The Real Numbers
In this video we characterise the real numbers in terms of the axioms they obey. The real line is an ordered field which obeys the least upper bound property. As such it has an addition and multiplication law defined on it which obey the axioms of field theory. It then has an ordering on it, for which addition and multiplication by a number greater than zero are order preserving. Finally unlike the rationals it obeys the LUB axiom, which means that there are no holes within it.
In this video we characterise the real numbers in terms of the axioms they obey. The real line is an ordered field which obeys the least upper bound property. As such it has an addition and multiplication law defined on it which obey the axioms of field theory. It then has an ordering on it, for which addition and multiplication by a number greater than zero are order preserving. Finally unlike the rationals it obeys the LUB axiom, which means that there are no holes within it.