Presentation on theme: "Complex Numbers Consider the quadratic equation x2 + 1 = 0."— Presentation transcript:
1Complex Numbers Consider the quadratic equation x2 + 1 = 0. Solving for x , gives x2 = – 1We make the following definition:
2Complex Numbers Note that squaring both sides yields: therefore and so And so on…
3Real numbers and imaginary numbers are subsets of the set of complex numbers.
4Definition of a Complex Number If a and b are real numbers, the number a + bi is a complex number, and it is said to be written in standard form.If b = 0, the number a + bi = a is a real number.If a = 0, the number a + bi is called an imaginary number.Write the complex number in standard form
5Addition and Subtraction of Complex Numbers If a + bi and c +di are two complex numbers written in standard form, their sum and difference are defined as follows.Sum:Difference:
6Perform the subtraction and write the answer in standard form. ( 3 + 2i ) – ( i )3 + 2i – 6 – 13i–3 – 11i4
7Multiplying Complex Numbers Multiplying complex numbers is similar to multiplying polynomials and combining like terms.Perform the operation and write the result in standard form. ( 6 – 2i )( 2 – 3i )F O I L12 – 18i – 4i + 6i212 – 22i + 6 ( -1 )6 – 22i
8Consider ( 3 + 2i )( 3 – 2i )9 – 6i + 6i – 4i29 – 4( -1 )9 + 413This is a real number. The product of two complex numbers can be a real number.This concept can be used to divide complex numbers.
9Complex Conjugates and Division Complex conjugates-a pair of complex numbers of the form a + bi and a – bi where a and b are real numbers.( a + bi )( a – bi )a 2 – abi + abi – b 2 i 2a 2 – b 2( -1 )a 2 + b 2The product of a complex conjugate pair is a positive real number.
10To find the quotient of two complex numbers multiply the numerator and denominator by the conjugate of the denominator.
11Perform the operation and write the result in standard form.
12Perform the operation and write the result in standard form.
13Expressing Complex Numbers in Polar Form Now, any Complex Number can be expressed as:X + Y iThat number can be plotted as on ordered pair inrectangular form like so…
14Expressing Complex Numbers in Polar Form Remember these relationships between polar andrectangular form:So any complex number, X + Yi, can be written inpolar form:Here is the shorthand way of writing polar form:
15Expressing Complex Numbers in Polar Form Rewrite the following complex number in polar form:4 - 2iRewrite the following complex number inrectangular form:
16Expressing Complex Numbers in Polar Form Express the following complex number inrectangular form:
17Expressing Complex Numbers in Polar Form Express the following complex number inpolar form: 5i
18Products and Quotients of Complex Numbers in Polar Form The product of two complex numbers,andCan be obtained by using the following formula:
19Products and Quotients of Complex Numbers in Polar Form The quotient of two complex numbers,andCan be obtained by using the following formula:
20Products and Quotients of Complex Numbers in Polar Form Find the product of 5cis30 and –2cis120Next, write that product in rectangular form
21Products and Quotients of Complex Numbers in Polar Form Find the quotient of 36cis300 divided by4cis120Next, write that quotient in rectangular form
22Products and Quotients of Complex Numbers in Polar Form Find the result ofLeave your answer in polar form.Based on how you answered this problem,what generalization can we make aboutraising a complex number in polar form toa given power?
23[r(cos F+isin F]n = rn(cos nF+isin nF) De Moivre’s TheoremDe Moivre's Theorem is the theorem which shows us how to take complex numbers to any power easily.De Moivre's Theorem – Let r(cos F+isin F) be a complex number and n be any real number. Then[r(cos F+isin F]n = rn(cos nF+isin nF)What is this saying?The resulting r value will be r to the nth power and the resulting angle will be n times the original angle.
24Remember to save space you can write it in compact form. De Moivre’s TheoremTry a sample problem: What is [3(cos 45°+isin45)]5?To do this take 3 to the 5th power, then multiply 45 times 5 and plug back into trigonometric form.35 = 243 and 45 * 5 =225 so the result is 243(cos 225°+isin 225°)Remember to save space you can write it in compact form.243(cos 225°+isin 225°)=243cis 225°
25De Moivre’s Theorem Find the result of: Because of the power involved, it would easier to change thiscomplex number into polar form and then use De Moivre’s Theorem.
26De Moivre’s TheoremDe Moivre's Theorem also works not only for integer values of powers, but also rational values (so we can determine roots of complex numbers).
28De Moivre’s TheoremEvery complex number has ‘p’ distinct ‘pth’ complex roots (2 square roots, 3 cube roots, etc.)To find the p distinct pth roots of a complex number, we use the following form of De Moivre’s Theorem…where ‘n’ is all integer values between 0 and p-1.Why the 360? Well, if we were to graph the complexroots on a polar graph, we would see that the p rootswould be evenly spaced about 360 degrees (360/p wouldtell us how far apart the roots would be).
29De Moivre’s TheoremFind the 4 distinct 4th roots of i
30De Moivre’s Theorem Solve the following equation for all complex number solutions (roots):