Cuckoo For Conjugates 2!

The square root of a negative pure imaginary number is equal to the conjugate of the square root of its additive inverse!

√(-bi) = the conjugate of √(bi)

Examples:

√(-i) = √(2)/2 - √(2)/2i & √(i) = √(2)/2 + √(2)/2i

√(-9i) = 3√(2)/2 - 3√(2)/2i & √(9i) = 3√(2)/2 + 3√(2)/2i

√(-5i) = √(5/2) - √(5/2)i & √(5i) = √(5/2) + √(5/2)i

√(-7i) = √(7/2) - √(7/2)i & √(7i) = √(7/2) + √(7/2)i

√(-1.5i) = √(3/2) - √(3/2)i & √(1.5i) = √(3/2) + √(3/2)i

√(-πi) = √(π/2) - √(π/2)i & √(πi) = √(π/2) + √(π/2)i

Notice how some of the pluses change to minuses & vice versa!

It doesn't matter what real number you plug in for the variable b, this trick will still work out! And remember: real × imaginary = imaginary!

Conjugates share the same absolute value!

According to the examples above, the real part & the imaginary part of each complex number is likely to be a fraction with the number 2 as the denominator! But if you pick an even number, then...

√(-2i) = 1 - i & √(2i) = 1 + i

√(-8i) = 2 - 2i & √(8i) = 2 + 2i

√(-36i) = 3√(2) - 3√(2)i & √(36i) = 3√(2) + 3√(2)i

...The real part & the imaginary part won't be fractions, but they're still likely to be multiples of the square root of 2. Furthermore, both parts of the complex number will always equal each other.

Finally, an extra fact to remember: If the real part & the imaginary part of a complex number are equal to each other, then the square of the complex number will be a pure imaginary number.

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