Presentation is loading. Please wait.

Presentation is loading. Please wait.

Introduction Finding sums and differences of complex numbers is similar to finding sums and differences of expressions involving numeric and algebraic.

Published byCassandra Ellis Modified over 9 years ago

Similar presentations

Presentation on theme: "Introduction Finding sums and differences of complex numbers is similar to finding sums and differences of expressions involving numeric and algebraic."— Presentation transcript:

1 Introduction Finding sums and differences of complex numbers is similar to finding sums and differences of expressions involving numeric and algebraic quantities. The real parts of the complex number are similar to the numeric quantities, and the imaginary parts of the complex number are similar to the algebraic quantities. Before finding sums or differences, each complex number should be in the form a + bi. If i is raised to a power n, use the remainder of n ÷ 4 to simplify i n. 1 4.3.2: Adding and Subtracting Complex Numbers

2 Key Concepts First, find the sum or difference of the real parts of the complex number. Then, to find the sum or difference of the imaginary numbers, add or subtract the coefficients of i. The resulting sum of the real parts and the imaginary parts is the solution. In the following equation, let a, b, c, and d be real numbers. (a + bi) + (c + di) = a + c + bi + di = (a + c) + (b + d)i 2 4.3.2: Adding and Subtracting Complex Numbers

3 Key Concepts, continued a + c is the real part of the sum, and (b + d)i is the imaginary part of the sum. When finding the difference, distribute the negative throughout both parts of the second complex number. (a + bi) – (c + di) = a + bi – c – di = (a – c) + (b – d)i a – c is the real part of the difference, and (b – d)i is the imaginary part of the difference. The sum or difference of two complex numbers can be wholly real (having only real parts), wholly imaginary (having only imaginary parts), or complex (having both real and imaginary parts). 3 4.3.2: Adding and Subtracting Complex Numbers

4 Common Errors/Misconceptions failing to distribute a negative throughout both the real and imaginary parts of a complex number before simplifying a difference adding the multiples of two powers of i when the powers of i are not equal, such as 2i 2 + 3i = 5i 4 4.3.2: Adding and Subtracting Complex Numbers

5 Guided Practice Example 2 Is (5 + 6i 9 ) – (5 + 3i 15 ) wholly real or wholly imaginary, or does it have both a real and an imaginary part? 5 4.3.2: Adding and Subtracting Complex Numbers

6 Guided Practice: Example 2, continued 1.Simplify any expressions containing i n. Two expressions, 6i 9 and 3i 15, contain i n. Divide each power of i by 4 and use the remainder to simplify i n. 9 ÷ 4 = 2 remainder 1, so 9 = 2 4 + 1. i 9 = i 2 4 i 1 = i 15 ÷ 4 = 3 remainder 3, so 15 = 3 4 + 3. i 15 = i 3 4 i 3 = –i 6 4.3.2: Adding and Subtracting Complex Numbers

7 Guided Practice: Example 2, continued Replace each occurrence of i n in the expressions with the simplified versions, and replace the original expressions in the difference with the simplified expressions. 6i 9 = 6 (i) = 6i 3i 15 = 3 (–i) = –3i (5 + 6i 9 ) – (5 + 3i 15 ) = (5 + 6i) – [5 + (–3i)] 7 4.3.2: Adding and Subtracting Complex Numbers

8 Guided Practice: Example 2, continued 2.Distribute the difference through both parts of the complex number. 8 4.3.2: Adding and Subtracting Complex Numbers

9 Guided Practice: Example 2, continued 3.Find the sum or difference of the real parts. 5 – 5 = 0 9 4.3.2: Adding and Subtracting Complex Numbers

10 Guided Practice: Example 2, continued 4.Find the sum or difference of the imaginary parts. 6i + 3i = 9i 10 4.3.2: Adding and Subtracting Complex Numbers

11 Guided Practice: Example 2, continued 5.Find the sum of the real and imaginary parts. 0 + 9i = 9i 11 4.3.2: Adding and Subtracting Complex Numbers

12 Guided Practice: Example 2, continued 6.Use the form of the sum to determine if it is wholly real or wholly imaginary, or if it has both a real and an imaginary part. 9i has only an imaginary part, 9i, so the difference is wholly imaginary. 12 4.3.2: Adding and Subtracting Complex Numbers ✔

13 Guided Practice: Example 2, continued 13 4.3.2: Adding and Subtracting Complex Numbers

14 Guided Practice Example 3 Is (12 – I 20 ) + (–18 – 4i 18 ) wholly real or wholly imaginary, or does it have both a real and an imaginary part? 14 4.3.2: Adding and Subtracting Complex Numbers

15 Guided Practice: Example 3, continued 1.Simplify any expressions containing i n. Two expressions, i 20 and 4i 18, contain i n. Divide each power of i by 4 and use the remainder to simplify i n. 20 ÷ 4 = 5 remainder 0, so 20 = 5 4 + 0. i 20 = i 5 4 = 1 18 ÷ 4 = 4 remainder 2, so 18 = 4 4 + 2. i 18 = i 4 4 i 2 = –1 15 4.3.2: Adding and Subtracting Complex Numbers

