Scheme Homework #1

This first section should be read and understood. The homework exercises are in the next section below it...

We're going to be using the "modulo" built-in Scheme function frequently. Recall that the modulo function is the same as what we think of as a "remainder", namely, (modulo 13 5) will divide 13 by 5, and give as an answer, the remainder, which is 3. We're only going to use modulo with positive numbers. So, try the little modulo practice below, by predicting what Scheme will output before typing it into Scheme to check:

(modulo 21 6)

(modulo 57 13)

(modulo 23 2)

(modulo 24 2)

(modulo 0 2)

Note 1: in a Scheme arithmetic calculation, if any one (or more) of the numbers contain a decimal point, then the calculation will be done using decimals (and will probably be approximate). If all of the numbers are integers or fractions, and none of them contain a decimal point, then the answer will be exact, and may contain a fractional part. For instance:

(/ 7 (+ 2 3)) ; -> 1 2/5

(/ 7 (+ 2.0 3)) ; -> 1.4

Note 2: Scientific notation can also be entered: for instance, 300.0 can be written 3e2

Note 3: Use the semi-colon character ";" to start a comment in the upper (definitions) pane. The comment will last unto the end of the line. Example:

; This is problem # 47.89B

(define (justin bieber) (* bieber bieber))

; That was problem # 7.89B

======================= Homework exercises to hand in ==================================

Save the following functions and expressions into a file (with suffix ".rkt") and submit the file to the homework server. In the exercises below, create the expression in Scheme, and for questions 1, 2 and 3, also submit (type in) the answers as comments-to-the-teacher when submitting the homework file.

1. Below, you'll see an object called a "continued fraction". Evaluate the fraction in Scheme, and get the result in decimal.
Place the decimal answer (rounded to 5 decimal places) in the comments-to-teacher section. (BTW, does the answer look familiar?)

2. Let's create a mathematical (not Scheme) notation: "A (mod B)" is the remainder when A is divided by B. Now let's notice a pattern found by the great mathematician Pierre de Fermat back in 1640. Evaluate the 4 examples below, each of which is in the form: X^Y (mod Z). There's nothing special about the choice of X, but there is something special about the choice of Z and of Y. Guess what's special about the relationship between Y and Z, and what's special about Z? Write a quick answer in the comments-to-teacher.
2a) evaluate 2¹² (mod 13)
2b) evaluate 5²² (mod 23)
2c) evaluate 55⁶ (mod 7)
2d) evaluate 18⁴²(mod 43)

3. One "Astronomical Unit" is defined to be the average distance from the Earth to the Sun. It is approximately 93 million miles. If light travels at approximately 186,000 miles/second, about how many minutes will it take light from the Sun to reach the Earth? Create the variables: c (speed of light in miles/sec), au (astronomical unit in miles). Write and evaluate a Scheme expression using these variables (in the upper-pane), and put the answer into the Comments-to-teacher..

4. Write two expressions to solve the quadratic equation below (use the quadratic formula). Put the two roots into the Comments-to-Teacher.

    X² - 6X + 13 = 0