Real numbers

Put the rational numbers Q and the irrational numbers Q′ together and you get the real numbers R = Q ∪ Q′. No number is in both, so Q ∩ Q′ = ∅. Inside Q sit the integers Z, and inside Z the natural numbers N (which include 0 in this book): N ⊂ Z ⊂ Q ⊂ R. Write ∈ when a number belongs to a set and ∉ when it does not: √2 ∈ Q′ and √2 ∉ Q. To order real numbers, write them as decimals, or square them when they are all positive: √12, 4, √40 and 6 become 12, 16, 40 and 36, so √12 < 4 < 6 < √40. To solve an equation in a set, find the numbers of that set that work. x² = 5 gives x = √5 or x = −√5; both are in R, but the solution set in Q is empty. x³ + 9 = 0 gives x = −∛9, which is in R but not in Q. And x² = −4 has no solution in R, because a square is never negative.