Lesson 11

In the previous lesson, students defined a number \(\sqrt{\text-1}\) as a solution to the equation \(x^2=\text-1\) and used that definition to build the imaginary number line. In this lesson, students are introduced to the traditional symbol \(i\) to denote the new number they defined in the previous lesson. Students solve various quadratic equations of the form \(x^2=\text-a\) that have imaginary solutions and they multiply imaginary numbers together. In future lessons, students will rewrite expressions like \(\sqrt{a}\), where \(a\) is a negative number, while using the quadratic formula to solve quadratic equations.

Students also see how the real and imaginary number lines can be used together to represent complex numbers — numbers that can be written as \(a+bi\), where \(a\) and \(b\) are real numbers and \(i^2=\text-1\). While a deep, geometric interpretation of complex numbers in the complex plane is beyond the scope of this course, some activities in this unit use the complex plane to support student understanding. The complex plane helps students conceptualize numbers that are not on the real number line and make sense of complex addition. This is similar to how the real number line can be used to understand signed numbers and signed number addition, but is not a topic itself. There are purposefully no assessment items related to the complex plane in this course.

Students attend to precision to understand the connection between solutions to \(x^2=\text-a\) for positive \(a\), the meaning of \(\sqrt{\text-a}\), the imaginary numbers \(i\sqrt{a}\) and \(\text-i\sqrt{a}\), and the phrase “square roots of \(\text-a\)” (MP6).