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Algebra 2 Complex Numbers Lesson 4-8 Part 1 Goals Goal • To identify, graph, and perform operations with complex numbers. Rubric Level 1 – Know the goals. Level 2 – Fully understand the goals. Level 3 – Use the goals to solve simple problems. Level 4 – Use the goals to solve more advanced problems. Level 5 – Adapts and applies the goals to different and more complex problems. Vocabulary • • • • • • • Imaginary Unit Imaginary Number Complex Number Pure Imaginary Number Complex Number Plane Absolute Value of a Complex Number Complex Conjugate Essential Question Big Idea: Solving Equations 1. What are complex numbers? 2. How do you perform basic operations on complex numbers? What is a imaginary number? • It is a tool to solve an equation. – Invented to solve quadratic equations of the form x 2 4 . • It has been used to solve equations for the last 200 years or so. • “Imaginary” is just a name, imaginary do indeed exist, they are numbers. Imaginary Numbers You can see in the graph of f(x) = x2 + 1 below that f has no real zeros. If you solve the corresponding equation 0 = x2 + 1, you find that x = ,which has no real solutions. However, you can find solutions if you define the square root of negative numbers, which is why imaginary numbers were invented. The imaginary unit i is defined as . You can use the imaginary unit to write the square root of any negative number. Powers of i ii i 1 2 We could continue but notice that they repeat every group of 4. For every i 4 it will = 1 i i i 1(i) i To simplify higher powers of i 4 2 2 then, we'll group all the i and i i i 1 1 1 see what is left. 5 4 8 33 4 8 i i i 1i i i i i 1 i i 6 4 2 i i i 1 1 1 4 will go into 33 8 times with 1 left. 7 4 3 i i i 1 i i 20 3 83 4 20 3 i i i 1 i i 8 4 4 i i i 11 1 3 2 4ths 4 will go into 83 20 times with 3 left. Powers of i Helpful Hint Notice the repeating pattern in each row of the table. The pattern allows you to express any power of i as one of four possible values: i, –1, –i, or 1. Powers of i • Always reduce powers of i to; i, -1, 1, or –i. • To reduce, divide the exponent by 4 and the remainder determines the reduced value. – – – – remainder remainder remainder remainder 1⇒ i 2 ⇒ -1 3 ⇒ -i 0 (no remainder) ⇒1 • Negative exponent, the term goes in the denominator and the exponent becomes positive. Powers of i For exponents larger than 4, divide the exponent by 4, then use the remainder as your exponent instead. Example: i ? 23 23 5 with a remainder of 3 4 So, use i which -i 3 i i 23 Your Turn: • Simplify each of the following powers. 1) i22 -1 2) i36 1 3) i13 i 4) i47 -i Example: Simplifying Square Roots of Negative Numbers Express the number in terms of i. Factor out –1. Product Property. Simplify. Multiply. Express in terms of i. Example: Simplifying Square Roots of Negative Numbers Express the number in terms of i. Factor out –1. Product Property. Simplify. 4 6i 4i 6 Express in terms of i. Your Turn: Express the number in terms of i. Factor out –1. Product Property. Product Property. Simplify. Express in terms of i. Your Turn: Express the number in terms of i. Factor out –1. Product Property. Simplify. Multiply. Express in terms of i. Your Turn: Express the number in terms of i. Factor out –1. Product Property. Simplify. Multiply. Express in terms of i. The Complex-Number System • The complex-number system is used to find zeros of functions that are not real numbers. • When looking at a graph of a function, if the graph does not cross the x-axis, it has no real-number zeros. Complex Numbers A complex number is a number that can be written in the form a + bi, where a and b are real numbers and i= . The set of real numbers is a subset of the set of complex numbers C. Every complex number has a real part a and an imaginary part b. The Complex-Number System Complex Numbers C Real Numbers R Integers Z Whole numbers W Natural Numbers N Rational Numbers Q R Irrational Numbers Q -bar Imaginary Numbers i Standard Form of Complex Numbers Complex numbers can be written in the form a + bi (called standard form), with both a and b as real numbers. a is a real number and bi would be an imaginary number. If b = 0, a + bi is a pure real number (2, -56, 34 etc.). If a = 0, a + bi is an pure imaginary number (2i, -56i, 34i etc.). Standard Form of Complex Numbers • Since all numbers belong to the Complex number system, C, all numbers can be classified as complex. Real numbers, R, and Imaginary numbers, i, are subsets of Complex Numbers. Complex Numbers a + bi Real Numbers a + 0i Imaginary Numbers 0 + bi Standard Form of Complex Numbers Zero is the only number