The Definitions Of Subtraction
The second of the two rules above can be taken as a definition of subtraction. That is "the difference of two numbers, a - b, is the number which must be added to the second number, b, to equal the first, a." Before the widespread use of electronic cash registers, grocery clerks and others, making change, would use this definition directly. If, for example, a customer bought groceries worth $3.70 and gave the clerk $5.00 to pay for them, the clerk would not subtract in order to compute the change. Instead he (or she) would count it out. Starting with "three seventy," he would give the customer a nickel, and say, "three seventy-five." He would give the customer a quarter and say, "four." Then he would give the customer a dollar and say, "five." This method is still in use when no calculator is available.
A second definition of subtraction is often given in algebra books after the introduction of negative numbers. There a - b is defined as a + (-b), i.e. "To subtract a number, add its opposite." This definition is mathematically efficient, but it hides a lot of the ways in which subtraction is used, as in the change-making example above.