What is a number that cannot be written as a ratio of two integers?

A number that cannot be expressed as a ratio of two integers is said to be irrational.

There are names for groups of numbers:

The natural numbers or the counting numbers are 1,2,3,...
The integers are whole numbers including zero and the negative numbers.
The rationals are numbers that can be expressed as a ratio of two integers.
The irrationals are numbers on the real number that cannot be expressed as a ratio of two integers.

There are subdivisions of the irrationals:

Algebraic numbers are solutions of equations involving addition, subtraction, multiplication, division, and the taking of roots (or rational powers.) Thus sqrt(2) is an algebraic irrational since it is the solution to x^2=2, but cannot be expressed as a ratio of integers.

Transcendental numbers are irrationals that are not algebraic. Pi, e (approximately 2.71828), sin(20) are transcendental. The vast majority of real numbers are transcendental.