# Complex Number Kookiness 7

## Observe the table below. Can you find something interesting in any 1 of the 2 columns?

Complex Numbers |
Absolute Values of Complex Numbers Squared |
---|

½+*i* |
1¼ |

½+2*i* |
4¼ |

½+3*i* |
9¼ |

½+4*i* |
16¼ |

½+5*i* |
25¼ |

½+6*i* |
36¼ |

½+7*i* |
49¼ |

½+8*i* |
64¼ |

The fraction **½** is added to consecutive integers multiplied by the imaginary unit *i* in the left column so that the squares of the absolute values
of the complex numbers are equal to **the squares of the consecutive integers + ¼!** If the real & imaginary parts of the complex numbers swapped values,
then their absolute values would stay exactly the same! The signs of the real part & imaginary part of each complex number doesn't matter because that won't
change the absolute value!

Complex Numbers |
Absolute Values of Complex Numbers Squared |
---|

1+½*i* |
1¼ |

2+½*i* |
4¼ |

3+½*i* |
9¼ |

4+½*i* |
16¼ |

5+½*i* |
25¼ |

6+½*i* |
36¼ |

7+½*i* |
49¼ |

8+½*i* |
64¼ |

## Below is a copy of the 1st table, except that this time, I printed the conjugates of the complex numbers!

Complex Numbers |
Absolute Values of Complex Numbers Squared |
---|

½-*i* |
1¼ |

½-2*i* |
4¼ |

½-3*i* |
9¼ |

½-4*i* |
16¼ |

½-5*i* |
25¼ |

½-6*i* |
36¼ |

½-7*i* |
49¼ |

½-8*i* |
64¼ |

## The main difference between a complex number & its conjugate is that the plus sign in the middle becomes a minus; or if it's already a minus, it becomes a plus!

__Here are the conjugates of the ones for the 2nd table:__

Complex Numbers |
Absolute Values of Complex Numbers Squared |
---|

1-½*i* |
1¼ |

2-½*i* |
4¼ |

3-½*i* |
9¼ |

4-½*i* |
16¼ |

5-½*i* |
25¼ |

6-½*i* |
36¼ |

7-½*i* |
49¼ |

8-½*i* |
64¼ |

## By the way, |**a+b***i*| = |**a-b***i*| = |**-a+b***i*| = |**-a-b***i*|;
all 4 of those are exactly the same!(In absolute value) You can have the variables a & b swap places without changing the absolute value!

__The absolute value of a complex number:__

## Finally, here's the appropriate formula for this math trick:

## |½ ± N*i*|^{2} = |N ± ½*i*|^{2} = N^{2} + ¼

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© Derek Cumberbatch