Hey I have this question due for thursday and cant figure anything out, (btw no calculus knowledge is required)

 

A) Express the number 25 as a product of two complex conjugates, a + bi and

a - bi in two different ways with a and b both natural numbers.

 

b) Find another perfect square that can be expressed as a product of two complex conjugates, a + bi and a - bi in two different ways, with a and b both natural numbers.

 

c) describe the most efficient method for finding numbers that satisfy the above relationship.

Common, it's not that hard. First of all, off course it's not calculus, second, before attempting any of these problems you should know the properties of the complex numbers. I'll solve the a) and the rest is pretty much the same.

 

[math]i=\sqrt{-1}[/math](this is kind of obvious, but just in case)

 

The problem is asking for a multiplication of two complex conjugates, a+bi and a-bi that equal 25. So...

[math]25=(a+bi)(a-bi)[/math]

[math]25=a^2-(bi)^2[/math]

[math]25=a^2-b^{2}i^2[/math]

[math]25=a^2-b^{2}(\sqrt{-1})^2[/math]

[math]25=a^2-b^{2}(-1)[/math]

[math]25=a^2+b^2[/math]

 

We know that a and b have to be positive whole numbers, so you think for one second... and come to the conclusion that niether one can be greater than four because... 5 squared equals 25 and adding anything to it will increase it making it not equal to 25,(which we don't want off course). So... if they have to be positive, whole numbers, less than five, it's easy to deduce: [math]25=4^2+3^2[/math]. Where one of the answers, a=4, b=3 and the other answer is backwards.

 

If there's anything wrong here, correction is appreciated.