Really Real

Algebra Level 2

Describe the subset of complex numbers such that

$\sqrt{x} \times \sqrt{x}$

is a real number.

Positive real numbers only All real numbers only Non-negative real numbers only Integers only

2 solutions

Andreas Wendler
Feb 17, 2016

The domain even contains negative real numbers because (after taking root) then the product of two imaginary numers gives real number -1.

I dont understand. The root x function is defined only for non negative real numbers. The given function is a product of two functions(say g(x) and h(x)). So when you evaluate the given function, you enter the x into g(x) and h(x). Therefore for x to be in the domain of the given function, it must be in the domain of g(x) and h(x). Please explain.

Shanthanu Rai - 5 years, 3 months ago

The square root function is defined on the complex numbers (including the negative real numbers, of course), and $\sqrt{x}\times\sqrt{x}=x$ , by definition of the (principal) square root. If we restrict this function to those complex numbers $x$ for which the value $\sqrt{x}\times\sqrt{x}=x$ is real, then the domain consists of all real numbers, as claimed. The problem asks, in effect: "Find all complex numbers $x$ such that $\sqrt{x}\times\sqrt{x}$ is real."

Otto Bretscher - 5 years, 3 months ago

Oh! I didn't know that. Thank you for the explanation.

Shanthanu Rai - 5 years, 3 months ago

Naman Kapoor
Feb 18, 2016

The name is the answer..

Really Real

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