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MIAMI-DADE COLLEGE, HIALEAH CAMPUS Project V-Coach, Title IV Basic Study of Imaginary & Complex Numbers Compiled and Edited by: Jean N. Alcime, STEM Specialist & Math Professor V-Coach Title IV & the Department of Mathematics Also Edited by: Ms. Ivonne Cruz, Administrative Supervisor Mr. Jared Blanchette, Math Professor Definition of Imaginary numbers Complex numbers are fundamentally based in the understanding of imaginary numbers. To fully comprehend complex numbers, you need to thoroughly grasp the concept of imaginary numbers. You might be wondering why you cannot take the square root of a negative number. In fact, you have been told that the square root of any negative number, say x, is not real. Applying the definition of the multiplication of sign, a number multiplies by itself produces a positive result; whether the number is positive or negative. Furthermore, when multiplying two numbers with the same sign, the result will always be positive. The rationale of this factual/proven definition is true in the basis of the definition of square root itself. With that in mind, you can truly say that the radicand (a number or quantity from which a root is to be extracted) of a square root must be positive in order for the root to be real. For example: √ √ Radicand Notice from the above example, 5 is the squares root of 25 where is the radicand. It should now be clear that the result of the square of a number is positive. All Rights Reserved. Now let’s try to take the square root of negative radicand, say √ . As previously mentioned when 5 or -5 is squared, the result is 25. That proves the fact that the radicand cannot be negative. However; we perceive the square root of a negative number as imaginary. This leads to the following definition: √ The next example demonstrates how to work with a square root containing a negative radicand. Example of square root with negative radicand: √( √ The result ) √ √ is an imaginary number, where indicates imaginary. The following shows the guidelines when working with imaginary numbers: √ (√ Definition of ) )) (( (√ ) Square both sides of the above equation. From above, multiply by -1 From the definition of we multiply -1 by -1 Simplifying imaginary numbers for n≥4 When simplifying an imaginary number, do the following: a. If where is a positive integer and indicates imaginary number ( ), Divide b. c. d. e. If the remainder is 0, then the answer is 1. If the remainder is 1, then the answer is . If the remainder is 2, then answer is -1 If the remainder is 3, then the answer - Example a: Solution All Rights Reserved. , we perform the division , the remainder is 3; therefore the answer – . Since Example b: Again we divide 57 by 4; the remainder is 1; therefore, the answer is Based on the examples above, the quotient of the fraction has no significance in terms of obtaining the result. In fact, we only care about the remainder. The remainder should always be between 0 and 3 where 0 and 3 are included. The remainder always tells you the answer. Now consider example b, after we divided 57 by 4, we found the quotient to be 14 with a remainder of 0. Since the remainder is 0, it follows that the answer is 1. Example c: Solution: . The quotient is 13 and the remainder is 0; therefore . Working with Complex Number A number written in the form part and is the imaginary part. is called a complex number where is the real Having a complex number written in the form is said to be in the standard form. You might have studied different types of numbers and their definitions in a previous class or prior to reading this booklet. Recall that the set of Rational (natural, whole, integer) and irrational numbers make up the set of the real numbers. Adding and subtracting complex numbers Complex numbers can be added and subtracted. When adding complex numbers you add the real parts and the imaginary parts separately. Example 1: ( ) ( ) Solution: ( ) ( ) ( All Rights Reserved. ) ( ) Eliminate the parenthesis and combine like terms Example 2: ( ) ( ) Solution ( ) ( ) ( Example 3: ( ) ( ) ( ) ) Solution ( ) ( ) ( ) ( ) Multiplying Complex Numbers Complex numbers are multiplied just like you would if you were to multiply a polynomial, with the exception that . Example 4: ( ) Solution Applying the distributive property, we end up with the following: ( ) ( ) Notice that before getting to the final answer, the complex number is written in descending power, and the final answer is written in the standard form. Example 5: ( )(11+6i) Solution Here we are applying the FOIL method ( )( ) ( ) ( ) ( ) ( ) ( ( ) ( ) ( ) ( ) ) As we do when we rationalize radical expressions, we apply the similar method with complex numbers. That is; complex numbers, say and are conjugates. That is the conjugate of a complex number is . Definition of Product of Conjugates All Rights Reserved. ( )( ) ; When we apply the FOIL method, we find that the product of a complex number and its conjugate is the sum of the real numbers . Example 6: ( )( ) Solution ( )( ) ( ) ( ) ( ) ( ( ) ) ( ) ( ) ( ) ( ) =13 )( ) Example 6 demonstrates the fact that( . Thus, the product of a complex number and its conjugate is the sum of the square of the real part and the square of the coefficient of the imaginary part. Dividing complex numbers Example 7: Write in standard form: Solution To write in the standard form, you need to multiply the numerator and the denominator by the conjugate of the denominator. The denominator should always be a real number. Notice that in the above example the fraction was separated into two fractions so that the result would be in standard form. Example 8: Write in standard form: Solution ( ) ( ) All Rights Reserved. ( ) ( ) Quadratic equations involving complex numbers: Example 9 solve for x: Solution: Re-write the quadratic equation in the standard form √( ( ) ( ) ( ) applying the quadratic formula: √ √ ( ) Factor by the greatest common divisor (6) The 6 is cancelled out, we left with The answer is * All Rights Reserved. √ +