Complex Numbers

How do we simplify complex numbers?  

Based on lesson by Milton Francis


Basic Information




Milton Francis


In this lesson, students will reinforce their previous knowledge of complex numbers by performing arithmetic operations on them. That is, they will add, subtract, multiply and divide using these imaginary/complex numbers. Their solution to these complex number problems will be written in the form of the sum or difference of a real number and an imaginary number, i.e. a + bi, where 'a' represents the real number and 'bi', the imaginary portion.



Time Frame

Two 48-minute class (i.e. 2 days)




How do we simplify Complex Numbers, leaving the answers in the form a + bi.


Even though this is a NY Regents Math B lesson, it could be modified to a Math A level. In teaching it as a Math A lesson, the teacher would not conjugate. That is, he/she would not introduce the binomial a + bi as the denominator. Instead, only the monomial bi, would be used in the denominators. By using this imaginary monomial, there would be no need to conjugate, as conjugation only works with binomial denominators. This modification, however, would cause the topic of the lesson to read thus: Simplifying Complex Numbers.



Standards and Key Concepts



NY- New York State Standards

Subject: Mathematics, Science, and Technology (1996, 2005 Math update)

Learning Standard 3: Mathematics (2005 update)
Students will:
•understand the concepts of and become proficient with the skills of mathematics;
•communicate and reason mathematically;
•become problem solvers by using appropriate tools and strategies;
through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.

Grade/Subject : Algebra 2 and Trigonometry

Area : Content Strands

Strand : Number Sense and Operations Strand

Standard : Students will understand meanings of operations and procedures, and how they relate to one

Performance Indicator : A2.N.9 Perform arithmetic operations on complex numbers and write
the answer in the form bi a + . Note: This includes simplifying
expressions with complex denominators.

Performance Indicator : A2.N.8 Determine the conjugate of a complex number

Performance Indicator : A2.N.7 Simplify powers of i


At the culmination of this lesson (Day 2), the students should be able to:
1. Recall the use of the imaginary number, i.
2. Determine the conjugate of a complex number.
3. Apply the four basic rules of arithmetic (addition, subtraction, multiplication, and division) in simplifying complex numbers.
4. Recognized that conjugation is only applied to the rule of division in complex numbers, and not to the other three rules.

Essential Questions

In a previous lesson on imaginary numbers, we rationalized the denominator of the fraction 5/2i by multiplying the fraction by 1 in the form of 2i/2i. How can we rationalize the denominator of the following fractions:
1. 6/(2 + i)and
2. (5 + 4i)/(7 - 3i),
being mindful of the fact that in rationalizing, we multiply the fraction by 1 in a certain form, and that an imaginary number cannot be part of a resulting denominator?

Knowledge and Skills

To fully comprehend the lesson, students would recall the knowledge applying the four basic arithmetic rules to algebraic expressions involving binomials. For example, the could use a few minutes prior to the lesson by reviewing these links:


1.               Adding Polynomials Review of addition of   


2.               Multiplying Binomials Multiplication of   


3.               Subtracting Polynomials Subtracting  


Performance Tasks and Assessment



Performance Task

This lesson is technology-related, so the students will use the Internet throughout.  After reviewing the polynomial links, they will be directed to a link on the topic.  This direction will serve to assist them in following instructions and the steps used in simplifying the complex numbers, while simultaneously discovering the wealth of information on the Internet with regards to the topic being learned.  The link is as follows:    


1.                Operations on Complex Numbers Addition,  

           subtraction, multiplication and division of   

           complex numbers.

Performance Prompt

The following interactive exercise will be done by the students as they work in pairs:                                    Links

1.               Practice Exercise on Simplifying Complex Numbers Addition, subtraction, and multiplication of complex numbers.

2.               Practice Multiplying and Dividing Complex Numbers Multiplication and division of complex numbers.


Complex Numbers 

Learning Experiences and Resources



Sequence of Activities

To comprehend the lesson by the culminating exercise, the students would have done the following:

1.  Differentiate between real and imaginary numbers.

2.  Understand the concept of the imaginary number i.

3.  Be able to simplify/not simplify like/unlike algebraic terms respectively through addition and subtraction.

4.  Able to multiply binomial algebraic expressions using the First-Outer- Inner-Last (FOIL) method.


Differentiated Instruction

For my English as a Second Language (ESL) students, certain terms of the lesson like conjugate, raionalize, imaginary, etc would be written as the equivalent to the Spanish terms and be placed on the chalkboard.  However, all four groups, ESL, At-Risk,Special Ed.and Advanced Learners would be working as a small group.  Each group will have at least one advanced learner assigned to it, whose task is to assist the other learners to comprehend the lesson as the teacher teaches.   


  • Materials and resources:
    Printed copy of homework sheet on the day's lesson
  • Technology resources:
    Internet Explorer
  • The number of computers required is 1 per student.
  • Students Familiarity with Software Tool:
    Students are familiar with the use of the Internet.
  • Computer use (Internet accessibilty)
    Use of notebooks and pens/pencils
    Distribution of printed copy of homework sheet.


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