Meta Description:

Explore more examples to master the SageMath interface. Dive deeper into calculations, graphing, interactive features, and customizations in this advanced walkthrough.

🔄 A Quick Recap

Previously, we explored how to launch SageMath, navigate its interface, and perform basic calculations. In this post, we’ll expand our skills with practical examples to help you unlock the full potential of SageMath.

🛠️ Beyond Basics: Advanced Features in the Toolbar

Highlight the additional functionalities of SageMath's interface:

Let's Practice more Examples:

1. Write python codes to input radius of a sphere and print its surface area and volume.

# Input radius from user

r = float(input("Enter the radius of the sphere: "))

# Surface area of a sphere = 4 * pi * r^2

surface_area = 4 * pi * r^2

# Volume of a sphere = (4/3) * pi * r^3

volume = (4/3) * pi * r^3

Output:

💬 Understanding the Output

Surface Area gives the amount of outer space the sphere occupies.
Volume tells us how much space is enclosed inside the sphere.

✅ Try entering values like r = 1, r = 5, or even a decimal like r = 2.5 and observe how the results scale.

🎯 Challenge:

Try modifying the code to calculate the area and volume for multiple spheres. Can you store the values in a list or a dictionary?

2. Input a positive integer n and integer k<=n. Verify the following properties

# Input values

n = int(input("Enter a positive integer n: "))

k = int(input("Enter an integer k (k <= n): "))

# Check for valid input

if n < 0 or k < 0 or k > n:

print("Invalid input. Please ensure n >= 0 and 0 <= k <= n.")

else:

# Define binomial coefficient function

C = binomial

# Identity i: nCk = nC(n−k)

lhs1 = C(n, k)

rhs1 = C(n, n - k)

print("\ni. nCk = nC(n-k)")

print(f"{lhs1} = {rhs1} ->", "Verified ✅" if lhs1 == rhs1 else "Not Verified ❌")

# Identity ii: k * nCk = n * (n-1)C(k-1)

lhs2 = k * C(n, k)

rhs2 = n * C(n - 1, k - 1)

print("\nii. k × nCk = n × (n−1)C(k−1)")

print(f"{lhs2} = {rhs2} ->", "Verified ✅" if lhs2 == rhs2 else "Not Verified ❌")

# Identity iii: nC(k−1) + nCk = (n+1)Ck

lhs3 = C(n, k - 1) + C(n, k)

rhs3 = C(n + 1, k)

print("\niii. nC(k−1) + nCk = (n+1)Ck")

print(f"{lhs3} = {rhs3} ->", "Verified ✅" if lhs3 == rhs3 else "Not Verified ❌")

Output :

💬 Understanding the Output

These are combinatorial identities often used in probability and algebra:

The first identity shows symmetry in combinations.
The second connects combinations and multiplication.
The third is Pascal's Rule, foundational for Pascal’s Triangle.

📸 Visual : Pascal’s Triangle highlighting how `nC(k-1) + nCk = (n+1)Ck`.

🎯 Challenge:

Can you build Pascal’s Triangle up to n = 10 using a loop? Try printing it in a triangular form!

3. Input a complex number z and explore some of the properties of z.

SageMath code that takes a complex number

$z = a + bi$ as input and explores the following common properties:

🔍 Explored Properties:

Real part
Imaginary part
Conjugate
Modulus (absolute value)
Argument (angle in radians)
Polar form

# Input complex number as string

z_input = input("Enter a complex number (e.g. 3+4i): ")

# Replace 'i' with 'I' to make it SageMath-friendly

z_input = z_input.replace('i', 'I')

# Evaluate the input string as a SageMath expression

z = sage_eval(z_input)

# Display the entered complex number

print(f"\nComplex number z = {z}")

# Real and Imaginary parts

print("Real part of z:", z.real())

print("Imaginary part of z:", z.imag())

# Conjugate

print("Conjugate of z:", z.conjugate())

# Modulus

print("Modulus (|z|):", abs(z))

# Argument (angle in radians)

print("Argument (arg(z)):", arg(z))

# Polar form

r = abs(z)

theta = arg(z)

print(f"Polar form: {r} * (cos({theta}) + i*sin({theta}))")

# Other computations

print("z squared (z^2):", z^2)

print("Reciprocal (1/z):", 1/z)

print("z * conjugate(z):", z * z.conjugate())