text: Palindrome Geoff is right about how to arrive at the root he described, however, that is not the whole story.\n\nJust as there are TWO numbers that satisfy the meaning of √16 (that is, 4 and -4) there are actually FOUR numbers that satisfy the meaning of √(√(-16)):\n● √2 + i√2\n● -√2 + i√2\n● √2 - i√2\n● -√2 - i√2\n\nGeoff's analysis about "turning the angle halfway" toward the positive x axis is a good intuitive way of understanding what is going on. However, what he left out is that it DOESN'T MATTER which direction you turn, clockwise or counter-clockwise. Both directions are equally valid and both should be taken into account.\n\nFor √(-16), turning clockwise gives you 4i, counter-clockwise gives you -4i\n\nFor √(4i), turning clockwise gives you √2 + i√2, counter-clockwise gives you -√2 - i√2 (remember, you are going halfway to the positive x axis the long way)\n\nFor √(-4i), turning clockwise gives you -√2 + i√2, counter-clockwise gives you √2 - i√2