Complex or imaginary shapes?

Algebra Level 5

The absolute value of a complex number would be the distance between the complex number to the origin $(0,0)$ in the complex plane. Or in other words, $|a+bi| =\sqrt { { a }^{ 2 }+{ b }^{ 2 } }$ So for $|z|=n$ , the possible values of $z$ would form a circle of radius $n$ centered at the origin on the complex plane.

Now, supposing I create a new function: $\ddagger a+bi\ddagger =|a|+|b|$

And all of the possible values of $z$ in $\ddagger z\ddagger =2015$ forms a shape of area $A$ on the complex plane.

Find $\left\lfloor A \right\rfloor$

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The answer is 8120450.

2 solutions

Pranjal Jain
Dec 22, 2014

Let $z=x+i y$ where $x,y\in\mathfrak{R}$ .

$\ddagger z\ddagger=|x|+|y|=2015$

Plotting graph in first quadrant, $x+y=2015$ , length of line between x-intercept and y-intersept= $2015\sqrt{2}$ . By plotting in other quadrants using symmetry, it would be a square.

Area= $(2015\sqrt{2})^{2}=\boxed{\boxed{8120450}}$

Here is diagram of your solution :

Deepanshu Gupta - 6 years, 5 months ago

Thanks for adding this! Being lazy, I skipped it.

Pranjal Jain - 6 years, 5 months ago

ur welcome :)

Deepanshu Gupta - 6 years, 5 months ago

Shams Jabin
Apr 19, 2015

Create a graph of that function and integrate from 0 to 2015.

Complex or imaginary shapes?

The answer is 8120450.

2 solutions

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