Because, as we describe it, mathematics is not inherently connected to the real world, we must find ways to make this connection in order to solve real world problems. Otherwise all mathematics would be pure mathematics: lots of fun, but no more relevant to the real world than the solution to a crossword puzzle.
One bridge between the real world and mathematics is called measurement. Measurement can be quite simple, such as counting the number dollar bills in your wallet to come up with an integer, or applying a ruler to a piece of string to come up with a rational number. Measurement can also be very complex, such as measuring the energy released by an earth quake.
No matter how simple or complex the measurement process is, the end result is one or more mathematical objects that correspond in some way to whatever we were trying to measure.
Every measurement has an uncertainty associated with it.
The uncertainty of a measurement is usually called error. This does not necessarily mean that there was some mistake or blunder in taking the measurement, although a mistake might have happened. Most often, error exists simply because our measurement methods are limited. This error is a mathematical object also and its effect can be very small, such as an error in counting your dollar bills, or it can be large enough that it must be accounted for whenever we use a measurement. In this discussion we will largely ignore measurement error and leave that topic for a later lesson.