Knowing pi as an expansion of digits is in a quite stronger sense not knowing pi at all; it's original geometric definition rests on the ratio of the circumference of a circle to its diameter; why is this even a useful definition - first because a circle is a simple geometric figure; look around you to see how many shapes are approximate circles or spheres; a square piece of paper when you increase its number of sides becomes a circle; it's also, more significantly given its provenance the shape that the sun moves around the sun.
But why should we note this ratio? Because knowing the circumference of a shape is significant; as it's diameter - so one might suppose its ratio might be significant; but then why not adding the two together or subtracting them?
Well, when we magnify a figure the ratio of its circumference multiplies by a certain value; as does its diameter; and it's this constancy that is recognised by seeing that the ratio is a constant.
But this is only the beginning of understanding the significance of the ratio pi.
In all this the digital expansion does not figure; it's of course important for computers since they represent everything digitally; but it is only a representation; and this representation is always approximate upto a precision determined by the designer.
There are systems of exact arithmetic that symbolically represent transcendental numbers (those numbers whose digits don't ever repeat); simply by representing this number by some additional symbol, as is normally done in mathematics by the Greek letter pi.
Computers only 'know' something in a manner of speaking; that is the Turing Test; which is an operational definition; which is good enough for practical and pragmatic purposes; but not good enough for philosophic purposes.
Knowing presupposes a knower; a thing cannot know; having no self to speak of; a computer no matter how closely it imitates knowing in a different sense does not know at all; consider for example a film starting Selma Hayek; it is recognisably her but one doesn't ordinarily confuse the image on the film with the person herself.
Finally, knowing pi as an unending string of digits is to only know that there is such a number; and that it is significant in some way; in a way it's a modern form of numerology; which is to mathematics what astrology is to astronomy; and in its own way, and a sociological way not uninsignificant either.