The little one written above the next column is the most mysterious mark in school arithmetic, and it has a plain meaning. It is a bundle of ten ones, tied up and moved to where bundles live. This chapter says what it is, proves the paper method gives the right answer, and shows why it is never more than one when two numbers are added.
On the table: 58 is five ten-bundles and 8 loose stones; 67 is six bundles and 7 loose. Push the loose stones together: 15. That is more than ten loose stones, which Book XI said you must never leave. So tie ten of them into a bundle and move it across to the bundles. That bundle is the little one. Five loose stones stay behind, and the bundles now number 5 + 6 + 1 = 12, so tie ten bundles into a hundred. The answer is 125.
For the teacher: Carrying.lean writes down column addition exactly as it is done on paper. In each column, add the two digits and the incoming carry, write the ones, and carry the tens. addc_correct proves the digits written name the true sum, for numbers of any length. The script also checks it directly on 1.1 million pairs COMPUTED. The procedure a child is taught is the same thing as the theorem.
Two digits are at most 9 + 9 = 18, and with a carry of one, 19. Nineteen loose stones make one bundle, not two. carry_small proves it: adding two digits and a carry of at most one never carries more than one. That is why the little one is always a 1 when you add two numbers. The script confirms it never exceeded 1 on the whole grid COMPUTED.
Add a longer column and the carry grows. With three numbers it can be 2; with ten numbers, 9; with eleven or more it can reach 10, and then the carry itself has two digits COMPUTED. That is why long columns of figures in account books were added in stages.
Of the hundred pairs of digits, 45 add to ten or more and need a carry COMPUTED. So a carry is almost as common as not, and a learner should expect to meet one in nearly every other column. It is not an exception to learn separately.