Number sets and square roots

Numbers form nested families, like Russian dolls. The natural numbers (0, 1, 2, 3...) form the set N. Adding the negatives gives the integers: the set Z. Then come decimal numbers (like 3.72) and the rationals, the set Q: all quotients of integers, like 3/4 or -5/2. Each family contains the previous one: every natural number is an integer, every integer is rational (for example -5 = -5/1).

But some numbers escape all these families! The square root of 2, which measures the diagonal of a square with side 1, cannot be written as any fraction of integers. It is an irrational number: its decimals 1.41421356... go on forever with no repeating pattern. Adding these rebels gives the set R of real numbers: the whole number line, with no gaps at all.

Speaking of square roots, let us dig in. For a positive number a, the square root of a, written √a, is the positive number whose square equals a. So √81 = 9 because 9² = 81, and (√5)² = 5 by definition. Knowing the perfect squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100...) gives you a real superpower.

Two reflexes to finish. First, the root of a product splits: √(a × b) = √a × √b, which lets you simplify √50 = √(25 × 2) = 5√2. Second, beware of the trap: the root of a sum does NOT split. √(16 + 9) = √25 = 5, while √16 + √9 = 4 + 3 = 7. The difference is plain to see!