§ 7.7 Complex Numbers

Complex Numbers In the next chapter we will study equation whose solutions involve the square roots of negative numbers. Because the square of a real number is never negative, there are no real number solutions to those equations. However, there is an expanded system of numbers in which the square root of a negative number is defined. This set is called the set of complex numbers. The imaginary number i is the basis of this new set. So come… now go with us to never-never land , a place where you have not been before… Blitzer, Intermediate Algebra, 5e – Slide #2 Section 7.7

The Square Root of a Negative Number Complex Numbers The Imaginary Unit i The imaginary unit i is defined as The Square Root of a Negative Number If b is a positive real number, then Blitzer, Intermediate Algebra, 5e – Slide #3 Section 7.7

Complex Numbers Write as a multiple of i: EXAMPLE SOLUTION Blitzer, Intermediate Algebra, 5e – Slide #4 Section 7.7

Complex Numbers & Imaginary Numbers The set of all numbers in the form with real numbers a and b, and i, the imaginary unit, is called the set of complex numbers. The real number a is called the real part, and the real number b is called the imaginary part of the complex number If , then the complex number is called an imaginary number. Blitzer, Intermediate Algebra, 5e – Slide #5 Section 7.7

Complex Numbers & Imaginary Numbers The set of all numbers in the form with real numbers a and b, and i, the imaginary unit, is called the set of complex numbers. The real number a is called the real part, and the real number b is called the imaginary part of the complex number If , then the complex number is called an imaginary number. Blitzer, Intermediate Algebra, 5e – Slide #6 Section 7.7

Complex Numbers & Imaginary Numbers The set of all numbers in the form with real numbers a and b, and i, the imaginary unit, is called the set of complex numbers. The real number a is called the real part, and the real number b is called the imaginary part of the complex number If , then the complex number is called an imaginary number. Blitzer, Intermediate Algebra, 5e – Slide #7 Section 7.7

Adding & Subtracting Complex Numbers 1) In words, this says that you add complex numbers by adding their real parts, adding their imaginary parts, and expressing the sum as a complex number. 2) In words, this says that you subtract complex numbers by subtracting their real parts, subtracting their imaginary parts, and expressing the difference as a complex number. Blitzer, Intermediate Algebra, 5e – Slide #8 Section 7.7

Complex Numbers EXAMPLE Perform the indicated operations, writing the result in the form a + bi: (a) (-9 + 2i) – (-17 – 6i) (b) (-2 + 6i) + (4 - i). SOLUTION (a) (-9 + 2i) – (-17 – 6i) = -9 + 2i + 17 + 6i Remove the parentheses. Change signs of the real and imaginary parts being subtracted. = -9 + 17 + 2i + 6i Group real and imaginary terms. = (-9 + 17) + (2 + 6)i Add real parts and imaginary parts. = 8 + 8i Simplify. Blitzer, Intermediate Algebra, 5e – Slide #9 Section 7.7

Complex Numbers (b) (-2 + 6i) + (4 - i) = -2 + 6i + 4 - i CONTINUED (b) (-2 + 6i) + (4 - i) = -2 + 6i + 4 - i Remove the parentheses. = -2 + 4 + 6i - i Group real and imaginary terms. = (-2 + 4) + (6 - 1)i Add real parts and imaginary parts. = 2 + 5i Simplify. Blitzer, Intermediate Algebra, 5e – Slide #10 Section 7.7

Complex Numbers EXAMPLE Find the products: (a) -6i(3 – 5i) (b) (-4 + 2i)(-4 - 2i). SOLUTION (a) -6i(3 – 5i) Distribute -6i through the parentheses. Multiply. Replace with -1. Simplify and write in a + bi form. (b) (-4 + 2i)(-4 – 2i) Use the FOIL method. Blitzer, Intermediate Algebra, 5e – Slide #11 Section 7.7

Complex Numbers Group real and imaginary terms. CONTINUED Group real and imaginary terms. Combine real and imaginary terms. Blitzer, Intermediate Algebra, 5e – Slide #12 Section 7.7

Multiplying Complex Numbers Because the product rule for radicals only applies to real numbers, multiplying radicands is incorrect. When performing operations with square roots of negative numbers, begin by expressing all square roots in terms of i. Then perform the indicated operation. Blitzer, Intermediate Algebra, 5e – Slide #13 Section 7.7

Complex Numbers Multiply: Express square roots in terms of i. EXAMPLE Multiply: SOLUTION Express square roots in terms of i. The square root of 64 is 8. Blitzer, Intermediate Algebra, 5e – Slide #14 Section 7.7

Complex Numbers Divide and simplify to the form a + bi: EXAMPLE Divide and simplify to the form a + bi: SOLUTION The conjugate of the denominator is 4 – 2i. Multiplication of both the numerator and the denominator by 4 – 2i will eliminate i from the denominator. Multiply by 1. Use FOIL in the numerator and in the denominator. Blitzer, Intermediate Algebra, 5e – Slide #15 Section 7.7

Complex Numbers Simplify. Perform the multiplications involving -1. CONTINUED Simplify. Perform the multiplications involving -1. Combine like terms in the numerator and denominator. Express answer in the form a + bi. Simplify. Blitzer, Intermediate Algebra, 5e – Slide #16 Section 7.7

Complex Numbers Divide and simplify to the form a + bi: EXAMPLE Divide and simplify to the form a + bi: SOLUTION The conjugate of the denominator, 0 - 4i, is 0 + 4i. Multiplication of both the numerator and the denominator by 4i will eliminate i from the denominator. Multiply by 1. Multiply. Use the distributive property in the numerator. Blitzer, Intermediate Algebra, 5e – Slide #17 Section 7.7

Complex Numbers Perform the multiplications involving -1. CONTINUED Perform the multiplications involving -1. Express the division in the form a + bi. Simplify real and imaginary parts. Blitzer, Intermediate Algebra, 5e – Slide #18 Section 7.7

Simplifying Powers of i Complex Numbers Simplifying Powers of i 1) Express the given power of i in terms of 2) Replace with -1 and simplify. Use the fact that -1 to an even power is 1 and -1 to an odd power is -1. Blitzer, Intermediate Algebra, 5e – Slide #19 Section 7.7

Complex Numbers Simplify: EXAMPLE SOLUTION Blitzer, Intermediate Algebra, 5e – Slide #20 Section 7.7

In Summary… To add or subtract complex numbers, add or subtract their real parts and then add or subtract their imaginary parts. Adding complex numbers is easy. To multiply complex numbers, use the rule for multiplying binomials. After you are done, remember that and make the substitution. In fact, if you can only remember one thing from this section – remember this fact, that is, when your square i, you get -1. To divide complex numbers, multiply numerator and denominator by the conjugate of the denominator. This gives you a real number in the denominator, and you will know how to proceed from that point. Blitzer, Intermediate Algebra, 5e – Slide #21 Section 7.7