This shape represents the mathematical function y = x raised to the x power, where x is a real number but y is complex.

The model was printed on the Dimension 3-dimensional printer.

For further information about this shape, continue reading below.

## Discussion

This discussion will involve taking exponents. If you are rusty on the rules for taking exponents, see the Exponent Rules section below.

x | y |
---|---|

1 | 1 |

2 | 4 |

3 | 27 |

4 | 256 |

5 | 3125 |

This looks like a very tame function. However, zero raised to the zero power is undefined. So x^{x} for x=0 will remain uncalculated. See the conclusions below.

Recalling that a negative exponent represents reciprocation, we can extend the table to negative values of x.

x | y |
---|---|

-1 | -1 |

-2 | 1/4 |

-3 | -1/27 |

-4 | 1/256 |

-5 | -1/3125 |

Notice that the function appears to oscillate for negative values of x. If this is the case, one might expect that there would be a value of x between -1 and -2 where y is zero.

Let’s do the math for x = -3/2.

(-3/2)^{(-3/2)} =

(-2/3)^{(3/2)} =

((-2/3)^{3})^{1/2} =

((-2^{3})/(3^{3}))^{1/2} =

(-8/27)^{1/2} =

(-0.2963)^{1/2}

Recalling that an exponent of 1/2 indicates a square root, then there are two solutions. Also, square root of a negative number causes a rotation of ninety degrees in the complex plane, indicated by the letter ‘i’.

(-8/27)^{1/2} =

(-0.2963)^{1/2} =

0.544331i, -0.544331i

So there’s the answer: The function y = x^{x} becomes complex for certain values. Here are a couple of examples for positive x values.

(3/2)^{(3/2)} =

((3^{3})/(2^{3}))^{1/2} =

(27/8)^{1/2} =

3.375^{1/2} =

1.837, -1.837

(3/4)^{(3/4)} =

((3^{3})/4^{3}))^{1/4} =

(27/64)^{1/4} =

.421875^{1/4} =

0.8059, 0.8059i, -0.8059, -0.8059i

The previous example has four roots. This can be shown to be a general rule that the number of complex roots equals value of the denominator of the exponent.

An alternative form is to use the polar form, y = R*exp(i*theta), where R is the magnitude, “exp” is the exponential function, and theta is the angle in the complex plane. Because x is real, x = X*exp(i*pi*(0+2n)) when positive, x = X*exp(i*pi*(1+2n)) when negative, where X = magnitude of x, n= 0,1,...

Then R = X^x and theta = pi*X*(0+2n) or pi*(-X)*(1+2n).

Using that form, the four roots of the previous example become:

R | n | theta |
---|---|---|

0.8059 | 0 | 0 |

0.8059 | 1 | 3/2 pi |

0.8059 | 2 | 3 pi |

0.8059 | 3 | 9/2 pi |

As another example, consider x = -1/3.

y = (1/3 exp (i*pi*(1 + 2n)))^{(-1/3)} =

3^{(1/3)} * exp(i*pi*(1+2n)*(-1/3))=

1.44225 * exp(i*pi*(1+2n)*(-1/3))

R | n | theta | principal theta |
---|---|---|---|

1.44225 | 0 | -1/3*pi | 5/3*pi |

1.44225 | 1 | -1*pi | 1*pi |

1.44225 | 2 | -5/3*pi | 1/3*pi |

(“principal theta” lists theta values between 0 and 2*pi.)

The only remaining thing to examine is the case of an irrational value of x. Let x = -sqrt(2). Find all the values for y in polar form.

y = (1.414*exp(i*pi*(1 + 2n)))^{(-1.414)}

y = (.7071^{1.414})*exp(i*pi*(-1.414)*(1 + 2n))

y = .6125*exp(i*pi*(-1.414)*(1 + 2n))

R | n | theta | principal theta |
---|---|---|---|

0.6125 | 0 | -1.414*pi | 0.586*pi |

0.6125 | 1 | -4.242*pi | 1.758*pi |

0.6125 | 2 | -7.071*pi | 0.930*pi |

0.6125 | 3 | -9.898*pi | 0.102*pi |

0.6125 | 4 | -12.73*pi | 1.274*pi |

0.6125 | 5 | -15.55*pi | 0.446*pi |

... | ... | ... | ... |

This list goes on forever and the principal theta values never repeat. So an irrational x results in a ring of an infinite number of points in the complex-y-plane with radius equal to |x|^{x}.

Because the irrational numbers outnumber the rationals, the graph of y=x^{x} is a 3-dimensional surface with its axis of symmetry along the x axis and with its radius equal to |x|^{x}. The surface is everywhere discontinuous except for the positive real part of y when x is positive.

Although 0^{0} is undefined, we can write: Limit of x^{x} as x approaches zero from positive values is equal to 1.

## Exponent Rules

x^{a} = x*x*.... (a times) = x to the ath power

Example: 2^{5} = 2*2*2*2*2 = 32 = two to the fifth power

x^{(a+b)} = (x^{a})(x^{b})

Example: 2^{(2+3)} = 2^{5} = 32 = 4*8 = (2^{2})(2^{3})

x^{(a-b)} = (x^{a})/(x^{b})

Example: 2^{(3-2)} = (2^{3})/(2^{2}) = 8/4 = 2 = 2^{1} = 2^{(3-2)}

x^{(-b)} = 1/(x^{b})

Example: 2^{(-3)} = 1/(2^{3}) = 1/8

x^{(a*b)} = (x^{a})^{b} = (x^{b})^{a}

Example: 2^{(2*3)} = 2^{6} = 64 = 4^{3} = (2^{2})^{3} = 8^{2 }= (2^{3})^{2}

if y = x^{a} then x = y^{(1/a)}

Example: 8 = 2^{3} so 2 = 8^{(1/3)}

x^{(1/b)} = bth root of x

Example: 81^{(1/4)} = (3*3*3*3)^{(1/4)} = 3 = 4th root of 81

x^{(a/b)} = (x^{a})^{(1/b)}

Example: 8^{(2/3)} = (8^{2})^{(1/3)} = 64^{(1/3)} = (4*4*4)^{(1/3)} = 4

Check: 4^{(3/2)} = (4*4*4)^{1/2} = 64^{(1/2)} = (8*8)^{1/2} = 8

Decimal example:

16^{(1.25)} = 16^{(1 + .25)} = (16^{1})(16^{(1/4)}) = 16*2 = 32

Alternate: 16^{(1.25)} = 16^{(5/4)} = (2^{4})^{(5/4)} = 2^{(4*5/4)} = 2^{5} = 32

Check: 32^{(.8)} = 32^{(4/5)} = (32^{4})^{1/5} = (32*32*32*32)^{1/5} = (16*2*16*2*16*2*16*2)^{1/5} = (16*16*16*16*16)^{1/5} = 16

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