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

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 MTH 55

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

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.

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”

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)

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

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)

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

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.

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

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

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

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

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,

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

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)

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)

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)

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

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)

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

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

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

MTH55_Lec-47_sec_7-7_Complex_Numbers.ppt 25 Bruce Mayer, PE Chabot College Mathematics Example  Complex Division  SOLN

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

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

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.

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)

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

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

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

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

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