Discussion 1: Intro + Python Basics

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Welcome! The problems shown below should be worked on on paper, since the quizzes and exams you take in this course will also be on paper.

We encourage you to complete this worksheet in a live discussion section. Solutions will be made available after all discussion sections have concluded. You don’t need to submit your answers anywhere.

Problem 1

prices is an array of prices, in dollars, of different products at the grocery store. Similarly, calories is an array of the calories in these same products, in the same order.

Problem 1.1

What does type(prices[0]) evaluate to?

Answer: float

prices[0] represents the price in dollars of some product at the grocery store. The data type should be a float because prices are numbers but not necessarily integers.

Problem 1.2

What does type(calories[0]) evaluate to?

Answer: int

Similarly, calories[0] represents the calories in some product at the grocery store. The data type should be int because calories in foods are always reported as integers.

Problem 1.3

When we divide two arrays of the same length, their corresponding elements get divided, and the result is a new array of the same length as the two originals. In one sentence, interpret the meaning of min(prices / calories).

Answer: This is the cost per calorie of the product which has the lowest cost per calorie, which you might say is the cheapest food to fuel up on (like instant ramen or pasta).

Problem 1.4

True or False: min(prices / calories) is the same as max(calories / price).

Answer: False

The former is measured in dollars per calories (a very small number), whereas the latter is measured in calories per dollar (a very big number).

However, there is a connection between these two values. The product that has the lowest price per calorie is the same product with the most calories per dollar. So these numbers refer to the same grocery store product, and we can convert one value into the other by taking the reciprocal, which swaps the numerator and denominator of a fraction. Therefore, it’s true that min(prices / calories) is the same as 1 / max(calories / price).

Problem 2

Suppose we have imported the math module using import math. Consider the nested expression below:

int(math.sqrt(math.pow(4 * 2 ** 4, min(9 % 4, 9 / 4, 9 - 4))))

Problem 2.1

How many function calls are there in this expression? How many arguments does each function have?

Answer: 4 function calls: one argument for int(), one for math.sqrt(), two for math.pow(), three for min().

There are four function calls. One is a call to the type-conversion function int(), which takes one argument. Another is a call to math.sqrt(), which takes one argument. Another is a call to math.pow(), which takes two arguments. Finally is a call to the built-in function min(), which in this case takes three arguments, but generally can take two or more arguments.

Problem 2.2

What does this expression evaluate to?

Answer: 8

For nested evaluation, it is helpful to work from the inside out. Let’s evaluate some arithmetic expressions first. 9 % 4 evaluates to 1 because when we divide 9 by 4, there is a remainder of 1. Additionally, 9 / 4 evaluate to 2.25, and 9 - 4 evaluates to 5. Starting with the inner most function call, we see min(9 % 4, 9 / 4, 9 - 4) is equiavlent to min(1, 2.25, 5) which evaluates to 1.

The next-most inner function call is the call to math.pow() which takes two arguments: a number for the base, and a number for the exponent. We’ve already evaluated the exponent, but we need to evaluate the base of 4 * 2 ** 4. Using the order of operations, we know we need to evaluate the exponent first. So 4 * 2 ** 4 is equivalent to 4 * 16 or 64.

Therefore, math.pow(4 * 2 ** 4, min(9 % 4, 9 / 4, 9 - 4)) simplifies to math.pow(64, 1), which Python evaluates to be 64.0, a float.

Next, math.sqrt(64.0) evaluates to 8.0. Finally, the type conversion function int(8.0) evaluates to 8.

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