H(x) is defined as x to the power of x. When you try to compute H(-0.5), you find yourself taking -0.5 to the -0.5 power. We recognize that

(-0.5)^(-0.5)

= 1/((-0.5)^(0.5))

= 1/√(-0.5) (the 1/2 power is the square root)

The square root of a negative number is an imaginary number, not a real number.

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Any number you can write without using "I" is a real number. -√2 is a real number, for example, as is 1, 85, -.354, or any other number you can think of.

H(-0.5) = 1/√(-.5) = 1/(I*√(1/2)) = -I*√2 (the "I" is √-1, signifying an imaginary number)

(-0.5)^(-0.5)

= 1/((-0.5)^(0.5))

= 1/√(-0.5) (the 1/2 power is the square root)

The square root of a negative number is an imaginary number, not a real number.

_____

Any number you can write without using "I" is a real number. -√2 is a real number, for example, as is 1, 85, -.354, or any other number you can think of.

H(-0.5) = 1/√(-.5) = 1/(I*√(1/2)) = -I*√2 (the "I" is √-1, signifying an imaginary number)