In this article, we classify gradually positive integers ≥2, and express each and every class of positive integers into a sum of 3 unit fractions. First, divide all positive integers ≥2 into 8 kinds, and then formulate each of 7 kinds of these 8 kinds into a sum of 3 unit fractions. For the unsolved kind, divide it into 3 genera, and then formulate each of 2 genera of these 3 genera into a sum of 3 unit fractions. For the unsolved genus, further divide it into 5 sorts, and formulate each of 3 sorts of these 5 sorts into a sum of 3 unit fractions. For two unsolved sorts, let each of them be expressed as a sum of an unit fraction plus a true fraction, and that take out the unit fraction as one of 3 unit fractions which express the sort as the sum. After that, if the true fraction can be transformed identically into an unit fraction, then we follow the formula that Ernst G. Straus made to transform either of these two unit fractions into a sum of two each other’s- distinct unit fractions, such that this part of the unsolved sort becomes a sum of 3 unit fractions. If the true fraction can not be transformed identically into an unit fraction, then we let it to equal the sum of an unit fraction plus another true fraction, and that take out the unit fraction as one of 3 unit fractions which express the sort as the sum. Next, prove that another proper fraction can be identically converted into an unit fraction. Due to c≥0, above two cases exist surely when c is taken different values.