Q: Why are complex numbers useful in today's society?

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The real numbers together with the imaginary numbers form a sort of vector. What these complex numbers (complex means the combination of real and imaginary numbers) represent depends on the specific situation. Just as there are situations where it doesn't make sense to use negative numbers, or fractional numbers, in many common situations it doesn't make sense to use complex numbers. In an electrical circuit (specifically, AC), the real numbers might represent resistance, while the imaginary number represent reactance - and voltages and currents are also represented by complex numbers, with the angle of the complex number representing how much one quantity is ahead or behind another quantity (the "phase angle"). In quantum mechanics, the complex numbers might not represent anything (perhaps they don't, I am not sure...), but they are useful for calculations.

Yes. Consider as the simplest example: i * i = -1. But there are others: (a + bi)(a - bi) = a² + b². When you multiply conjugates, the result is always real. This is useful when dividing to get a pure real number in the denominator.

Scientific notation is useful in economics to compute very large or very small numbers.

Mixed numbers can be useful for talking about quantities of things that are not whole numbers. Like in food recipes, heights of children, or number of gallons the gas station wants you to pay for.

It is useful when dealing with very small or very large numbers.

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Mainly engineering, particularly control theory and other electronic systems; complex numbers are useful to represent oscillation frequency and phase information.

The 16th century Italian mathematician, Gerolamo Cardano was the first to use imaginary and complex numbers in his work on cubic equations.

The real numbers together with the imaginary numbers form a sort of vector. What these complex numbers (complex means the combination of real and imaginary numbers) represent depends on the specific situation. Just as there are situations where it doesn't make sense to use negative numbers, or fractional numbers, in many common situations it doesn't make sense to use complex numbers. In an electrical circuit (specifically, AC), the real numbers might represent resistance, while the imaginary number represent reactance - and voltages and currents are also represented by complex numbers, with the angle of the complex number representing how much one quantity is ahead or behind another quantity (the "phase angle"). In quantum mechanics, the complex numbers might not represent anything (perhaps they don't, I am not sure...), but they are useful for calculations.

The concept of a complex number appears to be first conceived by Gerolamo Cardano (about 1545). See Related Link. The work of Euler in the 1700's helped to make them more useful, though.

Society for the Diffusion of Useful Knowledge was created in 1826.

Society for Establishing Useful Manufactures was created in 1791.

An inventor is someone or something that invents objects useful to todays modern needs or wants

Boston Society for the Diffusion of Useful Knowledge was created in 1829.

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Comparison of viscosity of liquids is used in assessment and treatment of infants with dysphagia which is very much useful in society.