Mathematical Constants

# Fundamental constants (accessed as mp.constant_name)
mp.pi        # π ≈ 3.14159265358979323846...
mp.e         # Euler's number e ≈ 2.71828182845904523536...
mp.ln2       # Natural logarithm of 2 ≈ 0.69314718055994530942...
mp.ln10      # Natural logarithm of 10 ≈ 2.30258509299404568402...
mp.phi       # Golden ratio φ = (1+√5)/2 ≈ 1.61803398874989484820...

# Number theory and analysis constants
mp.euler     # Euler-Mascheroni constant γ ≈ 0.57721566490153286061...
mp.catalan   # Catalan's constant G ≈ 0.91596559417721901505...
mp.khinchin  # Khinchin's constant K ≈ 2.68545200106530644531...
mp.glaisher  # Glaisher-Kinkelin constant A ≈ 1.28242712910062263688...
mp.apery     # Apéry's constant ζ(3) ≈ 1.20205690315959428540...

# Prime number constants  
mp.twinprime # Twin prime constant ≈ 0.66016181584686957392...
mp.mertens   # Mertens constant M ≈ 0.26149721284764278375...

# Unit conversion
mp.degree    # π/180 for degree to radian conversion ≈ 0.01745329251994329577...

# Special values
mp.inf       # Positive infinity
mp.ninf      # Negative infinity (-∞)
mp.nan       # Not a number (NaN)
mp.j         # Imaginary unit i = √(-1)
mp.eps       # Machine epsilon (smallest representable positive number)

# Constants adapt to current precision
with mp.workdps(50):
    high_precision_pi = mp.pi  # π with 50 decimal places

with mp.workdps(100):  
    very_high_precision_e = mp.e  # e with 100 decimal places

import mpmath
from mpmath import mp

# Set different precision levels
mp.dps = 30

# Display fundamental constants
print("Fundamental Constants:")
print(f"π = {mp.pi}")
print(f"e = {mp.e}")
print(f"Golden ratio φ = {mp.phi}")
print(f"ln(2) = {mp.ln2}")
print(f"ln(10) = {mp.ln10}")

# Special constants
print("\nSpecial Constants:")
print(f"Euler-Mascheroni γ = {mp.euler}")
print(f"Catalan's constant = {mp.catalan}")
print(f"Khinchin's constant = {mp.khinchin}")
print(f"Glaisher-Kinkelin A = {mp.glaisher}")
print(f"Apéry's constant ζ(3) = {mp.apery}")

# Prime-related constants
print("\nPrime-Related Constants:")
print(f"Twin prime constant = {mp.twinprime}")
print(f"Mertens constant = {mp.mertens}")

# Unit conversion
print("\nUnit Conversion:")
print(f"Degrees to radians: π/180 = {mp.degree}")
angle_deg = 90
angle_rad = angle_deg * mp.degree
print(f"{angle_deg}° = {angle_rad} radians")

# Verify relationships
print("\nConstant Relationships:")
print(f"φ² - φ - 1 = {mp.phi**2 - mp.phi - 1}")  # Should be 0
print(f"e^(iπ) + 1 = {mp.exp(mp.j * mp.pi) + 1}")  # Euler's identity, should be 0
print(f"ζ(2) = π²/6 = {mp.pi**2 / 6}")
print(f"Actual ζ(2) = {mp.zeta(2)}")

# Precision scaling
print("\nPrecision Scaling:")
with mp.workdps(10):
    pi_10 = mp.pi
    print(f"π with 10 digits: {pi_10}")

with mp.workdps(50):
    pi_50 = mp.pi  
    print(f"π with 50 digits: {pi_50}")

with mp.workdps(100):
    pi_100 = mp.pi
    print(f"π with 100 digits: {pi_100}")

# Special values
print("\nSpecial Values:")
print(f"Positive infinity: {mp.inf}")
print(f"Negative infinity: {mp.ninf}")
print(f"Not a number: {mp.nan}")
print(f"Imaginary unit: {mp.j}")
print(f"j² = {mp.j**2}")  # Should be -1

# Machine epsilon  
print(f"Machine epsilon: {mp.eps}")

# Using constants in calculations
print("\nCalculations with Constants:")
# Area of unit circle
area = mp.pi * 1**2
print(f"Area of unit circle: π = {area}")

# Compound interest formula: A = P*e^(rt)
principal = 1000
rate = 0.05  # 5%
time = 10
amount = principal * mp.exp(rate * time)
print(f"Compound interest: {principal} * e^({rate}*{time}) = {amount}")

# Golden ratio properties
print(f"φ = (1 + √5)/2 = {(1 + mp.sqrt(5))/2}")
print(f"1/φ = φ - 1 = {1/mp.phi} = {mp.phi - 1}")

# Trigonometric identities with high precision
print(f"sin²(π/6) + cos²(π/6) = {mp.sin(mp.pi/6)**2 + mp.cos(mp.pi/6)**2}")
print(f"e^(iπ/4) = {mp.exp(mp.j * mp.pi/4)}")
print(f"cos(π/4) + i*sin(π/4) = {mp.cos(mp.pi/4) + mp.j*mp.sin(mp.pi/4)}")

# Natural logarithm identities
print(f"e^(ln(2)) = {mp.exp(mp.ln2)}")  # Should be 2
print(f"ln(e) = {mp.ln(mp.e)}")        # Should be 1
print(f"ln(10) = {mp.ln10}")
print(f"ln(10)/ln(2) = {mp.ln10/mp.ln2}")  # log₂(10)