Natural numbers is a very intuitive mathematical construction, corresponding to counting, and this is why they are termed natural. Together with natural numbers, addition appears.
Then the need appears to solve problems like $3+x=7$, how much is $x$ ? This is solved by means of subtraction. But then you face frustrating problems such as $7+x=3$, how much is $x$ ? To cope with these, negative numbers are introduced, forming the integers. (A close friend of entire.)
The next step is to look at multiplication, i.e. repeated addition. All is fine until you want to solve problems like $4\times x=24$, how much is $x$ ?, then $7\times x=17$, how much is $x$ ? This is how division and the rational numbers are introduced. (From ratio.)
The ancient Greeks once discovered that rational numbers are not all, much to their resentment, when they asked the question $x\times x=2$, how much is $x$ ? Then came the real numbers. (Possibly evoking the continuous characteristic of our real world.)
Another step was reached in the Middle-Age when mathematicians started to deal with the equation $x\times x=-1$, also annoyingly unsolvable, and the imaginary and complex numbers were introduced.
You may attach a "romantic" meaning to these terms, but this is not the intent. On the opposite, they are conventional and universally adopted words with a well-defined understanding, and probably a more mnemonic intent.
Even though some of these number categories may look somewhat artificial, there are cases where they come handy just for intermediate results in solving a real-world problem. A famous example is the need for complex numbers to find the three real roots of some cubic equations.