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Imaginary Unit and Standard Complex Form The Imaginary Unit is defined as i= . The reason for the name "imaginary" numbers is that when these numbers were first proposed several hundred years ago, people could not "imagine" such a number. It is said that the term "imaginary" was coined by René Descartes in the seventeenth century and was meant to be a derogatory reference since, obviously, such numbers did not exist. Today, we find the imaginary unit being used in mathematics and science. Electrical engineers use the imaginary unit (which they represent as j ) in the study of electricity. Imaginary numbers occur when a quadratic equation has no roots in the set of real numbers. An imaginary number is a number whose square is negative. *i = or - i =- A pure imaginary number can be written in bi form where b is a real number and i is . Examples: pure imaginary numbers A complex number is any number that can be written in the standard form a + bi, where a and b are real numbers and i is the imaginary unit. A complex number is a real number a, or a pure imaginary number bi, or the sum of both. Note these examples of complex numbers written in standard a + bi form: 2 + 3i, -5 + 0i . Complex Number: standard a + bi form a bi 7 + 2i 7 2i 1 - 5i 1 - 5i 8i 0 8i The set of real numbers and the set of imaginary numbers are subsets of the set of complex numbers. Practice with Imaginary Unit and Standard Complex Form Try the following problems: 1. Solve the following Answer quadratic equation: x2 + 4 = 0 2. Solve the following expression for x: Answer 2 x = -3 3 Explain the circumstances under Answer which a square can be equal to a negative number. 4. State the values of the a and b Answer components of a + bi form for each complex number. State whether the value is the real part or the imaginary part of the number: a) 3 - 2i b) -2 - i c) 3i d) 0 + 0i e) -7 - 4i Simplifying Square Roots with Negative Numbers Remember: Consider: The Imaginary Unit is defined as i= . Whenever there is a negative under the radical sign, it comes out from underneath as i since the problem will contain a square root of -1. When simplifying these radical numbers in terms of i, follow the usual rules for simplifying radicals and treat the i with the rules for working with a variable. Examples: Problem Solution bi Form 1. 6i 2. 14i 3. 4. Practice with Simplifying Square Roots with Negative Numbers Try the following problems: 1. Answer Simplify: 2. Answer Simplify in terms of i: 3. Answer Simplify in terms of i: 4. Answer Simplify in terms of i: Answer 5. Simplify: Powers of i The powers of i repeat in a definite pattern: ( i, -1, -i, 1 ) Powers of i ... Simplified form i -1 -i 1 i -1 -i 1 ... Think about what happens when i is raised to a given power: You need to remember that: LOOK OUT!!! is not true when a and b are both negative. False TRUE: Whenever the exponent is greater than or equal to 5, you can use the fact that to simplify a power of i. Another way to think of this process of simplifying powers of i is to divide the exponent by 4, When raising i to - if the remainder is 0, the answer is 1 (i0). any integral power, - if the remainder is 1, the answer is i the answer is always (i1). i, -1, -i or 1. -if the remainder is 2, the answer is -1 (i2). -if the remainder is 3, the answer is -i (i3). Let's examine two ways to simplify : Using the patterns Looking at remainders shown in the robot table when dividing by 4: above: with a remainder of 3, which means the answer is 3 i = -i. Practice with Powers of i Topic Index | Algebra2/Trig Index | Regents Exam Prep Center Solve the following problems and simplify the answers. 1. Choose: Simplify: 1 i -i 2. Choose: Simplify: 0 1 2 Explanation 3. Choose: Find the product of: 8 4 2i Explanation 4. Choose: Find the sum of: 6 6i -6i Explanation 5. Choose: Find the value of: 25 -10 -25 Explanation Adding and Subtracting Complex Numbers *Add like terms* Find the sum of the real components. REMEMBER: Find the sum of the imaginary components Final answer must (the components with the i after them). be in simplest form. Adding Rule: Examples: 1. Add: (7 + 5i) + (8 - 3i) 2. Add: (2 + 3i) + (-8 - 6i) 3. Express the sum of and in the form . 4. Add and . *Subtract like terms* Find the difference of the real components. REMEMBER: Find the difference of the imaginary Final answer must be in components simplest (the components with the i after them). form. Subtracting Rule: Examples: 1. Subtract: (5 + 8i) - (2 + 2i) 2. Subtract: (4 + 10i) - (-12 + 20i) 3. Subtract from . 4. Subtract from . Practice with Arithmetic of Complex Numbers Practice makes perfect... Solve the following problems and express the result in form. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Add: , , and . 