Can a base be irrational?

A number which is a ratio of any two integers is a rational number. A number which is not the ratio of two integers is an IRRATIONAL number. HallsofIvy said: One can use any number as a base.

Are irrational numbers irrational in all bases?

No, the rationality or irrationality of a number is independent of the base that it is displayed in. A rational number can be represented as the division of one integer by another. An irrational number cannot. This does not change depending on whether the numbers are written in base 2, base 10, or some other base.

Do bases have to be integers?

Although most applications of bases restrict b to be a positive integer, it is worth considering what happens when b is, more generally, a positive real number.

Can an irrational number also be a real number?

Real numbers are, in fact, pretty much any number that you can think of. This can include whole numbers or integers, fractions, rational numbers and irrational numbers. Real numbers can be positive or negative, and include the number zero.

Is sqrt2 * pi irrational?

b x pi = a x sqrt(2). The right side is a root of a polynomial with integer coefficients (i.e. is algebraic) but the left side is not algebraic. This is impossible. Therefore pi/sqrt(2) is irrational.

Are there fractional bases?

An improper fractional base is a type of number base. Instead of using an integer for the base in our positional number system, we use an improper fraction for the base.

Is square rooted rational or irrational?

Approximating Square Roots Many square roots are irrational numbers, meaning there is no rational number equivalent. For example, 2 is the square root of 4 because \begin{align*}2 \times 2 = 4\end{align*}.

What is an irrational base number system?

irrational-base number system. A number system that is based on an irrational number or numbers, or is composed entirely of irrational numbers. i.e. pi, e, and the square root of 2.

What happens to rational/irrational numbers when you change base?

When it comes to properties like prime, irrational, rational, divisible by 2, etc., nothing changes when you change base. But I’m not sure about the rational/irrational one. The point is: is that number irrational, or it just has a lot of repeating decimals? Either way, how could you prove it?

Is the number 3 a rational number in base 3?

3 is a rational number and an integer, regardless of how it is written or in what base. It is true that a rational number will have a repeating or terminating representation in an integer base, and an irrational number will not, but that is separate from the question of whether it is a rational number.

Do rational numbers have infinite digits in base Pi?

Well, yes and no. 0 and 1 are still 0 and 1 in any base. And if you’re working with circle s, sphere s, and “spheres” in more than 3 dimensions, base pi is fairly useful. But it’s true that most rational numbers would have an infinite number of digits in base pi.