# Pure imaginary complex number

**URL:** <https://community.wolfram.com/t/pure-imaginary-complex-number/17480>\
**Category:** Questions\
**Tags:** algebra, graphics-and-visualization\
**Created:** [January 24, 2021, 5:03pm UTC](https://community.wolfram.com/t/pure-imaginary-complex-number/17480 "2021-01-24T17:03:56Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![mathieuauffray5](https://avatars.discourse-cdn.com/v4/letter/m/e9a140/32.png) [@mathieuauffray5](https://community.wolfram.com/u/mathieuauffray5)\
**Post date:** [January 24, 2021, 5:03pm UTC](https://community.wolfram.com/t/pure-imaginary-complex-number/17480/1 "2021-01-24T17:03:56Z")

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Hello,

How Wolfram can do this problem?  
Determine the set of complex numbers z such that (z-3i) / z (z + 1) is pure imaginary and after pure real  
Sorry for my English.

Best regards.

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<div class="post-metadata">

**Author:** ![jbaldwinpsw](https://avatars.discourse-cdn.com/v4/letter/j/ee7513/32.png) [@jbaldwinpsw](https://community.wolfram.com/u/jbaldwinpsw)\
**Post date:** [January 24, 2021, 6:46pm UTC](https://community.wolfram.com/t/pure-imaginary-complex-number/17480/2 "2021-01-24T18:46:54Z")

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Here is one way.

```
z = a + b I;
f = (z - 3 I)/(z (z + 1)) // ComplexExpand

```

 ![f.png](https://global.discourse-cdn.com/wolfram/original/3X/3/d/3d9a4a1041b9574b16661073b645d597f989820c.png)

Now solve for when the real part is zero:

```
sol = Reduce[Re[f] == 0, {a, b}, Reals]

```

 ![4370f.png](https://global.discourse-cdn.com/wolfram/original/2X/d/d192d086ff7fb84d921ce24df4dce0135f2fb154.png)

Here is a contour plot showing some of the solution:

```
ContourPlot[Re[f] == 0, {a, -5, 5}, {b, -10, 10}, 
  FrameLabel -> (Style[#, Bold, 18] &) /@ {"a", "b"}, PlotPoints -> 100]

```

 ![solution.png](https://global.discourse-cdn.com/wolfram/original/3X/3/7/370b1f0998b94aaab985a318afb02a972f66f9f2.png)

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<div class="post-metadata">

**Author:** ![shenghuiy](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/shenghuiy/32/36073_2.png) [@shenghuiy](https://community.wolfram.com/u/shenghuiy)\
**Post date:** [January 24, 2021, 7:03pm UTC](https://community.wolfram.com/t/pure-imaginary-complex-number/17480/3 "2021-01-24T19:03:09Z")

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