One function de ned on this set is the standard part function, which returns a real number which is an element of the monad of the input; that function is a ring homomorphism from the limited version of this set to the real numbers.
If a coordinate plane is de ned with both axes ranging over the limited ver- sion of this set, then the parallel postulate is violated. One function de ned on this set is the standard part function, which returns a real number which is an element of the monad of the input; that function is a ring homomorphism from the limited version of this set to the real numbers. This set can be constructed through ultrapowers on real sequences, and this set is subject to the transfer principle, which means statements in rst-order logic true about the real num- bers are true about it. Non-standard analysis is the study of, for ten points, what extension of the real numbers which includes in nitesimals? 
If a coordinate plane is de ned with both axes ranging over the limited ver- sion of this set, then the parallel postulate is violated.
hyperreal numbers [accept non-standard reals until non-standard ] 