COMPLEX NUMBER NUMBER OF THE FORM C= a+Jb a = real part of C b = imaginary part. 2
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
Real Numbers Imaginary Numbers Real numbers and imaginary numbers are subsets of the set of complex numbers. Complex Numbers
Conversion between Rectangular and polar form Convert Between Form C = a + jb (Rectangular Form) C = C<ø ( Polar Form) C is Magnitude a = C cos ø and b=C sin ø where C = √ a 2 + b 2 ø = tan -1 b/a 5
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.
Complex Plane A complex number can be plotted on a plane with two perpendicular coordinate axes The horizontal x-axis, called the real axis The vertical y-axis, called the imaginary axis Represent z = x + jy geometrically as the point P(x,y) in the x-y plane, or as the vector from the origin to P(x,y). The complex plane x-y plane is also known as the complex plane.
Complex plane, polar form of a complex number Geometrically, |z| is the distance of the point z from the origin while θ is the directed angle from the positive x-axis to OP in the above figure. From the figure,
θ is called the argument of z and is denoted by arg z. Thus, For z = 0, θ is undefined. A complex number z ≠ 0 has infinitely many possible arguments, each one differing from the rest by some multiple of 2π. In fact, arg z is actually The value of θ that lies in the interval (-π, π] is called the principle argument of z (≠ 0) and is denoted by Arg z.
Consider the quadratic equation x 2 + 1 = 0. Solving for x, gives x 2 = – 1 We make the following definition: Complex Numbers
Complex Numbers : power of j Note that squaring both sides yields: therefore and so and And so on…
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.
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 5cos30 and –2cos120 Next, write that product in rectangular form
Products and Quotients of Complex Numbers in Polar Form Find the quotient of 36cos300 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?