Are there any practical applications for imaginary numbers?

Yes! One of the simplest real-world applications is to represent two component numbers like a coordinate (x,y) or a vector (direction and magnitude). A complex number is a real quantity added to a imaginary quantity. Like 3*sqrt(-1) + 7.5, which would also be written out as 3i + 7.5. A number like this, cannot be reduced to existing on a one dimensional number line, like the kind introduced in grade school, with negative numbers on the left and positive numbers on the right and zero in the middle. A complex number has an "extra dimension" and to properly plot on a number line, the number requires an additional line, perpendicular to the first, through the zero point, stretching to the positive and negative imaginary infinities, up and down respectively. A complex number like 3i+7.5 would be plotted 3 points up and 7.5 points to the right. So, it just acted like the coordinate, (3,7.5)

It can be used to represent phase of electrical waveforms in circuits.

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