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Complex Numbers

Prerequisite Topics, Study Materials and Readiness Test

Introduction to Sets [Link]
Sets of Numbers [Link]

Geometric Representations of Sets of Numbers [Link]
Cartesian Coordinate System [Link]
Proficiency with Arithmetic Operations [Link]

Unit I. Algebraic Point of View

Introduction

The real component of 5+ 8i is 5  and its complex component is  8.

The real component of    58i  is 5  and its complex component is  8.

The real component of    23+ 78 i is  23  and it complex component is  78 .

Observe that i IS NOT part of the complex component.

Spaced and Interleaved Retrieval Practice

When answering questions in mathematics textbooks or classes, getting the answer is not your primary goal.
Your primary goals are:
(1) To learn concepts and processes using those concepts;
(2) To learn to write mathematics correctly and effectively;
(3) To convince yourself that you understand the concepts and procedures and;
(4) To convince yourself that you can properly communicate mathematics.

To properly answer the questions posed here requires that you write your reasoning (not just the numbers), probably you will rewite your reasoning several times before your presentation satisfactorly communicates the entire process. These questions are designed to force a review of previously learned material. Ensure that you do the necessary review. You should always reject first drafts or sloppy work not presented in a logical manner. Your response to questions should always include WORDS. The numbers are not the concepts!

No Decimals No mixed numbers No complex fractions No boxed or circled answers Put your calculator away

  1. Is a complex number a mathematical object?, a mathematical relation?, a binary operation?, a unary operation?
  2. Is equality of complex numbers a mathematical object?, a mathematical relation?, a binary operation?, a unary operation?
  3. What is a binary operation?
  4. Which of the following are complex numbers? Why? or Why not?
    If the complex number is not written in standard form rewrite it in standard form.
    3+4i,  7 34i,  5 ,  93i,    23i5 85 ,  3i + 5812
  5. Identify the real component in each of the following complex numbers? Classify the real component as a Natural number, a Whole number, an Integer, a Rational number, an Irrational number, a Real Number. 3+5i        2 3 8i       3 4 + 7 i        5+3i 7
  6. Identify the complex component in each of the following complex numbers? Classify the complex component as a Natural number, a Whole number, an Integer, a Rational number, an Irrational number, a Real number. 3+5i        2 3 8i       3 4 + 7 i        5+3i 7
  7. What is the universal symbol for the set of Rational numbers?
  8. Where would you picture/graph the real component of a complex number? Where would you picture/graph the complex component of a complex number?
  9. Why can't a non-real complex number be plotted on a real number line?
  10. Write each of the following real numbers as a complex number? What is its real component? What is its complex component? 3      2 3        7 8         5
  11. For each pair of complex numbers, determine whether they are equal or not. Explain how you arrived at your answer.

    3 + 4i and 4i + 3

    4 7i  and  2 - 7i

    5 7i  and  5 - 7i

    8 2 +3 i  and  4 + 6i

  12. Real Numbers are equal if they represent the same point on the real number line. Can you explain equality of complex numbers in the same manner?
  13. Can you explain equality of complex numbers by using the definition of equality of real numbers?
  14. Illustrate addition of real numbers on the real number line.
  15. Can you create a similar illustration for addition of complex numbers? Why not?
  16. What prerequiste topics did you use/review in order to answer these questions? Be specific.

Writing Conventions

The form a+bi as displayed in the above definition is called the standard form and is the prefered form for writing a complex number. For example each of the following are complex numbers written in standard form.

3+4i     7 8 5i     7 3 4 3 i     6 +  3 i

Remember the definition i= 1. From which it follows that i2=1.
Consequently i NEVER appears with an exponent greater than 1 in a complex number. All squared powers of i are replaced with 1.

It has become acceptable to occasionally write a complex number in the form a+ib. Presumably this is done to avoid confusion when the complex component contains a radical such as in 3+ 5i. By writing this complex number as 3+i5 we avoid posssible confusion about whether i is part of the radicand or not. However, because in a complex number, i is NEVER contained in a radicand, no possible confusion exists. Therefore although it has become acceptable to write a complex number like 3+i5 the prefered form is the standard form 3+ 5i.

If a the real component and the complex component both are fractions and they have the same denominator ac+bc i then there are three acceptable ways (the three forms are equal) to write that complex number: ac+bc i,    a+bic , and      1c( a+bi).

