Real numbers<\/strong><\/a> \u2014 generally we think of these numbers as numbers that can be written as decimal expansions and we denote it by R. It is beyond the scope of this text to go into the details of how to give a precise definition of real numbers, and the notion that a real number can be written as a decimal expansion will be sufficient. It took mathematicians quite a long time to realise that there were numbers that could not be written as ratios of integers . The first numbers that were shown to be not-rational are square-roots of rime numbers, like \u221a 2. Other well known examples are \u03c0 and e. Usually the fact that some numbers cannot be represented as ratios of integers is harmless because those numbers can be approximated by rational numbers to any desired precision.
\nThe reason that we can approximate real numbers in this way is the surprising fact that between any two real numbers, one can always find a rational number. So if we are interested in a particular real number we can always find a rational number that is extremely close. Mathematicians refer to this property by saying that Q is dense in R.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"