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Transcript

Complex Numbers Section 2.4 Objectives: • Use the imaginary unit i to write complex numbers. • Add, subtract, and multiply complex numbers. • Use complex conjugates to write the quotient of two complex numbers in standard form. • Plot complex numbers in the complex plane. 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. 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 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 Previously, when we encountered square roots of negative numbers in solving equations, we would say “no real solution” or “not a real number”. Imaginary Unit The imaginary unit i, is the number whose square is – 1. That is, i 2 1 and i 1 The Imaginary Unit, i Example: Write the following with the i notation. 25 25 1 5 i 32 32 1 16 2 1 4 2 i 4 i 2 121 121 1 11i Your Turn: Write as an imaginary number. (a) 16 116 i 16 4i (b) 80 i 80 i 16 5 4i 5 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 • 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 Powers of i Example: Simplify each of the following powers. i i i i 53 i 17 52 413 i 1i i 1 1 1 1 i i 1 2 17 16 i i i 1 i i i i i i i i 1 (1) Your Turn: • Simplify each of the following powers. 1) i22 -1 2) i36 1 3) i13 i 4) i47 -i Complex Numbers Real numbers and imaginary numbers are both subsets of a new set of numbers. Complex Numbers A complex number is a number that can be written in the form a + bi, where a and b are real numbers. Equity of Complex Numbers a+bi =c+di if and only if a=c and b=d. 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 real number. If a = 0, a + bi is an imaginary number. 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 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 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 Adding and Subtracting Complex Numbers Example: Add or subtract the following complex numbers. Write the answer in standard form a + bi. (4 + 6i) + (3 – 2i) = (4 + 3) + (6 – 2)i = 7 + 4i (8 + 2i) – (4i) = (8 – 0) + (2 – 4)i = 8 – 2i Your Turn: 1) (7 + 3i) + (5 – 4i) 12 – i 2) (3 + 4i) – (5 – 2i) -2 + 6i 3) 4i – (-3 + 5i) + (2 – 6i) 5 – 7i 4) (5 – i) + (3 + 4i) – (8 + 3i) 0 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. 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. Euler’s Blunder Multiplication of Complex Numbers 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 Multiplying Complex Numbers Example: Multiply the following complex numbers. 8i · 7i 56i2 56(1) 56 Multiplying Complex Numbers Example: Multiply the following complex numbers. Write the answer in standard form a + bi. 5i(4 – 7i) 20i – 35i2 20i – 35(–1) 20i + 35 35 + 20i Multiplying Complex Numbers Example: Multiply the following complex numbers. Write the answer in standard form a + bi. (6 – 3i)(7 + 4i) 42 + 24i – 21i – 12i2 42 + 3i – 12(–1) 42 + 3i + 12 54 + 3i Your Turn: 1) 8i(4 – 3i) 24 + 32i 2) (3 – i)(5 + 4i) 19 + 7i 3) (2 + 4i)(2 - 4i) 20 4) (5 – 3i)2 16 – 30i Complex Conjugate 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 The conjugate of a + bi is a – bi. The conjugate of a – bi is a + bi. The product of (a + bi) and (a – bi) is (a + bi)(a – bi) a2 – abi + abi – b2i2 a2 – b2(–1) a2 + b2, which is a real number. 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 Division of Complex Numbers 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: 4 3i 3 2i 4 3i 3 2i 12 8i 9i 6i 2 4 3i 2 3 2i 3 2i 3 2i 9 4i 6 17 12 8i 9i 6 6 17i i 13 13 13 94 Dividing Complex Numbers Example: Use complex conjugates to divide the following complex numbers. Write the answer in standard form. 24 18i 8i 6i 2 6 2i 6 2i 4 3i 2 4 3i 4 3i 4 3i 16 12i 12i 9i 18 26 24 26i 6(1) 18 26i i 25 25 25 16 9(1) Dividing Complex Numbers Example: Divide the following complex numbers. 5 6i 30i 30i 5 5 30i i 2 6i 6i 36i 36 6i 6 36(1) Your Turn: • Divide 4 3i by 1 8i. 4 3i 4 3i 1 8i 1 8i 1 8i 1 8i ( 4 3i )(1 8i ) (1 8i )(1 8i ) 4 29i 24i 2 1 64i 2 4 29i 24 65 28 29i 65 Your Turn: 2i 1) 2i 3 4 i 5 5 3i 2) i 1 3i Your Turn: 3 11i 3) 1 2i 5 i 7 4i 4) 2 5i 6 43 i 29 29 The Complex plane Real Axis Imaginary Axis Graphing in the complex plane . 2 5i 2 2i 4 3i 4 3i . Im . . Re Review • A number such as 3i is a purely imaginary number. Review • A number such as 3i is a purely imaginary number. • A number such as 6 is a purely real number. Review • A number such as 3i is a purely imaginary number. • A number such as 6 is a purely real number. • 6 + 3i is a complex number. Review • A number such as 3i is a purely imaginary number. • A number such as 6 is a purely real number. • 6 + 3i is a complex number. • a + bi is the standard form of a complex number. Review • A number such as 3i is a purely imaginary number. • A number such as 6 is a purely real number. • 6 + 3i is a complex number. • a + bi is the standard form of a complex number. • If a + bi = 6 – 4i then a = 6 and b = -4. Review • A number such as 3i is a purely imaginary number. • A number such as 6 is a purely real number. • 6 + 3i is a complex number. • a + bi is the standard form of a complex number. • If a + bi = 6 – 4i then a = 6 and b = – 4. • The ‘real part’ of 6 – 4i is 6. Review • A number such as 3i is a purely imaginary number. • A number such as 6 is a purely real number. • 6 + 3i is a complex number. • a + bi is the standard form of a complex number. • If a + bi = 6 – 4i then a = 6 and b = – 4. • The ‘real part’ of 6 – 4i is 6. • The ‘imaginary part’ of 6 – 4i is -4i. Assignment • Sec. 2.4, pg. 137 – 138: #1 – 75 odd