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 usually 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 # 12

(define pi 3.1415926535)
(define r 3.2)
(* pi r r)

; That was problem # 12

======================= 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.
When typing your numerical answers into the Comments-to-Teacher, only copy the number of significant digits requested.

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. a) 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 (round to 3 significant digits)? Write and evaluate a Scheme expression using these values (in the upper-pane), and put the answer into the Comments-to-teacher..

3. b) Once more, light travels at approximately 186,000 miles/second. How far does light go (in units of feet) in one billionth of a second (round answer to 2 significant digits)? Write and evaluate the Scheme expression and put the answer into the Comments-to-teacher.

3. c) Mr .Brooks asks Fred to calculate (and store, under his name) the distance in feet that light travels in a year (one light-year in feet), and then to give the answer to George, who should also store it under his name. (Note: I'd like you to use 365.242196 as the average number of days/year in your calculations). Fred, being mischievous, wants to keep the correct answer to himself, and so he adds one to his answer before giving it to George. Mr. Brooks suspects foulplay, and asks Fred and George for their answers and wants to calculate the difference between them. What is the answer that Mr. Brooks calculates? Put his answer into the Comments-to-Teacher.