In Grade 8, students evaluate square roots and use them to represent the solution of an equation in the form 𝑥² = 𝑝 where 𝑝 is a positive rational number. The complex numbers extend that knowledge to introduce the solution of equations in the form 𝑥² = 𝑝 where 𝑝 is negative rational number.
During the years from kindergarten to Grade 8, students must repeatedly extend their conception of number. With each extension of number, the meanings of addition, subtraction, multiplication, and division are extended. In each new number system-integers, rational numbers, real numbers, and complex numbers-the four operations stay the same in two important ways: They have the commutative, associative, and distributive properties and their new meanings are consistent with their previous meanings.