Question

Cube Comprehension: Use a list comprehension to generate a list of the first 10 cubes.

Short answer

Use the list comprehension: \( [n**3 \text{ for } n \text{ in range}(1, 11)] \).

Step-by-step solution

Understand the Problem

The task is to generate the first 10 cubes of natural numbers starting from 1. To do this, we'll be using Python's list comprehension, which is a concise way to create lists.

Set Up the List Comprehension

In Python, a list comprehension has the form \[ [expression for item in iterable] \]. In our case, the expression is the cube of each number, and the iterable is the range of numbers from 1 to 10.

Define the Expression

The expression we need is for generating cubes, which is \[ n^3 \]. In Python, this can be written as \[ n**3 \].

Define the Iterable

The iterable is the collection of numbers we're going to compute cubes for, which are the first ten natural numbers. In Python, you can use \[ range(1, 11) \] to generate numbers from 1 to 10.

Combine Expression and Iterable

Put everything together in a list comprehension: \[ [n**3 for n in range(1, 11)] \]. This iterates over each number \( n \) from 1 to 10, calculates the cube, and adds it to the list.

Execute and Verify

Execute the list comprehension code to generate the list. Verify that the output is a list containing the first ten cubes: \[ [1, 8, 27, 64, 125, 216, 343, 512, 729, 1000] \].

Key concepts

Cubes of Numbers

The concept of cubing a number is simple: multiplying a number by itself twice. For example, to cube the number 2, you compute \(2 \times 2 \times 2\), which equals 8. Cubing is denoted mathematically as \(n^3\). In Python, this operation is carried out with the exponentiation operator `**`. This gives us the syntax \(n^{**3}\).

Cubing numbers can reveal interesting patterns and is foundational in fields like algebra and geometry. Understanding cubes can help illustrate growth potentials as a number scales, since each increment grows the result exponentially. In this exercise, creating a list of cubes from 1 to 10 can show how quickly the values rise.

Iterables in Python

In Python, an iterable is any Python object capable of returning its members one at a time, allowing it to be looped over using a for-loop and other Python iterations. Examples of iterables include lists, tuples, and strings. These collections store sequences of data which can be efficiently accessed and manipulated.

Python's list comprehension harnesses the power of iterables to create new lists based on existing collections. It provides a syntactically compact way to apply expressions to each element in an iterable, such as transforming each item or filtering items based on a condition. Thus, recognizing an iterable's structure allows for efficient data manipulation in Python.

Expression Evaluation

Expressions are the core operations that are applied to the elements of an iterable in Python list comprehensions. An expression in Python can be as simple as a mathematical operation like addition, subtraction, multiplication, or exponentiation, and it provides the instruction for converting each item in an iterable.

In this exercise, the expression consists of cubing each number. Expression evaluation in the list comprehension transforms each number in the iterable \(n\) using \(n**3\), ultimately controlling the output of the transformation or calculation applied to each element.

• Ensures accurate calculations.
• Follows typical mathematical precedence and syntax.

Understanding how expressions are evaluated is key to making efficient computations in Python.

Range Function in Python

The range function is a versatile tool in Python for generating a specified sequence of numbers. It is particularly useful in for-loops and list comprehensions for iterating over a sequence of numbers without explicitly creating a list of numbers.

Using `range(start, stop)`, Python generates numbers starting from `start` up to, but not including, `stop`. For this exercise, `range(1, 11)` efficiently provides numbers from 1 through 10. It's a memory-friendly way of generating a sequence as it creates numbers on demand, as opposed to storing them in memory like a list would. This makes `range` exceptionally useful in large-scale iterations and loops where performance might otherwise be a concern.