  Assignment 39

 "I advise my students to listen carefully the moment they decide to take no more mathematics courses. They might be able to hear the sound of closing doors." -- (CAIP Quarterly 2 , Fall, 1989) Maria Agnesi (1718-1799): Agnesi was born into a wealthy family. As a result she was able to receive an excellent private education at a time when educational opportunities for women were extremely limited. A mathematician and a linguist, she was fluent in Italian, Latin, Greek, Hebrew, French, Spanish, and German. Her mathematical work Analytical Institutions (1748) covered mathematical topics ranging algebra and geometry to differential and integral calculus, and was used widely as a textbook. She held the position of mathematics chair at the University of Bologna and devoted much of her life to charitable works. Her name is associated with a curve known as the Witch of Agnesi . The curve looks nothing like a witch, but was so titled due to a mistranslation of a term used to describe cubic equations. What is the name of Herkimer' Arabian friend who wrote volumes and volumes about round objects? Answer: Sheik Sphere Herky's friends: CAL ANDER... this guy thinks his days are numbered. KAY BULL ... this girl installs wiring, etc. for TV sets. ASSIGNMENT #39 Reading: Review Chapter 6 sections, as necessary. Written: Problems on sheet provided in class. Mathematical word analysis:RADICAL: From jidr, the Arabic word meaning "plant-root." In historical Arabic mathematics texts, a square number, such as 25, was considered to grow out of a root number, 5. That is, 52 = 25. The reverse process involves determining a root from a square. A Latin translation of jidr produced radix (root).

NOTE: This is review material on complex numbers. It is not directly related to assignment #39.

There is no solution to the equation x2 = -1 in the real number system. The symbol ÷(-1)... that is, the square root of -1, does not exist on the number line. However, if we define a number (non-real) i = ÷(-1) and accept that i2 = -1, we get something called an imaginary number. If a and b are real numbers, then numbers of the form a+bi are called complex numbers. These numbers have some very meaningful applications in the real world.

We add, subtract, and multiply them in a manner similar to the way these operations are performed with algebraic expressions. Some examples:

(4 + 8i) + (7 - 3i) = 11 + 5i

(4 + 8i) - (7 - 3i) = -3 + 11i

(4 + 8i)(7 - 3i) = 28 - 12i + 56i -24i2 = 28 + 44i - 24(-1) = 52 + 44i. Remember that i2 = -1.

Your TI-83 can work with complex numbers. Hit the MODE button, then go down to Real, and then shift to a + bi. Your calculator is now in complex number mode. Notice the i above the period button at the bottom. If you now go to the home screen, you can now do complex number operations. You get a complex number such as 3 + 5i represented by typing 3+5, and then using 2nd to get the i. Try it... and then do the computations above, and a few others, in this mode.

We won't do a lot with complex numbers, but the fact that they can be represented on the TI-83 certainly suggests they are important and have meaningful applications.

 Problem: Explain how to determine how many years it would take for an investment of A dollars to increase 10-fold it the annual interest rate is 13%. Solution (with communication): It is necessary to solve for x in the equation A(1.13)x = 10A ==> (1.13)x = 10. We could solve this using graphics features on our calculators. However, using a calculator and simply checking values of x, we can quickly determine that (1.13)18 = 9.02 and (1.13)19 = 10.20. Conclusion: It would take 19 years for a \$1 investment to accumulate to \$10 if the annual interest rate is 13%.