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.
Table showing properties gained and lost when extending number systems
| SYSTEM | GAINED | LOST/CHANGED |
|---|---|---|
| N (naturals) | counting, +, x | no subtraction |
| Z (integers) | subtraction, negatives | no division |
| Q (rationals) | division, fractions | no sqrt(2) |
| R (reals) | all limits, sqrt(2), pi | no sqrt(-1) |
| C (complex) | all polynomial roots | algebraically closed |
| H (quaternions) | 3D rotations | ab not = ba |
| Each extension is a genuine enlargement, not just renaming |
Blue: natural numbers ℕ. Green adds 0. Purple extends to negative integers ℤ. Orange adds fractions ℚ. Red: irrationals fill the rest of ℝ.
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.