that is both real and imaginary. Standard Form of Complex Numbers Example: Write each of the following in the form of a complex number in standard form a + bi. 6 = 6 + 0i 8i = 0 + 8i 24 4 6 1 2 i 6 0 2 i 6 6 25 6 25 1 6 + 5i Your turn: • Write each of the following in the form of a complex number in standard form a + bi. • 9 5 • 24 5 3i 0 2i 6 Equity of Complex Numbers Two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal. Equity of Complex Numbers a+bi =c+di if and only if a=c and b=d. Example: Find the values of x and y that make the equation 4x + 10i = 2 – (4y)i true . Real parts 4x + 10i = 2 – (4y)i Imaginary parts 4x = 2 Equate the real parts. Solve for x. 10 = –4y Equate the imaginary parts. Solve for y. Your Turn: Find the values of x and y that make each equation true. 2x – 6i = –8 + (20y)i Real parts 2x – 6i = –8 + (20y)i Imaginary parts 2x = –8 Equate the real parts. x = –4 Solve for x. –6 = 20y Equate the imaginary parts. Solve for y. Your Turn: Find the values of x and y that make each equation true. –8 + (6y)i = 5x – i 6 Real parts –8 + (6y)i = 5x – i 6 Imaginary parts –8 = 5x Equate the real parts. Solve for x. Equate the imaginary parts. Solve for y. Complex Plane Just as you can represent real numbers graphically as points on a number line, you can represent complex numbers in a special coordinate plane. The complex plane is a set of coordinate axes in which the horizontal axis represents real numbers and the vertical axis represents imaginary numbers. Complex Plane Helpful Hint The real axis corresponds to the x-axis, and the imaginary axis corresponds to the y-axis. Think of a + bi as x + yi. Example: Graph each complex number. A. 2 – 3i –1+ 4i • B. –1 + 4i C. 4 + i D. –i 4+i • • –i • 2 – 3i Your Turn: Graph each complex number. a. 3 + 0i b. 2i c. –2 – i 2i • –2 – i d. 3 + 2i • 3 + 2i • • 3 + 0i Absolute Value of a Complex Number Recall that absolute value of a real number is its distance from 0 on the number line. Similarly, the absolute value of a complex number is its distance from the origin in the complex plane. Example: Find each absolute value. A. |3 + 5i| B. |–13| C. |–7i| |–13 + 0i| |0 +(–7)i| 13 7 Your Turn: Find each absolute value. a. |1 – 2i| b. c. |23i| |0 + 23i| 23 Adding and Subtracting Complex Numbers Sum or Difference of Complex Numbers If a + bi and c + di are complex numbers, then their sum is (a + bi) + (c + di) = (a + c) + (b + d)i Their difference is (a + bi) – (c + di) = (a – c) + (b – d)i Combine like Terms: add/sub. real parts and add/sub. Imaginary parts. Adding and Subtracting Complex Numbers Adding and subtracting complex numbers is similar to adding and subtracting variable expressions with like terms. Simply combine the real parts, and combine the imaginary parts. The set of complex numbers has all the properties of the set of real numbers. So you can use the Commutative, Associative, and Distributive Properties to simplify complex number expressions. Helpful Hint Complex numbers also have additive inverses. The additive inverse of a + bi is –(a + bi), or –a – bi. Procedure: 1. Pretend that “i” is a variable and that a complex number is a binomial 2. Add and subtract as you would binomials Example: (2 + i) – (-5 + 7i) + (4 – 3i) 2 + i + 5 – 7i + 4 – 3i 11 – 9i Example: Add or subtract. Write the result in the form a + bi. (4 + 2i) + (–6 – 7i) (4 – 6) + (2i – 7i) –2 – 5i Add real parts and imaginary parts. Example: Add or subtract. Write the result in the form a + bi. (5 –2i) – (–2 –3i) (5 – 2i) + 2 + 3i Distribute. (5 + 2) + (–2i + 3i) Add real parts and imaginary parts. 7+i Example: Add or subtract. Write the result in the form a + bi. (1 – 3i) + (–1 + 3i) (1 – 1) + (–3i + 3i) 0 Add real parts and imaginary parts. Your Turn: Add or subtract. Write the result in the form a + bi. (–3 + 5i) + (–6i) (–3) + (5i – 6i) –3 – i Add real parts and imaginary parts. Your Turn: Add or subtract. Write the result in the form a + bi. 2i – (3 + 5i) (2i) – 3 – 5i (–3) + (2i – 5i) –3 – 3i Distribute. Add real parts and imaginary parts. Your Turn: Add or subtract. Write the result in the form a + bi. (4 + 3i) + (4 – 3i) (4 + 4) + (3i – 3i) 8 Add real parts and imaginary parts. Multiplying Complex Numbers The technique for multiplying complex numbers varies depending on whether the numbers are written as single term (either the real or imaginary component is missing) or two terms. You can multiply complex numbers by using the Distributive Property and treating the imaginary parts as like terms. Simplify by using the fact i2 = –1. Multiplying Complex Numbers Note that the product rule for radicals does NOT apply for imaginary numbers. 