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 $x^2+1=0$. This has no real roots... hence $\sqrt{-1}$ has no real roots...

But then mathematicians never like to have no as an answer... so they have defined something like $i=\sqrt{-1}$. 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

Thank you - I kept feeling the answer was 2, but I couldn't work out how!

Michael