because

√(-9)=√(-1*9)=√(-1)*√(9)=3i

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: The imaginary unit (i).

Hi;

The square root function does not easily extend beyond the non-negative reals. There may be examples where what you did holds but for all complex numbers it would not. Also, when we say √ 4 we just mean +2, the principal square root, not the -2.

So

√( -1)*√( -1) = √[( -1)( -1)]

you are not allowed to do that with two complex numbers.

Read this http://math.stackexchange.com/questions … stribution
, especially Lee Mosher's answer.

That is the reason you are always told when working with a negative inside there to pull out a square root of -1 and to write is as i.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

hussam

Member

Registered: 2015-04-06

Posts: 46

Re: The imaginary unit (i).

/thank you /

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: The imaginary unit (i).

Hi;

I am sorry that I can not do better. There are people in here that would be better at answering questions of this type. Please wait for one of them to explain it better.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

hussam

Member

Registered: 2015-04-06

Posts: 46

Re: The imaginary unit (i).

Thank you anyway;)

Olinguito

Member

Registered: 2014-08-12

Posts: 649

Re: The imaginary unit (i).

√(-9)=√(-1*9)=√(-1)*√(9)=3i

That's not quite right; it should be

[list=*]
[*]

[/*]
[/list]

Note that

[list=*]
[*]

[/*]
[/list]

The function f(x)=√x only has the non-negative reals as domain. For negative reals, it is not a function but a one-to-many relation.

Last edited by Olinguito (2015-04-17 20:58:01)

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Bassaricyon neblina

Olinguito

Member

Registered: 2014-08-12

Posts: 649

Re: The imaginary unit (i).

So if you want to have

[list=*]
[*]

[/*]
[/list]

one of the √(−1) is supposed to be i and the other −i. Similarly if

[list=*]
[*]

[/*]
[/list]

one of the √(−1) is supposed to be i and the other two either both i or both −i. With a one–many relation you can't always tell.

Last edited by Olinguito (2015-04-17 21:05:57)

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Bassaricyon neblina

Bob

Administrator

Registered: 2010-06-20

Posts: 10,623

Re: The imaginary unit (i).

hi hussam

Thanks.  This has got me thinking.

The starting point for a mathematical theory will be a set of axioms; from which other properties can be worked out.

is not part of the axioms, so it has to be checked.  Just because something appears to be true doesn't make it true.  Here's an example.

generates primes numbers for n = 0, 1, 2, 3, 4, ......

so you might conclude that it is a prime number generator.  If you keep checking values of n, mostly it works.  But consider n = 41.  Now the 'generator' has produced a number that isn't prime.  It's a useful lesson in why mathematicians spend so long proving the apparently obvious.

So back to square roots.  Why should rule (1) apply?

Clearly we can find examples where it does work:

It is conventional to assume that the square root symbol means take the positive square root.  But supposing we take a more general meaning for square root such that

And let's choose √ 4 = -2 ; √ 9 = -3 and √36 = -6.  Now we find √ 4 x √ 9  ≠ √ 36.

When it comes to complex numbers, you cannot say 'by square root I mean the positive square root' because complex numbers lie somewhere in the Argand diagram not necessarily on a line.  The fundamental theorem of complex numbers tells us that every complex number will have two square roots (except zero)  so rule (1) won't work. 

What you have done is to produce a counter example (like n = 41) that demonstrates that the rule fails.  It just shows how careful you must be not to assume a rule holds without careful checking.

You have probably heard that you can 'prove' that 2 = 1 with some simple algebra.  It's another example of the need for careful checking.  If you haven't met this one have a look here:

http://en.wikipedia.org/wiki/Mathematic … on_by_zero

Bob

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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

hussam

Member

Registered: 2015-04-06

Posts: 46

Re: The imaginary unit (i).

hi Olinguito;

√(-9)=√(-1*9)=√(-1)*√(9)=3i

That's not quite right;

√4=2 ( not -2 )

but -√4=-2

so you cant write √4=-2/+2

just you can write this when you have for example:

X²=4     So X=√4=2 / X=-√4=-2

Hussam

Olinguito

Member

Registered: 2014-08-12

Posts: 649

Re: The imaginary unit (i).

√4=2 ( not -2 )

but -√4=-2

so you cant write √4=-2/+2

Correct. BUT:

[list=*]
[*]

[/*]
[/list]

Complex numbers are different.

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Bassaricyon neblina

hussam

Member

Registered: 2015-04-06

Posts: 46

Re: The imaginary unit (i).

hi Olinguito ; ok

what is that rule

So if you want to have

[list=*]
[*]

[/*]
[/list]

one of the √(−1) is supposed to be i and the other −i. Similarly if

[list=*]
[*]

[/*]
[/list]

one of the √(−1) is supposed to be i and the other two either both i or both −i. With a one–many relation you can't always tell.

Olinguito

Member

Registered: 2014-08-12

Posts: 649

Re: The imaginary unit (i).

It is not a rule. Just something I decided to put in to make things work.

Tell you what. Forget what I posted above. I didn't express myself well. Instead read Bob Bundy’s post #12. Bob has explained things much better than I did.

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Bassaricyon neblina

hussam

Member

Registered: 2015-04-06

Posts: 46

Re: The imaginary unit (i).

i have read  what he post

Thanks for you and thanks for him .

Last edited by hussam (2015-04-18 03:33:52)