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Introduction

Introduction

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Introduction

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  1. Introduction The product of two complex numbers is found using the same method for multiplying two binomials. As when multiplying binomials, both terms in the first complex number need to be multiplied by both terms in the second complex number. The product of the two binomials x + y and x – y is the difference of squares: x2– y2. If y is an imaginary number, this difference of squares will be a real number since i • i= –1: (a + bi)(a – bi) = a2– (bi)2= a2– b2(–1) = a2+ b2. 4.3.3: Multiplying Complex Numbers

  2. Key Concepts Simplify any powers of ibefore evaluating products of complex numbers. In the following equations, let a, b, c, and d be real numbers. 4.3.3: Multiplying Complex Numbers

  3. Key Concepts, continued Find the product of the first terms, outside terms, inside terms, and last terms. Note: The imaginary unit ifollows the product of real numbers. (a + bi) • (c + di) = ac (product of the first terms) + adi(product of the outside terms) + bci(product of the inside terms) + bidi(product of the last terms) = ac+ adi+ bci+ bdi2 = ac+ bd(–1) + adi+ bci = (ac– bd) + (ad + bc)i 4.3.3: Multiplying Complex Numbers

  4. Key Concepts, continued ac – bdis the real part of the product, and ad + bcis the multiple of the imaginary unit iin the imaginary part of the product. A complex conjugate is a complex number that when multiplied by another complex number produces a value that is wholly real. The product of a complex number and its conjugate is a real number. 4.3.3: Multiplying Complex Numbers

  5. Key Concepts, continued The complex conjugate of a + bi is a – bi, and the complex conjugate of a – bi is a + bi. The product of a complex number and its conjugate is the difference of squares, a2– (bi)2, which can be simplified. a2– b2i2= a2– b2• (–1) = a2+ b2 4.3.3: Multiplying Complex Numbers

  6. Common Errors/Misconceptions incorrectly finding the product of two complex numbers incorrectly identifying the complex conjugate of a + bi as a value such as –a + bi, a + bi, or –a – bi 4.3.3: Multiplying Complex Numbers

  7. Guided Practice Example 1 Find the result of i2• 5i. 4.3.3: Multiplying Complex Numbers

  8. Guided Practice: Example 1, continued Simplify any powers of i. 4.3.3: Multiplying Complex Numbers

  9. Guided Practice: Example 1, continued Multiply the two terms. Simplify the expression, if possible, by simplifying any remaining powers of i or combining like terms. 5(–i) = –5i ✔ 4.3.3: Multiplying Complex Numbers

  10. Guided Practice: Example 1, continued 4.3.3: Multiplying Complex Numbers

  11. Guided Practice Example 2 Find the result of (7 + 2i)(4 + 3i). 4.3.3: Multiplying Complex Numbers

  12. Guided Practice: Example 2, continued Multiply both terms in the first polynomial by both terms in the second polynomial. Find the product of the first terms, outside terms, inside terms, and last terms. 4.3.3: Multiplying Complex Numbers

  13. Guided Practice: Example 2, continued Evaluate or simplify each expression. 4.3.3: Multiplying Complex Numbers

  14. Guided Practice: Example 2, continued Combine any real parts and any imaginary parts. ✔ 4.3.3: Multiplying Complex Numbers

  15. Guided Practice: Example 2, continued 4.3.3: Multiplying Complex Numbers

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