Consider the quadratic equation x 2 + 1 = 0. Solving for x, gives x 2 = – 1 We make the following definition: Complex Numbers
Note that squaring both sides yields: therefore and so and And so on…
Real Numbers Imaginary Numbers Real numbers and imaginary numbers are subsets of the set of complex numbers. Complex Numbers
Definition 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
Addition 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:
Perform the subtraction and write the answer in standard form. ( 3 + 2i ) – ( 6 + 13i ) 3 + 2i – 6 – 13i –3 – 11i 4
Multiplying Complex Numbers Multiplying 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 ) FOILFOIL 12 – 18i – 4i + 6i 2 12 – 22i + 6 ( -1 ) 6 – 22i
Consider ( 3 + 2i )( 3 – 2i ) 9 – 6i + 6i – 4i 2 9 – 4( -1 ) 9 + 4 13 This is a real number. The product of two complex numbers can be a real number. This concept can be used to divide complex numbers.
Complex 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 2 a 2 – b 2 ( -1 ) a 2 + b 2 The product of a complex conjugate pair is a positive real number.
To find the quotient of two complex numbers multiply the numerator and denominator by the conjugate of the denominator.
Perform the operation and write the result in standard form.
Expressing Complex Numbers in Polar Form Now, any Complex Number can be expressed as: X + Y i That number can be plotted as on ordered pair in rectangular form like so…
Expressing Complex Numbers in Polar Form Remember these relationships between polar and rectangular form: So any complex number, X + Yi, can be written in polar form: Here is the shorthand way of writing polar form:
Expressing Complex Numbers in Polar Form Rewrite the following complex number in polar form: 4 - 2i Rewrite the following complex number in rectangular form:
Expressing Complex Numbers in Polar Form Express the following complex number in rectangular form:
Expressing Complex Numbers in Polar Form Express the following complex number in polar form: 5i
Products and Quotients of Complex Numbers in Polar Form The product of two complex numbers, and Can be obtained by using the following formula:
Products and Quotients of Complex Numbers in Polar Form The quotient of two complex numbers, and Can be obtained by using the following formula:
Products and Quotients of Complex Numbers in Polar Form Find the product of 5cis30 and –2cis120 Next, write that product in rectangular form
Products and Quotients of Complex Numbers in Polar Form Find the quotient of 36cis300 divided by 4cis120 Next, write that quotient in rectangular form
Products and Quotients of Complex Numbers in Polar Form Find the result of Leave your answer in polar form. Based on how you answered this problem, what generalization can we make about raising a complex number in polar form to a given power?
De Moivres Theorem De 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 +isin ) be a complex number and n be any real number. Then [r(cos +isin ] n = r n (cos n +isin n ) What is this saying? The resulting r value will be r to the n th power and the resulting angle will be n times the original angle.
De Moivres Theorem Try a sample problem: What is [3(cos 45 +isin45)] 5 ? To do this take 3 to the 5 th power, then multiply 45 times 5 and plug back into trigonometric form. 3 5 = 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
De Moivres Theorem Find the result of: Because of the power involved, it would easier to change this complex number into polar form and then use De Moivres Theorem.
De Moivres Theorem De Moivre's Theorem also works not only for integer values of powers, but also rational values (so we can determine roots of complex numbers).
De Moivres Theorem Every 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 Moivres Theorem …where n is all integer values between 0 and p-1. Why the 360? Well, if we were to graph the complex roots on a polar graph, we would see that the p roots would be evenly spaced about 360 degrees (360/p would tell us how far apart the roots would be).
De Moivres Theorem Find the 4 distinct 4th roots of -3 - 3i
De Moivres Theorem Solve the following equation for all complex number solutions (roots):