For Example:

34+7 4i=3+7 i4=14 (3+7i )


7 5+25i= 7+2i5 =15( 7 +2i)


3+2 57 +65 7i=( 3+2 )+6i5 7=( 157 )(( 3+2 )+6i )

Is it okay to write a complex number in the form bi+a ?
A beginning student might argue that becasue addition is commutative (i.e. 7 + 9 = 9 +7) then it follows that a+bi=b i+a.
However, the Commutative property refers to addition of Real numbers and the + symbol in the standard form for a complex number does not represent addition of Real numbers. In fact it does not mean addition! The + in the standard form of a complex number is simply a part of the symbol for a complex number. Just like is a part of the symbol for principle square root.

It is improper/incorrect to write bi+a.
However, I cannot give an example where writing a complex number in the form bi+a will lead to an error.

Operations

Spaced and Interleaved Retrieval Practice

When answering questions in mathematics textbooks or classes, getting the answer is not your primary goal.
Your primary goals are:
(1) To learn concepts and processes using those concepts;
(2) To learn to write mathematics correctly and effectively;
(3) To convince yourself that you understand the concepts and procedures and;
(4) To convince yourself that you can properly communicate mathematics.

To properly answer the questions posed here requires that you write your reasoning (not just the numbers), probably you will rewite your reasoning several times before your presentation satisfactorly communicates the entire process. These questions are designed to force a review of previously learned material. Ensure that you do the necessary review. You should always reject first drafts or sloppy work not presented in a logical manner. Your response to questions should always include WORDS. The numbers are not the concepts!

No Decimals No mixed numbers No complex fractions No boxed or circled answers Put away your calculator

  1. Perform each of the indicated additions. Write the sum in standard form with real and complex components reduced to simplest form. Identify the addends and the sum in each addition problem.
  2. Write the opposite of each complex number.
  3. Change each subtraction problem to an addition problem by adding the minuend and the opposite of the subtrahend. Perform the additions.
  4. Perform each of the indicated subtractions by changing them to addition problems. Write the difference in standard form with real and complex components reduced to simplest form.Identify the subtrahend, minuend, and difference in each problem.
  5. Perform each of the indicated multiplications as if multiplying binomials. Write the product in standard form with real and complex components reduced to simplest form. Identify the factors and the product in each multiplication problem.
  6. What kind of number is the product of two purely complex numbers?
  7. In each multiplication problem write the real number as a complex number with 0 complex component, then perform the multiplication as if multiplying two binomials. Formulate a type of distributive rule for multiplication of a real number and a complex number.
  8. Use your newly formulated distributive law to factor the largest common factor from the real component and the complex component. Write the original complexx number as a product of a real number and a complex number.
  9. Write the conjugate of each complex number. Identify the real and complex components of the original number and its conjugate. What is the relation between the real componets of the original number and the real component of its conjugate? What is the relation between the complex componets of the original number and the complex component of its conjugate?
  10. Calculate the norm of each complex number. Classify the norm as a Natural number, a Whole number, an Integer, a Rational number, an Irrational number, a Real number, a Complex number.
  11. Calculate the norm of each complex number. Calculate the product of each complex number and its conjugate. What do you observe? Is the product of a complex number and its conjugate a real number?
  12. Calculate the multiplicative inverse of each complex number. What kind of number is the multiplicative inverse? What is the product of the original complex number and its multiplicative inverse.
  13. Calculate each quotient by changing it to a multiplication problem consisting of the dividend times the mulitiplicative inverse of the divisor.
  14. Multiply both sides of each equation by the multiplicative inverse of the coefficient of the variable x. Have you solved the equation?
  15. What prerequiste topics did you use/review in order to answer these questions? Be specific.
  16. What new mathematics objects were introduced in this unit? How do they relate to previously known mathematical objects?
  17. What new mathematics operations were introduced in this unit? How do they relate to previously known mathematical operations? Which are unary operations? Which are binary operations?
  18. What new mathematics relations were introduced in this unit? How do they relate to previously known mathematical relations?
  19. Are there any mathematical relations absent from complex numbers that were prominent in the real numbers? Why are they absent?

About Complex Numbers

Minimal List of Exercises Page 135.

If you understand the previous material you should be able to answer the following questions.
Each of these questions should be answered. Most of these questions are included in the MyMathLab homework requirement. If a particular concept is difficult for you, you should study the related text material and then try to answer some additional questions from the list provided by the department. In each case you are expected to make an honest adult evaluation of your understanding of the concept. Your ability to answer these questions is one tool to help you make that evaluation.
Section 1.4 Exercise Set: 1, 5, 9, 13, 19, 21, 25, 27, 29, 41.

Unit II. Geometric Point of View

This is normally not included in Collge Algebra so its development is delayed.