is a compound of sub "from under" and trahere "to pull". Thus, to subtract is to draw from below, or to take away. Using the gerundive suffix -nd results in "subtrahend", "thing to be subtracted". Likewise, from minuere "to reduce or diminish", one gets "minuend", which means "thing to be diminished". Of integers and real numbers Integers Imagine a line segment of length b with the left end labeled a and the right end labeled c. Starting from a, it takes b steps to the right to reach c. This movement to the right is modeled mathematically by addition: a + b = c. From c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction: c − b = a. Now, a line segment labeled with the numbers , , and . From position 3, it takes no steps to the left to stay at 3, so . It takes 2 steps to the left to get to position 1, so . This picture is inadequate to describe what would happen after going 3 steps to the left of position 3. To represent such an operation, the line must be extended. To subtract arbitrary natural numbers, one begins with a line containing every natural number (0, 1, 2, 3, 4, 5, 6, ...). From 3, it takes 3 steps to the left to get to 0, so . But is still invalid, since it again leaves the line. The natural numbers are not a useful context for subtraction. The solution is to consider the integer number line (..., −3, −2, −1, 0, 1, 2, 3, ...). This way, it takes 4 steps to the left from 3 to get to −1: . Natural numbers Subtraction of natural numbers is not closed: the difference is not a natural number unless the minuend is greater than or equal to the subtrahend. For example, 26 cannot be subtracted from 11 to give a natural number. Such a case uses one of two approaches: Conclude that 26 cannot be subtracted from 11; subtraction becomes a partial function. Give the answer as an integer representing a negative number, so the result of subtracting 26 from 11 is −15. Real numbers The field of real numbers can be defined specifying only two binary operations, addition and multiplication, together with unary operations yielding additive and