How to multiply
Multiplication is one of the four basic arithmetic operations, with the other three being subtraction, addition, and division. Learning how to multiply is a necessary aspect of studying mathematics. For whole numbers, it can be thought of as repeated addition. Learning how to multiply largely involves memorizing a multiplication chart, also referred to as a times table or multiplication table. For larger values, a multiplication algorithm, sometimes referred to as "long multiplication," can be used.
Multiplication is an operation that, unlike addition and subtraction, can be indicated in a number of ways. The following are all multiplication symbols: ×, *, ·. Furthermore, multiplication can also be indicated using parenthesis; if there is no operator separating two numerals in parenthesis, the assumed operation is multiplication.
2 × 2 = 4
2 * 2 = 4
2 · 2 = 4
(2)(2) = 4
Using parenthesis to indicate multiplication can be particularly helpful for negative values:
(-2)(-2) = 4
The above example makes it clear that the -2's are being multiplied. Otherwise, something like -2 · -2 could be mistaken for a subtraction problem.
Multiplication as repeated addition
Multiplying whole numbers can be thought of as performing repeated addition, which is simply adding equal groups of objects together a given number of times. For example, the multiplication problem
2 × 4
can be read as "two groups of four," meaning that we would add a set of 4 objects twice to find the result. It is also equivalent to four groups of two.
Referencing the above figure,
4 × 2 = 4 + 4 = 8
2 × 4 = 2 + 2 + 2 + 2 = 8
In both cases, we can perform the multiplication problem by repeatedly adding. For larger values, or when multiplying decimals, we can use long multiplication.
Long multiplication is a multiplication algorithm we can use to multiply larger numbers or decimal numbers once we've memorized the multiplication chart. Use the following steps along with the example below to understand the process.
- Write the numerals being multiplied, aligning their ones places, with the largest numeral on top. Draw a line below the numerals being multiplied.
- Multiply the digit in the ones place in the bottom numeral by the digit in the ones place of the top numeral. Write the result below the line. If the product of the column is greater than 9, write the ones-place digit of the result below the line in the same column, and the tens-place digit above the top of the following column.
- Continue the process, multiplying the digit in the ones place in the bottom numeral with the digit in each column of the top numeral, moving from right to left; if there is a digit above the column, add it after multiplying the appropriate digits. The product of the last 2 digits, even if greater than 9, is simply written below the line in the appropriate result row.
- If the bottom numeral in the multiplication problem has more than one digit, continue the same process above for each digit, aligning the ones place of the result with the tens place of the previous result (shift the result one place value to the left).
- Once each digit in the top and bottom numerals have been multiplied, add the results of each row vertically. Refer to the addition algorithm if necessary.
Multiply 1437 × 28:
The problem above is separated just to make multiplication of the rows, as well as the carried digits, clearer. On top, shown in blue, is the first row of the result, obtained by multiplying 8 by each digit in the top numeral, and following the steps detailed above.
The bottom half shows the same process to multiply the digits shown in green.
The last line in black is the result of adding the blue and green rows, and is the solution. Therefore:
1437 × 28 = 40,236
How to multiply fractions
Multiplying fractions is similar to multiplying whole numbers, except that we need to pay attention to which digits we multiply. When multiplying fractions, multiply the numerators of all the fractions involved. Then multiply the denominators of all the fractions involved. The solution is the product of the numerators over the product of the denominators.
In the above example, we simplified the problem as we multiplied. Otherwise, we would've had to simplify the fraction after finding the solution.