Euler's constant (gamma or the Euler-Mascheroni constant) is a mathematical constant that appears in a wide variety of mathematical settings, including in Euler's famous formula for the sum of the reciprocal of the positive integers:

gamma = lim_{n->infty} (sum_{k=1}^{n} 1/k) - ln(n)

The constant gamma also arises in connection with the factorial function, and in particular with the Stirling's approximation for large factorials. In probability theory, gamma is the name given to the distribution of the sum of k independent exponential random variables with unit mean, and also appears in the negative binomial distribution.

The value of gamma can be numerically approximated to be 0.577216. Is the Euler-Mascheroni constant rational? No, the Euler-Mascheroni constant is not rational. This is because the constant is defined as the limit of the difference between the natural logarithm of n and the harmonic series. Both of these are irrational numbers, so the difference between them is also irrational.

#### How is Euler's constant defined?

Euler's constant is defined as the limit of (1 + 1/n)^n as n approaches infinity. This can be written as:

lim n→∞ (1 + 1/n)^n = e

where e is the natural logarithm of 2.

#### What is Euler-Mascheroni constant used for?

The Euler-Mascheroni constant is a mathematical constant that is often used in computer programming. It is named after Leonhard Euler and Lorenzo Mascheroni, and is also sometimes known as the Euler-Kronecker constant.

The Euler-Mascheroni constant is defined as the limit of the difference between the natural logarithm of n and the harmonic mean of 1, 2, ..., n as n goes to infinity. In other words, it is the difference between the sum of the reciprocals of the first n natural numbers and the natural logarithm of n.

The Euler-Mascheroni constant is often used in computer programming, particularly in numerical analysis and statistical mechanics. It is also used in some physical theories, such as quantum electrodynamics.

### How is Euler-Mascheroni constant calculated?

The Euler-Mascheroni constant is calculated by taking the limit of the difference between the natural logarithm of n and the harmonic series:

limn→∞(ln(n)−Hn)

This constant can also be expressed as the integral of the exponential function:

−∫∞0dt/t e−t

The Euler-Mascheroni constant is named after Leonhard Euler and Lorenzo Mascheroni.

### Is γ rational?

No, γ is not rational. γ is the Euler–Mascheroni constant, which is defined as the limit of the difference between the natural logarithm of n and the harmonic series:

γ = limn→∞(ln(n)−Hn)

where Hn is the nth harmonic number.

It is not known whether γ is algebraic or transcendental, but it is known that it is not rational. If γ were rational, then it would be possible to write it as a fraction p/q for some integers p and q, but it is not possible to do so.