Basically imagine numbers represented in scientific notation - some bits represent the mantissa, some bits the exponents. For more details, read the following Wikipedia article: http://en.wikipedia.org/wiki/Floating_point
It is also possible to represent numbers as a text (string) - this makes it possible to use numbers of an arbitrary size, with arbitrary precision, and without rounding errors (at least, for certain operations). For example, Java has a class called BigDecimal, precisely for this purpose.
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All real numbers can be represented in the decimal system. Complex numbers can be represented by a pair of numbers in the decimal system.
These number can also be represented on real line.
The set of real numbers is represented by a fancy script "R". However, real numbers are just represented by a number (be it integer, rational, irrational, or otherwise). The key here is that it is not a complex number (number containing the imaginary number "i").
Rational numbers are sometimes represented in tabular form, e.g. for proofs relating to different infinities. Irrational and real numbers are not because there is no way to do it.
The set of all real numbers usually represented by a capital R that is double vertical line in the left and bold.