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Euler's Number Calculator

Explore Euler's number e with Taylor series approximation, continuous compounding, and normal distribution.

Euler's Number (e)

e = 2.718281828459045

About this tool

Euler's number (e ≈ 2.71828) is one of the most important mathematical constants, appearing in calculus, statistics, finance, and physics. It represents the base of natural logarithms and describes continuous growth and decay. This calculator helps you explore e's behavior through Taylor series approximations, which show how mathematicians calculate this infinite constant to any desired precision.

Start by adjusting the number of Taylor series terms to see how the approximation converges toward the true value of e. The continuous compounding simulator shows how Euler's number emerges when interest compounds infinitely often—a core concept in financial mathematics. The normal distribution viewer displays the famous bell curve, which depends on e in its mathematical formula and is essential for statistics and data science.

This tool is ideal for students learning calculus and statistics, developers working with mathematical libraries, and anyone curious about how fundamental constants shape real-world phenomena. Each visualization updates instantly, letting you experiment with different parameters and build intuition for exponential growth, probability distributions, and the deep mathematics underlying everyday financial and scientific calculations.

Frequently Asked Questions

Code Implementation

import math
from decimal import Decimal, getcontext

# 1. Taylor series approximation of e
def euler_taylor(n_terms: int) -> float:
    """Approximate e using n terms of Taylor series: sum(1/k!) for k=0..n"""
    total = 0.0
    factorial = 1
    for k in range(n_terms):
        if k > 0:
            factorial *= k
        total += 1 / factorial
    return total

# 2. High-precision e using Python's decimal module
def euler_high_precision(decimal_places: int) -> Decimal:
    getcontext().prec = decimal_places + 10  # extra guard digits
    e = Decimal(0)
    factorial = Decimal(1)
    for k in range(200):  # 200 terms is enough for 100+ decimal places
        if k > 0:
            factorial *= k
        term = Decimal(1) / factorial
        e += term
        if term < Decimal(10) ** -(decimal_places + 5):
            break
    return +e  # re-apply precision

# 3. Common formulas involving e
def compound_continuous(principal: float, rate: float, years: float) -> float:
    """A = P × e^(r×t) — continuous compounding formula"""
    return principal * math.exp(rate * years)

def normal_pdf(x: float, mu: float = 0, sigma: float = 1) -> float:
    """Normal distribution PDF: f(x) = (1/σ√2π) × e^(-(x-µ)²/2σ²)"""
    return (1 / (sigma * math.sqrt(2 * math.pi))) * math.exp(-((x - mu)**2) / (2 * sigma**2))

# Examples
print(f"e (Python math):     {math.e}")
print(f"e (5 terms):         {euler_taylor(5):.8f}")
print(f"e (20 terms):        {euler_taylor(20):.15f}")

# Euler's identity: e^(iπ) + 1 = 0
e_to_ipi = math.exp(complex(0, math.pi))
print(f"\ne^(iπ) = {e_to_ipi.real:.10f} + {e_to_ipi.imag:.0f}i")
print(f"e^(iπ) + 1 = {e_to_ipi.real + 1:.2e}  (≈ 0, Euler's identity)")

# Continuous compounding
amount = compound_continuous(1000, 0.05, 10)
print(f"\n$1000 at 5% for 10 years (continuous): ${amount:.2f}")

# Normal distribution at x=0
print(f"Normal PDF at x=0: {normal_pdf(0):.6f}")

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