a fixed word length. A formerly widespread method to achieve a correct change amount, knowing the due and given amounts, is the counting up method, which does not explicitly generate the value of the difference. Suppose an amount P is given in order to pay the required amount Q, with P greater than Q. Rather than explicitly performing the subtraction P − Q = C and counting out that amount C in change, money is counted out starting with the successor of Q, and continuing in the steps of the currency, until P is reached. Although the amount counted out must equal the result of the subtraction P − Q, the subtraction was never really done and the value of P − Q is not supplied by this method. Multiplication Multiplication, denoted by the symbols or , is the second basic operation of arithmetic. Multiplication also combines two numbers into a single number, the product. The two original numbers are called the multiplier and the multiplicand, mostly both are called factors. Multiplication may be viewed as a scaling operation. If the numbers are imagined as lying in a line, multiplication by a number greater than 1, say x, is the same as stretching everything away from 0 uniformly, in such a way that the number 1 itself is stretched to where x was. Similarly, multiplying by a number less than 1 can be imagined as squeezing towards 0, in such a way that 1 goes to the multiplicand. Another view on multiplication of integer numbers (extendable to rationals but not very accessible for real numbers) is by considering it as repeated addition. For example. corresponds to either adding times a , or times a , giving the same result. There are different opinions on the advantageousness of these paradigmata in math education. Multiplication is commutative and associative; further, it is distributive over addition and subtraction. The multiplicative identity is 1, since multiplying any number by 1 yields that same number. The multiplicative inverse for any number except is the reciprocal of this number, because multiplying the reciprocal of any number by the number itself yields the multiplicative identity . is the only number without a multiplicative inverse, and the result of multiplying any number and is again One says that is not contained in the multiplicative group of the numbers. The product of a and b is written as or