“How do scientists and engineers represent complex numbers?”
When working in a technical subject, you will probably have to deal with numbers that are either extremely large or extremely small, or have to work with empirically obtained values. For example, the number 602000000000000000000000 is really to read, as well as 0.000027. So how can we make them simpler and more accurate?
Well, let’s try to tackle this problem ourselves. What if we took our understanding of significant figures, and applied it to this problem? Since the only numbers that we actually need to care about is significant figures, how about we just remove all of the unnecessary zeroes? And how can we accomplish this? Well since we know that if we multiply or divide anything by 10 we will just shift the zeroes behind the numbers, how about we simplify all of the extra zeroes into a power of ten? the For example, when we have the number 602000000000000000000000, we can turn this in to 6.02*10^23. And for the number 0.000027, we can shift it into 2.7*10^-5. This way of working so much simpler! Scientists and Engineers have termed this framework scientific notation.
Now how can we apply this system for calculations? Well, first let’s divided it into two cases, multiplication/division, and addition/subtraction. In the first case, we will multiply both numbers and round our final answer to the number of significant figures of the variable with the lowest amount of sig-figs. For example, 5.2*10.81 will be 56 instead of 56.212 since the former only has two significant figures. For addition and subtraction, We will simply put all of the numbers in terms of the highest amount of digits after the decimal place, and then round to the lowest amount of significant figures. Let’s do an example. Suppose we have the numbers 3.14 and 2.1, and 1. When we ad the numbers together, we will notice that 3.14 has the highest amount of digits (2), and then rewrite everything else accordingly to become 3.14 + 2.10 +1.00 = 6.24, and then we will round down to one significant figure to finally arrive at 6.
Scientific notation is an amazing tool for scientific accuracy, because when working with complicated systems such as mechanical engineering precision or particle physics, a single bit of inaccuracy could destroy all of our hard work!