# “The number of steps is infinite” or “The number of steps is infinity”?

In a mathematical paper about random-walks. Which is more correct: “The number of steps in the random-walk is infinite” or “The number of steps in the random-walk is infinity”?

The correct usage is “The number of steps is infinite”

You are qualifying the “number of steps”, here infinite is an adjective. You could say the number (or quantity) of steps is big, the number of steps is small, the number of steps in not important, the number of steps is mysterious, the number of steps is infinite.

Using “infinity” (a noun which represents a concept, infinity, that acts as a regular number in the sentence) would point at the exact value of the number of steps, like the number of steps is 3, the number of steps is 10, the number of steps is zero, the number of steps is infinity. (Here you use infinity like if it was a real number) That usage would sound strange but might be used for a comical effect.

Other possible valid sentences:

• There is an infinity of steps
• There is an infinite number of steps
• The steps are infinite

Hope that helps