This paper gives a brief account of the history of non-standard analgsis and introduces the hyper-real system, the nonstandard representation of deriate. We illustrate that the proof of mathematical proposition by nonstandard methods is simpler and more directly perceived through the senses than standard methods. In the last part of the paper, we introduce the loeb probability space, the integral in respect to p as *-finite summation and point out that the probability theory is radically elementary.