# Delicious Differences

## Here's a function to consider for this math trick:

## Y = X + √(X)

## Let X be equal to an integer that's a *perfect square.* If you list consecutive perfect squares in a table like this, then...

X | Y | Differences Between Y Values |

0 | 0 | 2 |

1 | 2 | 4 |

4 | 6 | 6 |

9 | 12 | 8 |

16 | 20 | 10 |

25 | 30 | 12 |

36 | 42 | 14 |

49 | 56 | 16 |

64 | 72 | 18 |

81 | 90 | 20 |

100 | 110 | 22 |

121 | 132 | 24 |

144 | 156 |

## The differences of the Y values are consecutive even numbers!

The next difference in the rightmost column would be **26** if I plugged the next perfect square into *X*!

## In this next table, their *additive inverses* are plugged into *X*. Now watch what happens in the rightmost column:

X | Y | Differences Between Y Values |

0 | 0 | -1 + *i* |

-1 | -1 + *i* | -3 + *i* |

-4 | -4 + 2*i* | -5 + *i* |

-9 | -9 + 3*i* | -7 + *i* |

-16 | -16 + 4*i* | -9 + *i* |

-25 | -25 + 5*i* | -11 + *i* |

-36 | -36 + 6*i* | -13 + *i* |

-49 | -49 + 7*i* | -15 + *i* |

-64 | -64 + 8*i* | -17 + *i* |

-81 | -81 + 9*i* | -19 + *i* |

-100 | -100 + 10*i* | -21 + *i* |

-121 | -121 + 11*i* | -23 + *i* |

-144 | -144 + 12*i* |

## The Y values & their differences are complex numbers! Each Y value is the additive inverse of a perfect square plus its square root multiplied by the imaginary unit *i*! The differences are consecutive negative odd numbers & each one is added to the imaginary unit *i*! (And of course, 0 + 0*i* = 0.)

Back to Index Page Back to Math Trick Menu

© Derek Cumberbatch