1. ## Complex number verification.

These are the questions that are leaving me slightly puzzled:

Which of the following are true?

i) Every real number is a complex number.
ii) The square of an imaginary number is a negative real number.
iii)there is a complex number which is both real and imaginary.

My second question is: "for which positive integral values of n is i^n real?

This is what I put:

i)I put false because every real number can be expressed as a+0i=a. The complex part does not exist.

ii)$\displaystyle (a+bi)^2=a^2+2abi-b^2=a^2-b^2+2abi$

Using that I put false because an imaginary part (2ab) exists.

iii) For this I put false. a+bi was my example of a complex number. Since both parts have to exist together to make it a complex number, a number cannot be both real and complex.

For the second question I did [tex](-1)^ \frac{n}{2} [\MATH]. Therefore n has to be even. This doesn't seem quite right since "values" implies I either have to write a number or an inequality.

It seemed a little fishy to get three false answers for each other the questions. I was hoping someone out there could verify that my answers are (hopefully!) correct.

2. i) is true as a + 0i is a complex number
ii) is also true, an imaginary number has no real part, so your squaring ai

EDIT: Sorry i misread the second question, yes your right n must be even.

Bobak

3. Hello
Originally Posted by Showcase_22
These are the questions that are leaving me slightly puzzled:

Which of the following are true?

i) Every real number is a complex number.
ii) The square of an imaginary number is a negative real number.
iii)there is a complex number which is both real and imaginary.

My second question is: "for which positive integral values of n is i^n real?

This is what I put:

i)I put false because every real number can be expressed as a+0i=a. The complex part does not exist.
Actually, the complex part does exist and you even wrote it.

A complex number is in the form $\displaystyle a+ib$ where a and b can be any real number (including 0)

ii)(a+bi)^2=a^2+2abi-b^2=a^2-b^2+2abi

Using that I put false because an imaginary part (2ab) exists.
Note the difference between imaginary number and complex number. An imaginary is in the form ai, whereas a complex number is in the form a+bi.
A real number is defined as a complex number having its imaginary part equal to 0.
An imaginary number is defined as a complex number having its real part equal to 0.

iii) For this I put false. a+bi was my example of a complex number. Since both parts have to exist together to make it a complex number, a number cannot be both real and complex.
0 is a real and an imaginary number.

For the second question I did $\displaystyle (-1)^ \frac{n}{2}$. Therefore n has to be even. This doesn't seem quite right since "values" implies I either have to write a number or an inequality.
Ok, I didn't see the second question...
Look the pattern...
$\displaystyle i^1=i$
$\displaystyle i^{2}=-1$
$\displaystyle i^{3}=-i$
$\displaystyle i^4=1$

It seemed a little fishy to get three false answers for each other the questions. I was hoping someone out there could verify that my answers are (hopefully!) correct.
yah cap'tain !

4. really? a+0i is complex????? You learn something new every day!

oh yeah! part ii) woulod be like that. I was using a "complex" number. :s

Are you thinking of 0+0i? I don't actually know if that's real or imaginary.....i'm a little shaky on this (must read up!)

The second question is false.
umm, that wasn't a true or false question.

The question was:

My second question is: "for which positive integral values of n is i^n real"?

and I put even numbers. It just doesn't seem right....

5. Hang on a second, let's just summarise what we've deduced so far:

i) Every real number is a complex number.

This is true because a complex number can be expressed as a+bi and b is allowed to be 0.

ii) The square of an imaginary number is a negative real number.

This is also true because $\displaystyle (bi)^2=b^2i^2=-b^2$. This ($\displaystyle -b^2$) is always negative since $\displaystyle b^2$ is always postive.

iii)there is a complex number which is both real and imaginary.

This is true since, quoting Moo, "0 is a real and imaginary number". Thanks Moo!

With regards to question 2:

My second question is: "for which positive integral values of n is i^n real?

The solution is n must be an even number (take a nice glance at Moo's pattern, thanks yet again).

Ps. I started writing this post after I noticed Moo and Bobak had edited some of theirs so there might not be any point in this. I guess it helps anyone else reading this thread!

yah cap'tain !
Ready the sails!! weigh the anchor!

I could have been a pirate, or a financier but then I thought "why not try a more challenging career?......."

6. Originally Posted by Showcase_22
Hang on a second, let's just summarise what we've deduced so far:

This is true because a complex number can be expressed as a+bi and b is allowed to be 0.

This is also true because $\displaystyle (bi)^2=b^2i^2=-b^2$. This ($\displaystyle -b^2$) is always negative since $\displaystyle b^2$ is always postive.

This is true since, quoting Moo, "0 is a real and imaginary number". Thanks Moo!

With regards to question 2:

The solution is n must be an even number (take a nice glance at Moo's pattern, thanks yet again).

Ps. I started writing this post after I noticed Moo and Bobak had edited some of theirs so there might not be any point in this. I guess it helps anyone else reading this thread!

Ready the sails!! weigh the anchor!

I could have been a pirate, or a financier but then I thought "why not try a more challenging career?......."
Perfect (even the song lol ! though i don't know it )

But because I'm picky, I would add the definitions of complex, real and imaginary numbers (in this order), because it's the key of the solutions, especially the third

7. song is from Muppets treasure island!! Every lucky child has seen it, You must be one of the unlucky ones!

Okay then. I'll hit google for the definitions.