16 Guided Practice: Example 3, continued Replace each occurrence of i n in the expressions with the simplified versions, and replace the original expressions in the difference with the simplified expressions. i 20 = 1 4i 18 = 4 (–1) = –4 (12 – i 20 ) + (–18 – 4i 18 ) = (12 – 1) + [–18 – (–4)] 16 4.3.2: Adding and Subtracting Complex Numbers

17 Guided Practice: Example 3, continued 2.Find the sum or difference of the real parts. 17 4.3.2: Adding and Subtracting Complex Numbers

18 Guided Practice: Example 3, continued 3.Find the sum or difference of the imaginary parts. The expression contains no multiples of i, so there are no imaginary parts, and the multiple of i is 0: 0i. 18 4.3.2: Adding and Subtracting Complex Numbers

19 Guided Practice: Example 3, continued 4.Find the sum of the real and imaginary parts. –3 + 0i = –3 19 4.3.2: Adding and Subtracting Complex Numbers

20 Guided Practice: Example 3, continued 5.Use the form of the sum to determine if it is wholly real or wholly imaginary, or if it has both a real and an imaginary part. –3 has only a real part, –3, so the sum is wholly real. 20 4.3.2: Adding and Subtracting Complex Numbers ✔

21 Guided Practice: Example 3, continued 21 4.3.2: Adding and Subtracting Complex Numbers

Download ppt "Introduction Finding sums and differences of complex numbers is similar to finding sums and differences of expressions involving numeric and algebraic."

Similar presentations

Complex Numbers Objectives Students will learn:

Complex Numbers Objectives Students will learn:
Complex Numbers.

Complex Numbers.
Complex Numbers; Quadratic Equations in the Complex Number System

Complex Numbers; Quadratic Equations in the Complex Number System
Operations with Complex Numbers

Operations with Complex Numbers
Introduction Recall that the imaginary unit i is equal to. A fraction with i in the denominator does not have a rational denominator, since is not a rational.

Introduction Recall that the imaginary unit i is equal to. A fraction with i in the denominator does not have a rational denominator, since is not a rational.
Introduction Functions are relations in which each element in the domain is mapped to exactly one element in the range; that is, for every value of x,

Introduction Functions are relations in which each element in the domain is mapped to exactly one element in the range; that is, for every value of x,
Adapted from Walch Education  First, find the sum or difference of the real parts of the complex number. Then, to find the sum or difference of the.

Adapted from Walch Education  First, find the sum or difference of the real parts of the complex number. Then, to find the sum or difference of the.
Introduction A trinomial of the form that can be written as the square of a binomial is called a perfect square trinomial. We can solve quadratic equations.

Introduction A trinomial of the form that can be written as the square of a binomial is called a perfect square trinomial. We can solve quadratic equations.
Introduction You have learned several methods for solving polynomial equations by determining the factors, but not all equations are factorable. In this.

Introduction You have learned several methods for solving polynomial equations by determining the factors, but not all equations are factorable. In this.
Introduction Polynomial identities can be used to find the product of complex numbers. A complex number is a number of the form a + bi, where a and b are.

Introduction Polynomial identities can be used to find the product of complex numbers. A complex number is a number of the form a + bi, where a and b are.
6.2 – Simplified Form for Radicals

6.2 – Simplified Form for Radicals
Introduction You can determine how far a ladder will extend from the base of a wall by creating a quadratic equation and then taking the square root. To.

Introduction You can determine how far a ladder will extend from the base of a wall by creating a quadratic equation and then taking the square root. To.
Introduction Solving inequalities is similar to solving equations. To find the solution to an inequality, use methods similar to those used in solving.

Introduction Solving inequalities is similar to solving equations. To find the solution to an inequality, use methods similar to those used in solving.
Review and Examples: 7.4 – Adding, Subtracting, Multiplying Radical Expressions.

Review and Examples: 7.4 – Adding, Subtracting, Multiplying Radical Expressions.
Complex Numbers OBJECTIVES Use the imaginary unit i to write complex numbers Add, subtract, and multiply complex numbers Use quadratic formula to find.

Complex Numbers OBJECTIVES Use the imaginary unit i to write complex numbers Add, subtract, and multiply complex numbers Use quadratic formula to find.
Introduction While it may not be efficient to write out the justification for each step when solving equations, it is important to remember that the properties.

Introduction While it may not be efficient to write out the justification for each step when solving equations, it is important to remember that the properties.
Introduction Two equations that are solved together are called systems of equations. The solution to a system of equations is the point or points that.

Introduction Two equations that are solved together are called systems of equations. The solution to a system of equations is the point or points that.
Introduction Until now, you have been told that you cannot take the square of –1 because there is no number that when squared will result in a negative.

Introduction Until now, you have been told that you cannot take the square of –1 because there is no number that when squared will result in a negative.
Introduction Algebraic expressions are mathematical statements that include numbers, operations, and variables to represent a number or quantity. We know.

Introduction Algebraic expressions are mathematical statements that include numbers, operations, and variables to represent a number or quantity. We know.
Standardized Test Practice

Standardized Test Practice

Similar presentations

Ads by Google