Lecture 15: Section 2.4 Complex Numbers Imaginary unit Complex numbers Operations with complex numbers Complex conjugate Rationalize the denominator Square roots of negative numbers Complex solutions of quadratic equations L15 - 1 Consider the equation x2 = −1. Def. The imaginary unit, i, is the number such that p i2 = −1 or i = −1 Power of i p i1 = −1 = i i2 = −1 i3 = i4 = i5 = i6 = i7 = i8 = Therefore, every integer power of i can be written as i; −1; −i; 1. In general, divide the exponent by 4 and rewrite: ex. 1) i85 2) (−i)85 3) i100 4) (−i)−18 L15 - 2 Def. Complex numbers are numbers of the form a + bi, where a and b are real numbers. a is the real part and b is the imaginary part of the complex number a + bi. a + bi is called the standard form of a complex number. ex. Write the number −5 as a complex number in standard form. NOTE: The set of real numbers is a subset of the set of complex numbers. If b = 0, the number a + 0i = a is a real number. If a = 0, the number 0 + bi = bi, is called a pure imaginary number. Equality of Complex Numbers a + bi = c + di if and only if L15 - 3 Operations with Complex Numbers Sum: (a + bi) + (c + di) = Difference: (a + bi) − (c + di) = Multiplication: (a + bi)(c + di) = NOTE: Use the distributive property (FOIL) and remember that i2 = −1. ex. Write in standard form: 1) 3(2 − 5i) − (4 − 6i) 2) (2 + 3i)(4 + 5i) L15 - 4 Complex Conjugates Def. The conjugate (complex conjugate) of a complex number a + bi is ex. Find the conjugate of the following: 1) −1 − 3i 2) −2i 3) 5 NOTE: A real number is its own conjugate. Theorem: The product of a complex number and its conjugate is a nonnegative real number: L15 - 5 Rationalize the denominator To express a fraction with a complex number in the denominator in standard form, multiply the numerator and denominator by the conjugate of the denominator. ex. Rationalize and write in standard form. 1 3 + 2i 1 − 3i ex. Perform the operation and write in 4 + 2i standard form. L15 - 6 Square Roots of Negative Numbers Def. If a is a positive real number then the principal square root of −a is given by p p −a = = ai ex. Evaluate: p 1) −64 p 2) −200 NOTE: Convert the negative radicand of a square root to a complex number before multiplying. p p p ( a b = ab is NOT valid if both a and b are negative numbers) p p ex. 1) −3 −3 2) p(−3)(−3) L15 - 7 ex. Perform the operations and simplify: p p 1) −8 + −18 p p 2) −5 −15 p 3) (1 + −2)2 L15 - 8 Complex Solutions of Quadratic Equations In the complex number system, the solutions of the quadratic equation ax2 +bx+c = 0, where a, b and c are real numbers and a =6 0 are given by the formula p −b ± b2 − 4ac x = 2a ex. Solve the equation x2 + 4x + 10 = 0 in the complex number system. NOTE: The complex solutions are always conjugates of each other! L15 - 9 To summarize: For the equation ax2 + bx + c = 0 with real coeffi- cients, find the discriminant D = b2 − 4ac. 1. If D > 0, the equation has two unequal real number solutions. 2. If D = 0, the equation has a repeated real solution. 3. If D < 0, the equation has two complex conjugate solutions that are not real numbers. ex. Determine the character of the solution(s) of the following equation: 1) 2x(6 − x) = 18 2) x−2 − 3x−1 + 3 = 0 L15 - 10.