Multiplication is the process of counting a number two or more times. It can be considered a shortened form of addition. Thus, to add the number 4 three times, 4 + 4 + 4, we can use multiplication terms, that is, 4 multiplied by 3. When numbers are multiplied, the result is called the product. The numbers multiplied are called factors. One factor is called the multiplicand; the other is called the multiplier. Multiplication is indicated by the times or multiplication sign (x), by a raised dot (^.), or by an asterick (*).

In multiplying several numbers, the same product is obtained even if the numbers are multiplied in a different order or even if some of the numbers are multiplied together before the final multiplication is made. These properties are called the commutative and associative laws for multiplication.

The commutative law for multiplication states that numbers can be multiplied in any order, and the result is the same product. In equation form:

axb=bxa (1-4)

Thus, the product of 8 x 3 is the same as 3 x 8.

The associative law for multiplication states that factors can be associated in any order, and the result is the same product. In equation form:

ax(bxc)=(axb)xc (1-5)

Thus, the numbers 2, 3, and 5 can be multiplied by first multiplying 2 x 3 to equal 6 and then multiplying 6 x 5 to equal 30. The equation may also be solved by first multiplying 3x5 to equal 15, and then multiplying 15 x 2 to equal 30. In either case, the product is 30.

In multiplying two numbers, one number is placed under the other with the digits arranged in columns placing units under the units place, tens under the tens place, and so on. Usually, the larger number is considered the multiplicand and the smaller number is considered the multiplier. The digit in the units place of the multiplier is multiplied first, the digit in the tens place of the multiplier next, and so on.

In multiplying the multiplier in the units column to the multiplicand, 8 x 2 = 16. A 6 is placed under the units column, and 1 ten is carried. Then, 8 x 3 = 24, plus the 1 carried over equals 25.

A 5 is placed under the tens column, and 2 hundreds are carried over. Next, 8 x 4 = 32, plus 2 carried over, equals 34. A 4 is placed under the hundreds column and a 3 under the thousands column.

Start by multiplying the digit in the units place of the multiplier, 9 x 6 = 54. A 4 is placed under the units column, and 5 tens are carried over.

Next, 9 x 7 = 63, plus the 5 carried over, equals 68. An 8 is placed under the tens column, and 6 hundreds are carried over. Then, 9 x 1 = 9, plus 6 carried over, equals 15. A 5 is placed under the hundreds column and a 1 under the thousands column.

The digit in the tens place of the multiplier is multiplied now: 5 x 6 = 30.Since the 5 in 59 is in the tens column, the zero is placed under the tens column, and 3 tens are carried over. Next, 5 x 7 = 35, plus the 3 carried over, equals 38. An 8 is placed under the hundreds column, and 3 hundreds are carried over.

Then, 5 x 1 = 5, plus 3 carried over, equals 8. An 8 is placed under the thousands column. The results of 176 multiplied by 9 and 50 are then added to give the final product.