Common Mathematical Constants
Reference table of the most widely used mathematical constants — π, e, φ, √2, the Euler–Mascheroni constant, and common natural and base-10 logarithms. Each entry includes a precise decimal approximation and where it shows up in algebra, geometry, calculus, statistics, physics, and engineering.
What Are Mathematical Constants?
A mathematical constant is a number that has a fixed, well-defined value across all of mathematics. Unlike variables or parameters, constants do not change — π is always the same ratio, regardless of context. Many of the most useful constants are irrational (their decimal expansions never terminate or repeat) and appear in multiple unrelated areas of math, which is part of what makes them important.
- Irrational constants: π, e, φ, √2, √3, √5, γ. Their decimals continue forever without repeating.
- Transcendental constants: π and e are not roots of any polynomial with rational coefficients.
- Algebraic constants: φ and √2 are irrational but satisfy simple polynomial equations.
Common Constants Table
Practical approximations sufficient for everyday calculations. For exact work, prefer the symbolic form (e.g., π, (1 + √5)/2).
| Symbol | Name | Approx. value | Notes |
|---|---|---|---|
π | Pi | 3.141592653589793 | Circle constant: circumference / diameter. |
τ | Tau | 6.283185307179586 | τ = 2π. One full turn in radians. |
e | Euler's number | 2.718281828459045 | Base of natural logarithms; continuous growth. |
φ | Golden ratio | 1.618033988749895 | φ = (1 + √5) / 2. |
√2 | Square root of 2 | 1.4142135623730951 | Diagonal of a unit square. |
√3 | Square root of 3 | 1.7320508075688772 | Appears in 30–60–90 triangles. |
√5 | Square root of 5 | 2.23606797749979 | Used in φ and geometry. |
γ | Euler–Mascheroni constant | 0.5772156649015329 | Appears in analysis and harmonic sums. |
ln(2) | Natural log of 2 | 0.6931471805599453 | Common in doubling/half-life calculations. |
ln(10) | Natural log of 10 | 2.302585092994046 | Useful for base-10 log conversions. |
log₁₀(e) | Base-10 log of e | 0.4342944819032518 | Conversion factor: log10(x) = ln(x)·log10(e). |
log₁₀(2) | Base-10 log of 2 | 0.3010299956639812 | Useful in orders of magnitude and digits/bit relations. |
Key Constants Explained
π — Pi (3.14159…)
The ratio of a circle's circumference to its diameter. π is irrational and transcendental. Beyond geometry, π appears in trigonometry (radian measure), Fourier analysis, probability (the normal distribution), and quantum mechanics.
- Common identities: circumference = 2πr, area of circle = πr², eiπ + 1 = 0.
- Approximations: 22/7 ≈ 3.1428 (off by ~0.04%), 355/113 ≈ 3.14159292 (off by < 0.0001%).
e — Euler's number (2.71828…)
The base of the natural logarithm. e arises naturally in continuous growth, compound interest, and the derivative rule for exponential functions: d/dx(ex) = ex.
- Definition: e = limn→∞ (1 + 1/n)n.
- Series form: e = Σ 1/n! = 1 + 1 + 1/2 + 1/6 + 1/24 + …
φ — Golden ratio (1.61803…)
The unique positive number satisfying φ² = φ + 1. The golden ratio appears in Fibonacci sequences, regular pentagons, and continued fractions, and is widely cited in art and design as an aesthetically pleasing proportion.
- Closed form: φ = (1 + √5) / 2.
- Continued fraction: φ = 1 + 1/(1 + 1/(1 + 1/(1 + …))).
γ — Euler–Mascheroni constant (0.57721…)
Defined as the limit of the difference between the harmonic series and the natural log: γ = limn→∞ (Hn − ln n), where Hn = 1 + 1/2 + 1/3 + … + 1/n. Whether γ is rational or irrational is one of the famous unsolved problems in mathematics.
Useful Formulas with These Constants
| Formula | Meaning |
|---|---|
C = 2πr | Circumference of a circle of radius r. |
A = πr² | Area of a circle of radius r. |
eiπ + 1 = 0 | Euler's identity, linking e, i, π, 1, and 0. |
log₁₀(x) = ln(x) / ln(10) | Base-10 logarithm in terms of natural log. |
t½ = ln(2) / λ | Half-life from exponential decay constant λ. |
Fn ≈ φn / √5 | Binet's approximation for the n-th Fibonacci number. |
How to Use These Values
- For numeric work: use the decimal approximations directly. 15 digits is enough for double-precision floating point.
- For exact work: prefer symbolic forms like
π,e, or(1 + √5) / 2to avoid round-off error. - Log base conversion:
log₁₀(x) = ln(x) / ln(10)andln(x) = log₁₀(x) · ln(10). - Mental approximations: π ≈ 3.14, e ≈ 2.72, φ ≈ 1.62, √2 ≈ 1.41, √3 ≈ 1.73.
Frequently Asked Questions
Why is π irrational?
π cannot be expressed as a ratio of two integers. This was first proved by Johann Lambert in 1768. π is also transcendental (proved by Lindemann in 1882), meaning it is not the root of any non-zero polynomial with rational coefficients.
What's the difference between e and π?
Both are irrational and transcendental, but they arise in different ways. π is fundamentally a geometric constant (circles), while e is fundamentally an analytic constant (continuous growth and natural logarithms). Euler's identity eiπ + 1 = 0 shows they are deeply connected.
How many digits of π do I actually need?
For most engineering work, 6–10 digits is more than sufficient. NASA uses only 15 digits for interplanetary navigation. 39 digits would be enough to calculate the circumference of the observable universe to the width of a hydrogen atom.
Is the golden ratio really "everywhere" in nature?
φ does appear in some natural patterns (sunflower seed arrangements, pinecone spirals, Fibonacci-related growth), but claims that it appears in classical art, architecture, and the human body are often overstated. Real instances are common in plant phyllotaxis, where φ produces optimal packing.