i2 =-1
Another question: my mum and I can't decide whether there is
such a thing as the square root of minus one (as worked out by
Marvin in the Hitchhiker's Guide). A positive number multiplied
by a positive number gives a positive number; a negative number
multiplied by a negative number gives a positive number. So I
think I have worked out that every positive number must have two
square roots; and every negative number has no square root. Mum
doesn't agree that there is no such thing as the square root of a
minus number, even if such a number is indefinable.
Michael Brooker
age 10
home-educated
Michael,
Yes!!your mother is actually correct.There is no such thing as
square root of a negative number....(i.e if you are trying to
find its root in real number set).
This simple looking but complicated fact arises from this equation x2+1=0.
This has no real roots... hence
has no real roots...
But then mathematicians never like to have no as an answer... so they have
defined something like
. This `i' is called the imaginary unit...
This concept is now well knit into a theory called the theory of complex numbers.
i am not going deep into things and trying to be as superificial
as possible....if any doubts please feel free to write
back...
love arun
I was interested to find out about these 'imaginary numbers'
and that they are referred to as 'i' when mathematicians deal
with this sort of thing. What a complex branch of algebra!
Michael
Michael,
Since you find imaginary and complex numbers interesting, I have
something you might find fun to try: Multiply
(1+i)(1-i)
This is the product of two complex numbers, but not just any two
complex numbers -- they are "conjugates" of one another. That
means they have the same real part and opposite imaginary
parts.
Use "FOIL" (first, outer, inner, last) or the distributive
property, three times, to get the answer.
You're not done if you leave an "i^2" (i squared) in the answer.
It can be simplified further. Write back when you have the
answer!
Yes, your problem is fascinating, but I can only half solve
it. No matter how hard I try, I always end up with an i²
term in the answer!
Michael
Try using the fact (actually, the definition of i) that
i2 =-1
Sorry, I'm hopelessly bamboozled!
Michael
You should find
Now as Brad suggests we need to think about what i means. We defined i to
be
. But then
So we can simplify our answer to
Thank you - I kept feeling the answer was 2, but I couldn't
work out how!
Michael