Complex Numbers

Enrichment Project #2Introduction to Complex NumbersWhen dealing with generic quadratic equations back in Algebra, you probably ran across the non-real value, . Since this value showed up a great deal in the study of quadratics, it was determined that it was important, even though it was not a part of .As a result, mathematicians decided to give this imaginary number a name (i) and create a new batch of numbers called the complex numbers (), which could be thought of as the set of linear combinations of the real number 1 and the imaginary number i.Arithmetic of the complex numbersExpressing this linear combination, we get a + bi, where . To effectively use the calculator to manipulate complex numbers, look in MODE for this number format, and get familiar with the location of i on the keypad.Practice entering several complex numbers into the calculator. You may want to practice storing the numbers into variables so that arithmetic with the value is easier than keeping track of all the pieces.Arithmetic on complex numbers, because the imaginary piece is incomparable to the real piece, is similar to arithmetic on polynomials. You have “like terms” with the real and imaginary parts, and so addition and multiplication of complex numbers should seem familiar.Create and store several complex numbers and practice adding them together.What conjecture can you make about the process for adding two complex numbers together?Create and store several imaginary numbers (complex numbers with the real part = 0). Make some guesses about the multiplication processby trying several examples as outlined below.What conjecture can you make about the product of two imaginary numbers?What conjecture can you make about the product of a real number and a complex number (i.e., a scalar multiple)?What conjecture can you make about the product of two generic complex numbers?One of the functions in the calculator is the conj( ) function. This will take any complex number z and return its conjugate which is denoted as .For several of your stored complex numbers, use the conj( ) function to determine the conjugate of that number.In general, how do you find the conjugate of a numberPractice multiplying several complex numbers with their own conjugate. What do you notice about the product and why do you think this occurs?What is the conjugate of the conjugate of a complex number?Geometry of Complex NumbersIn addition to the real and imaginary coordinates, complex numbers can be described by their magnitude (or distance from the origin) and their argument (the angle made between the positive real axis and a ray from the origin to the number). In the TI calculator, these characteristics can be found using the abs( ) function and the angle( ) function. The angles should be expressed in radian form. If not, adjust the settings so that they do.Find the magnitude of 5, i, and 3 + 4i. How do you think it could be calculated given a generic complex number?Find the argument of 5, i, and 3 + 4i. How might that be found for a generic complex number?To determine the effects of multiplication on complex numbers, it is usually easier to convert the numbers into polar formConvert 5, i, and 3 + 4i into Polar form. What is the basic format for this form?Using results from #11 and #12, what do the constants in polar form seem to stand for?Take two generic complex numbers in rectangular form and multiply them together. Then convert all three (the factors and the product) into polar form. Do you notice a pattern due to multiplication in polar form?Take two generic complex numbers in polar form and multiply them. Does your pattern hold?In general, what effect does multiplication have on a pair of complex numbers?Algebra in Complex NumbersIn real number algebra, we have two operations: addition and multiplication. Each of these operations has an identity (i.e., a value which has no effect on another value when the operation is performed: 0 for addition and 1 for multiplication).Use several complex numbers to verify that the additive and multiplicative identities for real numbers (0 and 1 respectively) are also the additive and multiplicative identities for complex numbers.In algebra, once you know an operations identity, you typically look to see if that operation then has an inverse for each object in the set. This inverse and its original would result in the identity. For example, in addition, if you have 5, you can add -5 (the opposite, or additive inverse, of 5), and get 5 + (-5) = 0 (the additive identity). The algebraic concept of “adding the opposite” is identical to the arithmetic concept of “subtraction.”Using a generic complex number, make a guess about its opposite. Verify that your number plus your opposite does in fact give you 0.Use your guess about complex opposites to solve the equation for z :For multiplication, the inverse is the reciprocal of the original value. The algebraic concept of “multiplying the reciprocal” is identical to the arithmetic concept, “division”.While subtraction of a complex number is relatively easy to define, division of a complex number is a little bit more complicated.Division by a rational number is well defined but dividing by irrational or imaginary numbers requires some adjustments.Rationalizing the denominatorIf you have a root in the denominator, such as , you typically multiply the numerator and denominator by some value which will turn the denominator into a rational number (in this case, multiply both byRecall earlier in the project where a certain multiplication always seemed to result in a real value rather than imaginary. What was that function called?Suppose you had the “reciprocal of i”, . Multiply the top and bottom by a value so that the denominator becomes a real number. What is the “rationalized” reciprocal of i?Try rationalizing . Were you able to get a complex number where all imaginary pieces are in the numerator? Try using the inverse key to check your work.Using this definition of reciprocal, attempt to solve the complex equation, .

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