Minus signs undo addition but scramble order rules
Subtraction, written with −, removes quantity and inverts addition: a − b equals c exactly when c + b equals a. It is anti-commutative—order flips sign—and not associative. Latin roots say pull from under; minuend, subtrahend, and difference name the parts.
The Latin behind the vocabulary is vivid: subtrahere joins sub, from under, with trahere, to pull, so subtracting means drawing something out from below. The subtrahend is the thing to be taken away, and the minuend, from minuere, to diminish, is the thing made smaller. Taking away zero leaves a number unchanged, and for any integer, subtracting one gives the next smaller integer, its predecessor. Accountants sometimes skip the sign altogether, printing the lower figure of a column in red to show it should be deducted.
On a number line, adding means stepping right and subtracting stepping left. Among natural numbers alone, 3 minus 4 walks off the end of the line, so subtraction there is not closed: 11 minus 26 either has no answer, making the operation a partial function, or the line is extended to integers and the answer is negative 15. Mathematicians can even define the real numbers with only addition and multiplication plus inverses, treating 3 minus π as 3 plus negative π.
Units must match before subtracting, and percentages invite confusion. If defects fall from 30 percent to 20 percent, that is a drop of 10 percentage points but a relative change of minus one third. Mechanical calculators and modern computers subtract by adding complements; in binary, flipping every bit and adding one yields the two's complement, and discarding the leading 1 of the sum gives the answer.
Nearly all American schools teach borrowing with memory marks called crutches, a practice that spread after William A. Brownell published a study praising them. Some European schools use the Austrian method, which avoids borrowing by raising the next subtrahend digit instead.
Source: Subtraction