12. Subtract from . Multiplying and Dividing Complex Numbers Multiplication: Multiplying two complex numbers is accomplished in a manner similar to multiplying two binomials. You can use the FOIL process of multiplication, distributive multiplication, or your personal favorite means of multiplication. Distributive Multiplication: (2 + 3i) • (4 + 5i) = 2(4 + 5i) + 3i(4 + 5i) = 8 + 10i + 12i + 15i2 = 8 + 22i + 15(-1) = 8 + 22i -15 = -7 + 22i Answer Be sure to replace i2 with (-1) and proceed with the simplification. Answer should be in a + bi form. The product of two complex numbers is a complex number. (a+bi)(c+di) = a(c+di) + bi(c+di) = ac + adi + bci + bdi2 = ac + adi + bci + bd(-1) = ac + adi + bci - bd = (ac - bd) + (adi + bci) = (ac -bd) + (ad + bc)i answer in a+bi form The conjugate of a complex number a + bi is the complex number a - bi. For example, the conjugate of 4 + 2i is 4 - 2i. (Notice that only the sign of the bi term is changed.) The product of a complex number and its conjugate is a real number, and is always positive. (a + bi)(a - bi) = a2 + abi - abi - b2i2 = a2 - b2 (-1) (the middle terms drop out) = a2 + b2 Answer This is a real number ( no i's ) and since both values are squared, the answer is positive. Division: When dividing two complex numbers, 1.write the problem in fractional form, 2.rationalize the denominator by multiplying the numerator and the denominator by the conjugate of the denominator. (Remember that a complex number times its conjugate will give a real number. This process will remove the i from the denominator.) Example: Dividing using the conjugate: Answer Practice with Multiplying and Dividing Complex Numbers Solve the following problems. Answers are to be in simplest a+bi form. You should be able to solve the problems both with, and without, your graphing calculator. 1. Choose: Multiply: (3 + 5i)(3 - 9 - 25i 25 5i) 34 2. Choose: 15 - 12i Multiply: (8 + 9i)(7 - 3i) 29 - 39i 83 + 39i Explanation 3. Choose: 25 Multiply: (4 - 3i)(3 - 4i) -25i 12 - 12i Explanation 4. Choose: 21 + 20i Simplify: (2 + -21 + 5i)2 20i 29 + 20i Explanation 5. Choose: 7 + 8i Simplify: 8 + i(8 - i) 8 + 8i 9 + 8i Explanation 6. Choose: 5 - 2i 3 + 2i Simplify: 15 + 10i Explanation 7. Choose: 35/37 + (12/37)i 35 + 12i Simplify: 35/36 + (12/36)i Explanation 8. Choose: 5 + 3i -5 - 3i Simplify: 5 - 3i Explanation 9. Choose: 2/15 + i/15 2/15 - i/15 Simplify: 1/45 + i/15 Explanation 10. Choose: What is the multiplicative inverse 2/(1 + i) (1/2) - of (1/2)i (1 + i)/2 Explanation Representing Complex Numbers Graphically (+ and -) Due to their unique nature, complex numbers cannot be represented on a normal set of coordinate axes. In 1806, J. R. Argand developed a method for displaying complex numbers graphically as a point in a coordinate plane. His method, called the Argand diagram, establishes a relationship between the x-axis (real axis) with real numbers and the y- axis (imaginary axis) with imaginary numbers. In the Argand diagram, a complex number a + bi is the point (a,b) or the vector from the origin to the point (a,b). Graph the complex numbers: 1. 3 + 4i (3,4) 2. 2 - 3i (2,-3) 3. -4 + 2i (-4,2) 4. 3 (which is really 3 + 0i) (3,0) 5. 4i (which is really 0 + 4i) (0,4) The complex number is represented by the point, or by the vector from the origin to the point. Absolute Value of Complex Numbers The absolute value of a complex number is written as . It is a nonnegative real number defined as: . Geometrically, the absolute value of a complex number is the number's distance from the origin in the complex plane. In the diagram at the left, the complex number 8 + 6i is plotted in the complex plane on an Argand diagram (where the vertical axis is the imaginary axis). For this problem, the distance from the point 8 + 6i to the origin is 10 units. Distance is a positive measure. Notice the Pythagorean Theorem at work in this problem. A complex number can be represented by a point, or by a vector from the origin to the point. When thinking of a complex number as a vector, the absolute value of the complex number is simply the length of the vector, called the magnitude. The formula for finding the absolute value of a complex number, can be derived from the Pythagorean theorem, (see example 2 below). In the Pythagorean Theorem, c is the hypotenuse and when represented in the coordinate plane, is always positive. This same idea holds true for the distance from the origin in the complex plane. Using the absolute value in the formula will always yield a positive result. To find the absolute value of a complex number a + bi: 1. Be sure your number is expressed in a + bi form 2. Pick out the coefficients for a and b 3. Substitute into the formula Example 1: Plot z = 8 + 6i on the complex plane, connect the graph of z to the origin (see graph below), then find | z | by appropriate use of the definition of the absolute value of a complex number. Example 2: Find the | z | by appropriate use of the Pythagorean Theorem when z = 2 - 3i. You can find the distance | z | by using the Pythagorean theorem. Consider the graph of 2 - 3i shown at the left. The horizontal side of the triangle has length | a |, the vertical side has length | b |, and the hypotenuse has length | z |. By applying the Pythagorean Theorem, you have, | z |2 = a2 + b2 . Notice: you can drop the absolute value symbols for a and b since | a |2 = a2 and | b |2 = b2. You must keep the absolute value symbol for z to insure that the final answer will be positive. Solving this equation for | z |, you have just derived the formula for the absolute value of a complex number: Practice with Absolute Value of Complex Numbers Solve the following problems dealing with the absolute value of complex numbers. 1. Answer Find the absolute value of the complex number: 2. Answer Find the distance from the origin to this point. 3. Answer If , find | z|. 4. Answer Find the absolute value of the complex number 5. Answer Find the distance from the origin to this point.