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 i before 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 i follows 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 – bd is the real part of the product, and ad + bc is the multiple of the imaginary unit i in 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 – b 2i 2 = 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 i 2 • 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