Question for the Clever: what is the square root of -64??

Answer 1

Split into it's base components the Square roots of -1 and 64.

Square root of 64 is 8 or -8
Square root of -1 is i

Thus 8i or -8i

Answer 2

The Square Root of Complex Conjugation - A Puzzle
John Baez
October 24, 1997

When I teach complex analysis, I like to motivate the complex numbers as follows: multiplication by the real number x is a linear transformation of the real line which amounts to dilating by a factor of x when x is positive, but dilating and reflecting when x is negative.

Thus, if we seek a square root of -1, we seek a linear transformation of the real line which, when done twice, amounts to a reflection.

But there is no such thing.

To save the day, we need to go to the plane and consider rotations. A rotation by 90 degrees, done twice, amounts to a reflection through the origin. Voila: i is born!

Unfortunately, when i was born, so was its evil twin, -i. One corresponds to rotation 90 degrees clockwise, while the other corresponds to rotation 90 degrees counterclockwise. For some strange reason the usual convention is that the sinister -i corresponds to a clockwise rotation, while i is condemned to rotate things counterclockwise. Perhaps there was a mixup of some kind. Of course, there is an obvious excuse for this error: there is a symmetry of the complex numbers, namely complex conjugation, which interchanges i and -i, so it's a bit tricky to tell which is which.

In fact, this clockwise/counterclockwise business is probably a indication that we've made a horrible mistake all along: what we've been calling "-i" is really i, while "i" is -i! But I'm afraid it's too late to undo the damage: if we switched notations now, the confusion would be immense.

Answer 3

It's amazing how many people will answer a question and have no idea what they are talking about.

To find the correct answer among the garbage, check it yourself. Those who say 8, is 8 x 8 = -64? No?

Those who say -8: is -8 x -8 = -64? No?

(The answer is plus or minus 8i, where i is an "imaginary" number equal to the square root of -1.)

Answer 4

Mathamaticians use the letter 'i' to represent the square root of minus 1, so, since -64 is 64 x -1, the square root of -64 is i8.

Answer 5

8

Answer 6

8 i

Answer 7

The square root of -8 is an imaginary number, since no number multiplied by itself (whether that number is positive or negative) can have a negative result.

Answer 8

Most people only think of numbers in one dimension, but numbers actually occur in three dimensions in the x,y,z planes. The square root of -64 does not lie on the X plane but it does exist in three dimensional space.

Answer 9

V64 = V(-1. 64) = 8i or -8i
where V is 'square root'
8i is the same as 8(cos pi/2 + i sin pi/2) = 8.e^i.pi/2
-8i is 8(cos 3pi/2 + i sin 3pi/2) = 8.e^i.3pi/2 for 0=< arg z < 2pi
where arg z is the angle.
else for -pi< arg z =< pi
-8i = 8(cos pi/2 - i sin pi/2)
(by replacing the angle with its negative value)

Answer 10

its 8