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**Complex Numbers Consider the quadratic equation x2 + 1 = 0.**

Solving for x , gives x2 = – 1 We make the following definition:

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**Complex Numbers Note that squaring both sides yields: therefore and so**

And so on…

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**Real numbers and imaginary numbers are subsets of the set of complex numbers.**

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**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

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**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:

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**Perform the subtraction and write the answer in standard form.**

( 3 + 2i ) – ( i ) 3 + 2i – 6 – 13i –3 – 11i 4

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**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 ) F O I L 12 – 18i – 4i + 6i2 12 – 22i + 6 ( -1 ) 6 – 22i

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Consider ( 3 + 2i )( 3 – 2i ) 9 – 6i + 6i – 4i2 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.

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**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.

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To find the quotient of two complex numbers multiply the numerator and denominator by the conjugate of the denominator.

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**Perform the operation and write the result in standard form.**

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**Perform the operation and write the result in standard form.**

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**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…

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**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:

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**Expressing Complex Numbers in Polar Form**

Rewrite the following complex number in polar form: 4 - 2i Rewrite the following complex number in rectangular form:

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**Expressing Complex Numbers in Polar Form**

Express the following complex number in rectangular form:

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**Expressing Complex Numbers in Polar Form**

Express the following complex number in polar form: 5i

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**Products and Quotients of Complex Numbers in Polar Form**

The product of two complex numbers, and Can be obtained by using the following formula:

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**Products and Quotients of Complex Numbers in Polar Form**

The quotient of two complex numbers, and Can be obtained by using the following formula:

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**Products and Quotients of Complex Numbers in Polar Form**

Find the product of 5cis30 and –2cis120 Next, write that product in rectangular form

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**Products and Quotients of Complex Numbers in Polar Form**

Find the quotient of 36cis300 divided by 4cis120 Next, write that quotient in rectangular form

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**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?

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**[r(cos F+isin F]n = rn(cos nF+isin nF)**

De Moivre’s 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 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.

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**Remember to save space you can write it in compact form.**

De Moivre’s Theorem Try 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°

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**De Moivre’s 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 Moivre’s Theorem.

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De Moivre’s 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).

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De Moivre’s Theorem Simplify the following:

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De Moivre’s 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 Moivre’s 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).

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De Moivre’s Theorem Find the 4 distinct 4th roots of i

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**De Moivre’s Theorem Solve the following equation for all complex**

number solutions (roots):

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Solve the quadratic equation x 2 + 1 = 0. Solving for x, gives x 2 = – 1 We make the following definition: Bell Work #1.

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