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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §7.7 Complex Numbers

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review § Any QUESTIONS About §7.6 → Radical Equations Any QUESTIONS About HomeWork §7.6 → HW-29 7.6 MTH 55

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 3 Bruce Mayer, PE Chabot College Mathematics Imaginary & Complex Numbers Negative numbers do not have square roots in the real-number system. A larger number system that contains the real-number system is designed so that negative numbers do have square roots. That system is called the complex-number system. The complex-number system makes use of i, a number that with the property (i) 2 = −1

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 4 Bruce Mayer, PE Chabot College Mathematics The “Number” i i is the unique number for which i 2 = −1 and so Thus for any positive number p we can now define the square root of a negative number using the product-rule as follows.

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 5 Bruce Mayer, PE Chabot College Mathematics Imaginary Numbers An imaginary number is a number that can be written in the form bi, where b is a real number that is not equal to zero Some Examples i is called the “imaginary unit”

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 6 Bruce Mayer, PE Chabot College Mathematics Example Imaginary Numbers Write each imaginary number as a product of a real number and i a)b)c) SOLUTION a)b)c)

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 7 Bruce Mayer, PE Chabot College Mathematics ReWriting Imaginary Numbers To write an imaginary number in terms of the imaginary unit i: 1.Separate the radical into two factors 2.Replace with i 3.Simplify

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example Imaginary Numbers Express in terms of i: a)b) SOLUTION a) b)

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 9 Bruce Mayer, PE Chabot College Mathematics Complex Numbers The union of the set of all imaginary numbers and the set of all real numbers is the set of all complex numbers A complex number is any number that can be written in the form a + bi, where a and b are real numbers. Note that a and b both can be 0

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 10 Bruce Mayer, PE Chabot College Mathematics Complex Number Examples The following are examples of Complex numbers Here a = 7, b =2.

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 11 Bruce Mayer, PE Chabot College Mathematics The complex numbers: a = bi Complex numbers that are real numbers: a + bi, b = 0 Rational numbers: Complex numbers that are not real numbers: a + bi, b ≠ 0 Irrational numbers: Complex numbers (Imaginary) Complex numbers

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 12 Bruce Mayer, PE Chabot College Mathematics Add/Subtract Complex No.s Complex numbers obey the commutative, associative, and distributive laws. Thus we can add and subtract them as we do binomials; i.e., Add Reals-to-Reals Add Imaginaries-to-Imaginaries

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example Complex Add & Sub Add or subtract and simplify a+bi (−3 + 4i) − (4 − 12i) SOLUTION: We subtract complex numbers just like we subtract polynomials. That is, add/sub LIKE Terms → Add Reals & Imag’s Separately (−3 + 4i) − (4 − 12i) = (−3 + 4i) + (−4 + 12i) = −7 + 16i

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example Complex Add & Sub Add or subtract and simplify to a+bi a)b) SOLUTION a) b) Combining real and imaginary parts

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 15 Bruce Mayer, PE Chabot College Mathematics Complex Multiplication To multiply square roots of negative real numbers, we first express them in terms of i. For example,

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 16 Bruce Mayer, PE Chabot College Mathematics Caveat Complex-Multiplication CAUTION With complex numbers, simply multiplying radicands is incorrect when both radicands are negative: The Correct Multiplicative Operation

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 17 Bruce Mayer, PE Chabot College Mathematics Example Complex Multiply Multiply & Simplify to a+bi form a) b)c) SOLUTION a)

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example Complex Multiply Multiply & Simplify to a+bi form a) b)c) SOLUTION: Perform Distribution b)

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 19 Bruce Mayer, PE Chabot College Mathematics Example Complex Multiply Multiply & Simplify to a+bi form a) b)c) SOLUTION : Use F.O.I.L. c)

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 20 Bruce Mayer, PE Chabot College Mathematics Complex Number CONJUGATE The CONJUGATE of a complex number a + bi is a – bi, and the conjugate of a – bi is a + bi Some Examples

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 21 Bruce Mayer, PE Chabot College Mathematics Example Complex Conjugate Find the conjugate of each number a) 4 + 3i b) −6 − 9i c) i SOLUTION: a) The conjugate is 4 − 3i b) The conjugate is −6 + 9i c) The conjugate is −i (think: 0 + i)

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 22 Bruce Mayer, PE Chabot College Mathematics Conjugates and Division Conjugates are used when dividing complex numbers. The procedure is much like that used to rationalize denominators. Note the Standard Form for Complex Numbers does NOT permit i to appear in the DENOMINATOR To put a complex division into Std Form, Multiply the Numerator and Denominator by the Conjugate of the DENOMINATOR

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 23 Bruce Mayer, PE Chabot College Mathematics Example Complex Division Divide & Simplify to a+bi form SOLUTION: Eliminate i from DeNom by multiplying the Numer & DeNom by the Conjugate of i

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 24 Bruce Mayer, PE Chabot College Mathematics Example Complex Division Divide & Simplify to a+bi form SOLUTION: Eliminate i from DeNom by multiplying the Numer & DeNom by the Conjugate of 2−3i NEXT SLIDE for Reduction

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 25 Bruce Mayer, PE Chabot College Mathematics Example Complex Division SOLN

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 26 Bruce Mayer, PE Chabot College Mathematics Example Complex Division Divide & Simplify to a+bi form SOLUTION: Rationalize DeNom by Conjugate of 5−i

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 27 Bruce Mayer, PE Chabot College Mathematics Powers of i → i n Simplifying powers of i can be done by using the fact that i 2 = −1 and expressing the given power of i in terms of i 2. The First 12 Powers of i Note that (i 4 ) n = +1 for any integer n

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 28 Bruce Mayer, PE Chabot College Mathematics Example Powers of i Simplify using Powers of i a) b) SOLUTION : Use (i 4 ) n = 1 a) b) = 1 Write i 40 as (i 4 ) 10. Write i 32 as (i 4 ) 8. Replace i 4 with 1.

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 29 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work Problems From §7.7 Exercise Set 32, 50, 62, 78, 100, 116 Ohm’s Law of Electrical Resistance in the Frequency Domain uses Complex Numbers (See ENGR43)

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 30 Bruce Mayer, PE Chabot College Mathematics All Done for Today Electrical Engrs Use j instead of i

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 31 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 32 Bruce Mayer, PE Chabot College Mathematics

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 33 Bruce Mayer, PE Chabot College Mathematics Graph y = |x| Make T-table

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BMayer@ChabotCollege.edu MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 34 Bruce Mayer, PE Chabot College Mathematics

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