One of the more interesting properties of the real numbers is their notation: when we write "2.3543", we actually mean a number that is closer to 2.3543 than it is to 2.3542 or 2.3544 . Similarly, when we write "2", it actually means any real number closer to 2 than it is to 1 or 3 ("2.0" would represent a number closer to 2 than it is to 2.1 or 1.9) .
This property means that it is possible that the numeral 2 can refer to a real number close to 2.4 , and so ~2.4 + ~2.4 = ~4.8 . ~4.8 is closer to 5 than it is to 4 or 6, and therefore 2 + 2 = 5, for high values of 2, and 2 + 2 = 3 for low values of 2.
This notation only applies to real numbers, not to integers, or rational numbers. This often causes confusion in high school maths classes, as the concept of inaccurate real numbers is often only briefly touched upon, and never seen again, and the distinction between the meaning of decimal fractions when dealing with inaccurate real numbers, and the meaning of decimal fractions in the rest of the syllabus is not made clear enough.
In The Real World, accuracy is hardly ever an issue unless you're doing engineering or physics, and even then, it is often ignored. If the accuracy of a number is not mentioned, the following rules usually apply:
- If accuracy is not mentioned at all, it is usually ignored. All numbers are assumed to be completely accurate
- Numbers which could represent integers do.
- Numbers in which the majority of the figures are significant are assumed to represent completely accurate rational numbers.
- Numbers in which few of the figures are significant are as accurate as the number of significant figures
For example, "2" represents the
integer 2 (and is therefore completely accurate). "2.124" represents the
rational number 2124/1000 (and is therefore also completely accurate). 3x10
8 has only one significant figure, and could represent any real number from 2.5x10
8 to 3.5x10
8.
In the real world, 2+2=4. In physics and engineering, (2±0.5) + (2±0.5) = (4±1)