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Which set of points are defined by the relation $x/|x|=y/|y|$?

I think the answer is a straight line bisecting the first and third quadrants through the origin ( ie the line x=y). However wolfram alpha gives a very different result. Where am I going wrong?

mikoyan
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  • Hint: $x/|x|=\text{sgn} x$, the sign of $x$, so it's the same as $xy=|xy|$ (apart from either being $0$) – Adam Hughes Jul 22 '14 at 19:21
  • Without showing your goings, it's hard to say how it's going wrong. This problem does have a simple approach, though: there is a very simple description of the function $f(z) = z/|z|$. –  Jul 22 '14 at 19:22
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    You are on the right track, think of entire sets of numbers though not just a line in the quadrants. – kleineg Jul 22 '14 at 19:31

2 Answers2

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Let's have some questions:

1- Is one of $x=0$ or $y=0$ allowed to be happen?

No

2- So let $x\neq 0$and $y\neq0$. What we have if $x>0$ and $y<0$?

Indeed, $1=-1$ which is forbidden.

3- what we will be there if $x>0$ and $y>0$ or $x<0$ and $y<0$.

Indeed, we will face to the identity $1=1$.

Conclusion: All $(x,y)$ wherein $x>0,y>0$ or $x<0,y<0$ are the solutions.

enter image description here

Mikasa
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What if $x > 0$ and $y > 0$? Then we have $1=1$. This means that $(x,y)$ is a point in your set provided $x$ and $y$ are positive.

If both are negative, then we find $-1=-1$.

This does not work if $x>0$ and $y<0$ or the other way around since $1\neq -1$.

Neither $x$ nor $y$ are allowed to be zero. Since the relation would be undefined.

Thus we have the set you are looking for is the union of the first and third quadrants of the plane.

Joel
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  • thank you for your clear answer. So to check I understand, any point that is in the first or third quadrant Is defined by the relation $x/|x|=y/|y|$? Which would explain the graph wolfram alpha drew. – mikoyan Jul 22 '14 at 19:35
  • Yep pretty much. – Joel Jul 22 '14 at 19:41