To manage errors from trying to calculate square roots of negative numbers without modifying your data, you can use Excel's IF and SQRT functions together with ABS. A formula like '=IF(number<0, "Imaginary", SQRT(number))' can differentiate between negative and positive inputs, providing "Imaginary" for negative values and calculating the square root for non-negative values. This method ensures your formulas remain error-free and alerts you to negative inputs.