Rules of Multiplication: Definition and Examples

Updated March 10, 2023

Multiplication is one of the four basic arithmetic operations, alongside addition, subtraction and division. People in many industries use multiplication daily. The ability to multiply figures quickly and accurately can help you solve problems on the job, perform complex calculations or even advance in your current employment. In this article, we discuss multiplication, its uses, its rules and its place in the mathematical order of operations.

What is multiplication?

Multiplication is a mathematical process that adds a number to itself repeatedly a specific number of times. For example, you can express the multiplication problem 10 x 3 as 10 + 10 + 10, as you have three groups of 10. In every multiplication expression, there are factors and a product. The factors are the numbers you multiply together to arrive at the product. A factor can be either a multiplier or a multiplicand. The multiplicand represents the number of objects within each group, and the multiplier is the total number of groups you're multiplying.

Identifying which number is the multiplicand or multiplier can be useful in word problems but does not affect the product. In the above example, the product is 30 regardless of whether 10 or 3 is the multiplier. Being adept at mental multiplication processes like these can contribute to your numeracy skills in both professional and personal problem-solving situations.

What are the rules of multiplication?

Like the other basic arithmetic operations, multiplication follows certain rules. You can use the following rules to multiply numbers quickly:

Any number times zero is always zero. The multiplier is the number of times that a multiplicand appears. Therefore, if a multiplicand appears 0 times, it does not exist.
Any number times one is always the same number. Similar to the rule of zero, if a number appears only once, it doesn't change. In the problem 4 x 1, for example, the product is always four.
Add a zero onto the original number when multiplying by 10. This shortcut allows for quick solutions of expressions involving 10. For example, to solve 34 x 10, simply add a zero onto the end of 34 to get an answer of 340. This rule applies to all multiples of 10, including 100, 1,000 and so on.
The order of factors does not affect the product. Switching the roles of multiplier and multiplicand results in the same answer. For instance, having three groups of five oranges results in 15 oranges, and so does having five groups of three oranges.
Products are always positive when multiplying numbers with the same signs. Therefore, in the expression −2 x −4, the product would be positive eight. The same is true when the factors are positive two and positive four.
Products are always negative when multiplying numbers with different signs. This means that when you multiply a negative number by a positive number, the result is negative.

Where does multiplication fit in the order of operations?

In mathematics, the order of operations refers to the sequence of steps to follow for simplifying a math expression that includes a mix of all four of the mathematical operations. A widely accepted mnemonic device for remembering the order of operations is the acronym PEMDAS, which stands for "parentheses, exponents, multiplication and division, addition and subtraction." The order of operations groups multiplication with division and addition with subtraction, where you solve a math expression from left to right, completing any multiplication and division you see before solving addition and subtraction.

For example, in the expression 8 ÷ 2 + 3 x 4, you would first address the multiplication and division elements. Since you perform the operation from the left and division shows up first, divide 8 and two to get four. Then, perform the multiplication operation of 3 x 4 = 12. The last step is 4 + 12, which is 16.

What industries use multiplication rules?

Basic math operations such as multiplication are among the top functional skills to have. Most professions that involve finance, accounting or bookkeeping may rely on multiplication to keep accurate business records. Multiplication can also apply to personal planning, scheduling and budgeting. Consider the following industries that rely on multiplication to complete important jobs:

Architecture: Architects use multiplication to plan building designs and draw blueprints, making multiplication necessary to determine the area of a room.
Business: Business owners are likely to use multiplication to determine estimates for monthly overhead costs or product pricing. Employees of large companies might use multiplication to calculate the overall cost of a project.
Culinary arts: With multiplication, a baker can determine how much flour they need to bake 100 loaves of bread, while a chef could multiply the number of ingredients they need to double a batch of soup.
Engineering: The operations of arithmetic, including multiplication, are integral to all fields of engineering. A civil engineer, for example, often uses multiplication to determine the number of materials needed when designing and constructing structural features like bridges and roadways.
Retail: Managers and sales associates use retail math, including multiplication, to conduct transactions, calculate profits and perform other sales calculations.

Tips for improving multiplication skills

If you want to improve your multiplication skills, consider the following options:

Memorize the multiplication table

The multiplication table, or times table, is a list that shows the products of factors, normally between one and 12. It's common to learn and apply the multiplication table earlier in your education, however, it's can be an effective tool for the continuous practice and development of your ability to multiply quickly. By memorizing the products of small numbers, you can easily recall products of common factors when needed.

Learn multiplication tricks

There are several multiplication tricks you can use to solve multiplication expressions quickly and easily. A well-known trick involves the number nine. For example, to solve the expression 4 x 9, hold out both your hands in front of you and bend down the fourth finger from the left. There should be three fingers up to the left of the bent finger and six fingers up to the right. Together, the number is 36, which is the answer to the example problem. This strategy works for multipliers up to 10.

Practice daily

Consistent practice can train your brain to recall knowledge quickly. When you encounter multiplication expressions, try to find the product without a calculator. Instead, use a pencil and paper to solve the problem or try to work the calculation out using mental math. Practicing consistently like this can help you strengthen your multiplication skills and ability to solve problems quickly.

Apply real-world examples

Be mindful of how multiplication fits into your everyday life. For example, when you go to the store or a restaurant, practice your skills by multiplying money values or doubling and tripling recipes you make. Additionally, practice visualizing multiplication expressions mentally to improve the way you approach solving problems quickly. For instance, if you order three sodas at a restaurant at $1.99 each, visualize the transaction as a written expression.