Real+Transcendental+Irrational+Numbers

=Real Transcendental Irrational Numbers=

==**Depict π as the Ratio of a Circle's Circumference to its Diameter.** == (a) Draw the line connecting the three points. (b) What is the slope of the line? ==**Additive Identity and Inverse:** == Answer the following questions in the rightmost cells below. Identify group members participating in this section of the lesson. The logarithmic expression to "cancel" the base of 3 is log ﻿3 3 x //=// log 3 27 || log 3 27 = ? || log 10 (1000) = //y//. So 1000 = 10^//y// || log 10 (1000) = ? || log 2 (10)﻿=y || To the nearest 0.01, find a number, y, such that 2^y = 10. ||
 * 1. Draw circles of diameter = 1, 2, and 3. Compute their circumferences.
 * You may draw your circles and write your computations by hand, scan your figure, and upload the file into the cell below.
 * Include the names of group members participating in this section of the lesson. ||
 * 2. Sketch the circumferences of the three circles drawn above vs. their diameters.
 * 2. Sketch the circumferences of the three circles drawn above vs. their diameters.
 * You may graph your line by hand, scan your graph, and upload the file into the cell below.
 * Include the names of group members participating in this section of the lesson. ||
 * Suppose you need to solve the following for //x//: 3 x = 27.
 * Similarly, the exponential expression 10 x cancels log 10 :
 * What if the number you are "logging" is not a rational power of the logarithmic base?

**Write 3 Properties:**
Next to the following 4 properties of exponents, write a similar property of logarithms. Identify group members participating in this section of the lesson.
 * (a^m)*(a^n) = a^(m+n) |||| log(mn)= ||
 * (a^m)/(a^n) = a^(m-n) |||| log(m/n)= ||
 * a^0 = 1 |||| log(1)= ||
 * (a^m)^p = a^(mp) |||| log(a^p)= ||

**Euler's Number, //e//**
You learned about the number, //e//, when you studied continuous compounding. Jacob Bernoulli estimated //e// using the following formula: .
 * 1) When and where did Jacob Bernoulli live? Consider using the following site: []
 * 2) Evaluate the above expression for //n = 1, 5, 10, 100, 1000, 10,000//?
 * 3) To what value do these expressions seem to be converging?

Create an attractive illustration (8.5"x11") of the convergence of //e// to add to the class timeline/map on the back wall of the classroom. Include answers to the questions above. Cite any sources used and identify group members participating in this section of the lesson. ==**John Napier and Johannes Kepler** == German astronomer/mathematician Johannes Kepler was enthusiastic when he learned of the work of John Napier (//Mirifici logarithmorum canonis descriptio).// At a minimum, research and answer the following questions in the space below. Consider using the following site: []
 * 1) When and where did John Napier live?
 * 2) When was this work published?
 * 3) Why might Kepler have been so enthusiastic about logarithms?

Use the guidelines provided in class to collaborate using Google Docs to create an attractive trading card (half of an 8.5"x11" sheet of paper) including a picture of John Napier and a brief synopsis of his life. Cite any sources used and identify group members participating in this section of the lesson.