Mathematical constants (pi, e, golden ratio) in SciPy - Time & Space Complexity

Time Complexity: Mathematical constants (pi, e, golden ratio)

O(n)

Understanding Time Complexity

We want to understand how fast we can get values of famous math constants using scipy.

How does the time to get these constants change if we ask for many of them?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.

from scipy import constants

values = []
for _ in range(n):
    values.append(constants.pi)
    values.append(constants.e)
    values.append(constants.golden)

This code collects the constants pi, e, and the golden ratio n times into a list.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

• Primary operation: Loop running n times, each time fetching 3 constants.
• How many times: The loop runs exactly n times, repeating the constant retrieval 3 times per loop.

How Execution Grows With Input

Each time we increase n, the number of operations grows directly with n.

Input Size (n)	Approx. Operations
10	30 (10 loops x 3 constants)
100	300 (100 loops x 3 constants)
1000	3000 (1000 loops x 3 constants)

Pattern observation: The operations increase evenly as n grows; doubling n doubles the work.

Final Time Complexity

Time Complexity: O(n)

This means the time to get all constants grows in a straight line as we ask for more.

Common Mistake

[X] Wrong: "Getting constants like pi or e takes a long time or grows with their value."

[OK] Correct: These constants are stored values in scipy, so fetching them is very fast and does not depend on their size or value.

Interview Connect

Knowing how simple operations like fetching constants scale helps you understand bigger data tasks better.

Self-Check

"What if instead of fetching constants, we computed them with a complex formula inside the loop? How would the time complexity change?"

Practice

1. Which of the following constants is NOT directly available in scipy.constants?

easy

A. pi

B. e

C. golden ratio

D. Avogadro's number

Solution

1. Step 1: Recall scipy.constants common constants

   Scipy provides many constants like pi, e, and the golden ratio directly accessible.

2. Step 2: Check Avogadro's number availability

   Avogadro's number is not included in scipy.constants; it is usually found in other libraries like scipy.constants.physical_constants.

3. Final Answer:

   Avogadro's number -> Option D

4. Quick Check:

   Avogadro's number not in scipy.constants [OK]

Hint: Remember scipy.constants has math constants, not all physical constants [OK]

Common Mistakes:

• Assuming all physical constants are in scipy.constants
• Confusing golden ratio with Avogadro's number
• Thinking Avogadro's number is a math constant

2. Which is the correct way to import the constant pi from scipy?

easy

A. from scipy.constants import pi

B. import scipy.pi

C. from scipy import pi

D. import pi from scipy.constants

Solution

1. Step 1: Understand Python import syntax

   To import a specific item from a module, use from module import item.

2. Step 2: Apply correct import for pi

   Since pi is inside scipy.constants, the correct import is from scipy.constants import pi.

3. Final Answer:

   from scipy.constants import pi -> Option A

4. Quick Check:

   Correct import syntax for pi [OK]

Hint: Use 'from module import item' for constants [OK]

Common Mistakes:

• Using 'import scipy.pi' which is invalid
• Trying 'from scipy import pi' without constants
• Incorrect import order like 'import pi from ...'

3. What is the output of this code?

from scipy.constants import golden_ratio
print(round(golden_ratio, 3))

medium

A. 3.142

B. 1.618

C. 2.718

D. 1.414

Solution

1. Step 1: Identify the constant used

   The code imports golden_ratio from scipy.constants, which is approximately 1.6180339887.

2. Step 2: Apply rounding to 3 decimals

   Rounding 1.6180339887 to 3 decimals gives 1.618.

3. Final Answer:

   1.618 -> Option B

4. Quick Check:

   golden_ratio ≈ 1.618 [OK]

Hint: Golden ratio rounds to 1.618 [OK]

Common Mistakes:

• Confusing golden ratio with pi or e
• Rounding incorrectly to 3 decimals
• Printing without importing golden_ratio

4. Find the error in this code snippet:

from scipy.constants import e
print(e())

medium

A. e is a constant, not a function; remove parentheses

B. Import statement is incorrect

C. e needs to be converted to string before printing

D. Parentheses are required to call e

Solution

1. Step 1: Understand what e represents

   In scipy.constants, e is a constant value (Euler's number), not a function.

2. Step 2: Identify misuse of parentheses

   Calling e with parentheses e() tries to call a float as a function, causing an error.

3. Final Answer:

   e is a constant, not a function; remove parentheses -> Option A

4. Quick Check:

   Constants are values, not callable functions [OK]

Hint: Constants are values, don't add () [OK]

Common Mistakes:

• Trying to call constants as functions
• Incorrect import syntax
• Thinking constants need conversion before print

5. You want to calculate the area of a circle with radius 3 using scipy constants. Which code snippet correctly uses the constant pi for this calculation?

hard

A. import scipy.constants as sc
area = sc.pi(3) ** 2
print(area)

B. from scipy.constants import e
area = e * 3 ** 2
print(area)

C. from scipy.constants import pi
area = pi * 3 ** 2
print(area)

D. from scipy.constants import golden_ratio
area = golden_ratio * 3 ** 2
print(area)

Solution

1. Step 1: Identify the correct constant for circle area

   The area of a circle formula is pi times radius squared, so we need pi.

2. Step 2: Check code correctness

   from scipy.constants import pi area = pi * 3 ** 2 print(area) imports pi correctly and calculates area as pi * 3^2, which is correct.

3. Step 3: Verify other options

   import scipy.constants as sc area = sc.pi(3) ** 2 print(area) tries to call pi as a function, which is wrong. Options C and D use wrong constants (e and golden_ratio) for area.

4. Final Answer:

   from scipy.constants import pi area = pi * 3 ** 2 print(area) -> Option C

5. Quick Check:

   Use pi * r^2 for circle area [OK]

Hint: Use pi constant directly for circle area [OK]

Common Mistakes:

• Calling pi as a function
• Using wrong constants for area
• Incorrect import or calculation syntax