If you take a college-level math course or solve mathematical problems as part of your job, you know that certain numbers have special uses and will appear in your solutions more often than other numbers. Several of these constants such as pi and the square root of 2 have been used in higher mathematics since antiquity; many more have been around for several hundred years. Most calculators have the numbers, pi, e, and ln(2) built-in, but if you have a programmable calculator (such as a graphing calculator) you can add these other important constants to your calculator's memory.
Euler-Mascheroni Constant, γ
The Euler-Mascheroni constant is a transcendental number, an irrational number that is not the root of any polynomial function with rational number coefficients. It is usually represented by the lower-case Greek letter γ and its decimal expansion is
This constant is defined as the value of the limit
lim n→ ∞ [ H(n) - ln(n)],
where H(n) is the sum of the first n integer reciprocals, 1/1 + 1/2 + 1/3 + ... + 1/n. The E-M constant also appears in the values of some integrals, for example, the following integrals evaluated from 0 to ∞.
γ = ∫e-x ln(1/x) dx
γ2 + π2/6 = ∫e-x ln(x)2 dx
Natural Logarithm of 2, ln(2)
The natural log of 2 is the number that satisfies the relation eln(2) = 2. It is also a transcendental number and its decimal value is
Remarkably, the sum of the infinite alternating series 1/1 - 1/2 + 1/3 - 1/4 + ... converges to ln(2).
Apéry's Constant ζ(3)
Apéry's constant is the sum of the infinite series
(1/1)3 + (1/2)3 + (1/3)3 + (1/4)3 + (1/5)3 + ...
and is equal to
Unlike the sum of the square reciprocals, the sum of cubed reciprocals does not have a nice neat closed form involving π or any other known constants.
Pythagoras's Constant √2
The square root of 2 is sometime's called Pythagoras's constant since he was the first to prove that the number is irrational, that it cannot be expressed as a fraction p/q where p and q are integers. Its decimal expression is
The square root of 2 figures in some interesting identities, for example, the continued fraction
sqrt(2) = 1 + 1/( 2 + 1/( 2 + 1/( 2 + 1/( 2 + ...
Another relation involves infinite exponentials (power towers):
2 = sqrt(2)^[sqrt(2)^[sqrt(2)^[sqrt(2)^ ...
Golden Section φ
The golden section φ is equal to [1 + sqrt(5)]/2 or about
This number is a solution of the following equations:
- x2 = x + 1
- 1/x = x - 1
- x3 = 2x + 1
If a rectangle has sides that are in a ratio of φ:1 and you divide that rectangle into a square and a smaller rectangle, then the aspect ratio of the smaller rectangle will also be φ:1.
The golden section is also the limiting ratio of consecutive Fibonacci numbers. For example, the 15th, 16th, and 17th Fibonacci numbers are 610, 987, and 1597. Their ratios are
987/610 ≈ 1.6180328 and 1597/987 ≈1.6180344
The larger the Fibonacci numbers, the closer their ratio is to φ. Some other expressions that converge to φ are the following continued fraction and nested radical:
φ = 1 + 1/( 1 + 1/( 1 + 1/( 1 + 1/( 1 + ...
φ = sqrt(1 + sqrt(1 + sqrt(1 + sqrt( 1 + ...
Nested Radical Constant
The nested radical constant is the value of the infinitely nested square root
sqrt(1 + sqrt(2 + sqrt(3 + sqrt(4 + sqrt(5 + sqrt(6 + ....
and its decimal value is
Euler's Number e
Euler's constant e is defined as the limit
lim n→ ∞ (1 + 1/n)n
or as the value of the infinite sum of factorial reciprocals
1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + ...
= 1/1 + 1/1 + 1/2 + 1/6 + 1/24 + 1/120 + ...
The value of e is
and is often used in continuous compounding or continuous exponential growth problems.
Archimedes's Constant π
Pi is the ratio of a circle's circumference to its diameter. This is sometimes called the Archimedes constant since Archimedes was the first to develop an algorithm for computing the digits of π to a high degree of accuracy. Since Archimedes's time, much more efficient algorithms have been developed and we now know the value of π up to more than 17 million digits. The first digits are
The constant π appears in many mathematical expressions besides formulas for circles and spheres. For example, the infinite series and integral
π/4 = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - ...
π2/6 = (1/1)2 + (1/2)2 + (1/3)2 + (1/4)2 + (1/5)2 + ...
sqrt(π) = ∫e-x2 dx from -∞ to ∞
The constants e and π and the imaginary unit i = sqrt(-1) are all related by the formula
eπi + 1 = 0
Another curious identity involving these three constants is
ii = e-π/2