Month: March 2024 (Page 1 of 3)

Announcements

WebWork:

• Radical Expressions-Complex Numbers – due Mon April 1 (Sec 1.4.6)
• Quadratic Equations-Square Root – due Wed April 3

Topics

We covered division of complex numbers, and then shifted to discussing quadratic equations and how to solve them.

For division of complex numbers, we first reviewed the idea of complex conjugates:

We can use the complex conjugate (of the given denominator) to do division of complex numbers:

We then discussed quadratic equations. We showed how we can solve certain quadratic equations by factoring (and using what’s called “the zero product property”–see Sec 2.2.1):

If the quadratic equation is given in a certain form, we can solve via square roots (using what’s called “the square root property”, i.e., that we need to account for both the positive and negative square roots–see Sec 2.2.2):

We also showed how we can solve this equation using the quadratic formula:

We did some examples from the “Quadratic Equations-Square Root” WebWork:

Next time we will discuss “completing the square”, which will show us how the quadratic formula comes from:

Announcements

WebWork:

• Radical Expressions-Multiplying – due Thurs March 28 (Sec 1.4.4)
• Radical Expressions-Complex Numbers – due Mon April 1 (Sec 1.4.6)

Topics

We revisited “complex numbers” using the new number symbol “i” to represent the square root of -1. A complex number is anything of the form “a + bi” (where a and b are regular real numbers). We call “a” the real part and “bi” the imaginary part.

(Complex numbers are covered in Sec 1.4.6 of the textbook, where you can find additional examples.)

We then reviewed addition/subtraction of complex by “combining like terms” (here the real and imaginary parts of the two given complex numbers):

For multiplication we use “FOIL”–together with the fact that i^2 = -1 in order to simplify:

We then did a multiplication example from the WebWork which introduced the idea of two complex numbers which are “complex conjugates”–they differ just by the sign between the real and imaginary parts–in which case their product is just a real number:

Next time we will show how to use this feature of the complex conjugate to do division of complex numbers, as explained in the WebWork set:

Announcements

We will have a quiz on Monday , covering the basics of simplifying and adding/subtracting radicals (square roots), as well as complex fractions.

I have reopened/extended some of the WebWork sets so you can work on these if you didn’t complete them earlier:

• Radical Expressions-Adding and Subtracting – due Sun March 24 (Sec 1.4.4)
• Rational Expressions-Complex Fractions 2 – due Mon March 25 (Sec 1.3.4)
• Radical Expressions-Multiplying – due Wed March 27 (Sec 1.4.4)
• Radical Expressions-Complex Numbers – due Fri March 29 (Sec 1.4.6)

Topics

We did a couple more examples of simplifying complex fractions, using the “keep-change-flip” technique:

We then did a complex fractions example (from the “Complex Fractions 2” WebWork) where we first have to do addition/subtraction using least common denominators (to get it into a form where we can “keep-change-flip”):

We did an example from the WebWork of multiplying square roots epxresions:

We then introduced “imaginary number” symbol i, to represent the square root of -1, and showed how we can use this to represent the square root of a negative number, such as in the solutions of a quadratic equation from the quadratic formula:

We introduced operations on these “complex numbers” by verifying that checking that one of the results we got from the quadratic formula is actually a solution of the quadratic equation:

We went back and defined such complex numbers as expressions of the form “a+bi” (where a, b are real numbers), how to do addition and multiplication on complex numbers: