1. A complex formula like (z+1/z) is used to study and design airplane wings.
2. I use complex numbers to make math related art. The LINK below shows artwork
based on the formula (z-1)/(z+i)
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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.
Counting statistics experiments are useful in math. This is what helps with numbers.