2 16 25 4i 5i 20i 20(1) 20 16 25 16 25 400 20 Before performing operations with imaginary numbers, be sure to rewrite the terms or factors in i-form first and then proceed with the operations. Procedure: 1. Pretend that “i” is a variable and that a complex number is a binomial 2. Multiply as you would binomials 3. Simplify by changing “i2” to “-1” and combining like terms Example: (-4 + 3i)(5 – i) -20 + 4i + 15i - 3i2 -20 + 4i + 15i + 3 -17 + 19i Example: Multiply. Write the result in the form a + bi. –2i(2 – 4i) –4i + 8i2 Distribute. –4i + 8(–1) Use i2 = –1. –8 – 4i Write in a + bi form. Example: Multiply. Write the result in the form a + bi. (3 + 6i)(4 – i) 12 + 24i – 3i – 6i2 Multiply. 12 + 21i – 6(–1) Use i2 = –1. 18 + 21i Write in a + bi form. Example: Multiply. Write the result in the form a + bi. (2 + 9i)(2 – 9i) 4 – 18i + 18i – 81i2 Multiply. 4 – 81(–1) Use i2 = –1. 85 Write in a + bi form. Example: Multiply. Write the result in the form a + bi. (–5i)(6i) –30i2 Multiply. –30(–1) Use i2 = –1 30 Write in a + bi form. Your Turn: Multiply. Write the result in the form a + bi. 2i(3 – 5i) 6i – 10i2 Distribute. 6i – 10(–1) Use i2 = –1. 10 + 6i Write in a + bi form. Your Turn: Multiply. Write the result in the form a + bi. (4 – 4i)(6 – i) 24 – 4i – 24i + 4i2 Distribute. 24 – 28i + 4(–1) Use i2 = –1. 20 – 28i Write in a + bi form. Your Turn: Multiply. Write the result in the form a + bi. (3 + 2i)(3 – 2i) 9 + 6i – 6i – 4i2 Distribute. 9 – 4(–1) Use i2 = –1. 13 Write in a + bi form. Complex Conjugates The complex numbers and are related. These complex numbers are a complex conjugate pair. Their real parts are equal and their imaginary parts are opposites. The complex conjugate of any complex number a + bi is the complex number a – bi. If a quadratic equation with real coefficients has nonreal solutions, those solutions are complex conjugates. Helpful Hint When given one complex solution, you can always find the other by finding its conjugate. Example: Find each complex conjugate. B. 6i A. 8 + 5i 8 + 5i 8 – 5i Write as a + bi. Find a – bi. 0 + 6i 0 – 6i –6i Write as a + bi. Find a – bi. Simplify. Your Turn: Find each complex conjugate. B. A. 9 – i 9 + (–i) Write as a + bi. Write as a + bi. 9 – (–i) 9+i Find a – bi. Find a – bi. Simplify. C. –8i 0 + (–8)i 0 – (–8)i 8i Write as a + bi. Find a – bi. Simplify. Product of Complex Conjugates Complex Conjugates The complex numbers (a + bi) and (a – bi) are complex conjugates of each other, and (a + bi)(a – bi) = a2 + b2 Complex Conjugate Examples: 6 + 7i and 6 7i 8 3i and 8 + 3i 14i and 14i • The product of a complex number and its conjugate is a real number. Example: (4 + 9i)(4 9i) = 42 (9i)2 = 16 81i2 = 16 81(1) = 97 Example: (7i)(7i) = 49i2 = 49(1) = 49 Your Turn: • Multiply each complex number by its complex conjugate. 1) 3 + i 10 2) 5 – 4i 41 Dividing Complex Numbers Recall that expressions in simplest form cannot have square roots in the denominator. Because the imaginary unit represents a square root, you must rationalize any denominator that contains an imaginary unit. To do this, multiply the numerator and denominator by the complex conjugate of the denominator. Helpful Hint The complex conjugate of a complex number a + bi is a – bi. Procedure: 1. Write the division problem in fraction form. 2. The trick is to make the denominator real, this is done using the complex conjugate. 3. Multiply the fraction by a special “1” where “1” is the conjugate of the denominator over itself. 4. Simplify and write answer in standard form: a + bi. Example: Simplify. Multiply by the conjugate. Distribute. Use i2 = –1. Simplify. Example: Simplify. Multiply by the conjugate. Distribute. Use i2 = –1. Simplify. Your Turn: Simplify. Multiply by the conjugate. Distribute. Use i2 = –1. Simplify. Your Turn: Simplify. Multiply by the conjugate. Distribute. Use i2 = –1. Simplify. Your Turn: 3 11i 1) 1 2i 5 i 7 4i 2) 2 5i 6 43 i 29 29 Essential Question Big Idea: Solving Equations 1. What are complex numbers? – The set of complex numbers includes real numbers and imaginary numbers. The basis for the complex numbers is the imaginary unit i, whose square is -1. Use a complex number of the form a + bi to simplify the square root of a negative number. 2. How do perform basic operations on complex numbers? – To perform basic operations on complex numbers, treat the imaginary parts like variable terms of binomials. Add, subtract, or multiply as you would with algebraic expressions. To divide complex numbers, multiply numerator and denominator by the complex conjugate of the denominator and simplify. Assignment • Section 4-8 Part 1, Pg 273;# 1 – 6 all, 8 – 30 even.