Mathematics has built five main number systems, each one an extension of the previous. Every extension was motivated by an equation that had no solution: "what is 3-5?" forced the integers; "what is 1/3?" forced the rationals; "what is sqrt(2)?" forced the reals; "what is sqrt(-1)?" forced the complex numbers.
The five number systems: what problem each one solved
What each extension gains and what it loses
Where each type of number lives on (and off) the number line
The rationals are dense: between any two rationals there is another rational - yet they still leave gaps. The irrationals (√2, e, π…) fill those gaps. Together they make up the real line. Complex numbers add an entirely new dimension perpendicular to it.
Mathematics has five main number systems: natural numbers N (counting, no subtraction), integers Z (add subtraction and negatives), rationals Q (add division), reals R (add limits, irrationals), complex numbers C (add sqrt(-1)). Each extension solved an equation unsolvable in the previous system. Complex numbers are algebraically closed: every polynomial equation has a solution within C. The inclusion is strict: N inside Z inside Q inside R inside C, with transcendentals filling the outer ring of R.