Curate, connect, and discover
If you look through definitions of “number” most will say that numbers are used to represent quantities (amounts or measures). Whole numbers 0, 1, 2, 3, … are probably the first numbers that come to mind and they are often used to count things like say how may watermelons jimmy has. But when quantifying things like money, whole numbers are not always enough and so we have rational numbers (which include the whole numbers but also fractions and numbers with finite or repeating decimal expansions). Yet, sometimes even these numbers are not enough to express certain quantities. Pi, for example, is not a rational number but is certainly a number as it represents the quantity that is the ratio of a circle’s circumference to its diameter. It can be shown that the number pi has an infinite decimal expansion with no repeating patterns, and so a number like pi is called an irrational number. (Because they are silly? Although at first some thought so, the term irrational just means not rational.) More specifically, pi is a transcendental number as it is not the root of any polynomial. (Transcendental, because they transcend the usual notion of number? Idk. Again, strange names.) So, the rational numbers were extended to the real numbers to include both rational and irrational numbers. Either way, we see that both rational and irrational numbers are truly numbers since they can be used to represent quantities.
What about complex numbers though? Are they really numbers, or do people just call them “numbers”? So, we should ask, can complex numbers represent some amount or measure of something? Can jimmy have i watermelons? No, but jimmy can’t really have pi watermelons either and pi is a number. Jimmy may have a watermelon that weighs pi pounds though (the only way to know this would be if jimmy had a scale with infinite accuracy, which, turns out, he does). Okay but can jimmy have a watermelon weighing i pounds? That doesn’t seem to make sense. To see if complex numbers can represent quantities we need to elaborate on what complex numbers are exactly.
The complex numbers are the real numbers extended to include the square root of negative 1 (i) and all its multiples. They have the form a+bi where a and b are real numbers. i is called an imaginary number (named imaginary because, i is not a real number, but this implies numbers like i are somehow not “real”, in the usual English sense of the word (are any numbers really “real”?) again, with the names). What truly makes complex numbers different than the other numbers we have discussed is that they “live” in 2 dimensions (the complex plane); complex numbers (e.g., 7+2i) have a real part (7) and an imaginary part (2i). While real numbers (which include whole, rational, and irrational numbers) “live” in one dimension (they can be found anywhere on the number line).
So, a complex number is a sort of two-dimensional quantity, it has a real measure and an imaginary measure. This makes them strange as numbers. We know 12 is bigger than 11 and that there are a bunch of numbers in-between 11 and 12, but which is bigger 2-8i or 3+i? Complex numbers cannot be compared in the same way i.e., there is no way to order complex numbers from smallest to largest.
These properties make complex numbers more abstract than typical numbers we encounter day to day. Nevertheless, “complex numbers are useful abstract quantities that can be used in calculations and result in physically meaningful solutions. However, recognition of this fact is one that took a long time for mathematicians to accept.”—Wolfram MathWorld http://mathworld.wolfram.com/ComplexNumber.html