Multiplication

Multiplication is one of the four basic operations in arithmetic, along with addition, subtraction, and division. It helps us find the total number of items when we have several equal groups of them. For example, if you have 3 bags with 4 apples in each bag, multiplication lets you quickly figure out that you have 12 apples in total by calculating 3 × 4.

We often think of multiplication as repeated addition. This means that multiplying two numbers is the same as adding one number to itself as many times as the value of the other number. In the example above, 3 × 4 is the same as adding 4 + 4 + 4. Multiplication can also be shown using symbols like ×, ⋅, or an asterisk (*).

Besides counting groups, multiplication helps us measure areas. If you know the length and width of a rectangle, multiplying these two numbers gives you the area. This works the same whether you multiply length × width or width × length, showing that multiplication does not change the result based on the order of the numbers. The result of multiplying two numbers is called a product, and division is the operation that undoes multiplication.

Notation

See also: Multiplier (linguistics)

In arithmetic, multiplication is shown with the multiplication sign (×) between numbers. For example, 2 × 3 = 6 means two times three equals six.

There are other ways to show multiplication. One way is to use a dot, like 5 ⋅ 2. This is common in the United States. In algebra, we often write multiplication without a sign, like xy for x times y. In computer programming, we use an asterisk (), like 52, because old computers did not have a multiplication sign on their keyboards.

The numbers being multiplied are called factors. The answer is called the product. For example, in 2 × 3 = 6, 2 and 3 are factors, and 6 is the product.

Definitions

Multiplication is one of the basic operations in math, along with addition, subtraction, and division. When we multiply two numbers, the result is called a product.

We can multiply different types of numbers, like whole numbers, fractions, and more complex numbers.

For whole numbers, multiplication can be thought of as repeated addition. For example, multiplying 3 by 4 means adding 3 to itself 4 times: 3 + 3 + 3 + 3 = 12. When multiplying fractions, we multiply the top numbers together and the bottom numbers together. For example, multiplying 1/2 by 1/3 gives us (1×1)/(2×3) = 1/6.

Computation

To multiply numbers by hand, many people learn a multiplication table for numbers from 0 to 9. One special way to multiply is called "peasant multiplication," which doesn’t need a table. The example below shows "long multiplication," a common way to multiply big numbers:

  23958233

× 5830 ——————————————— 00000000 ( = 23,958,233 × 0) 71874699 ( = 23,958,233 × 30) 191665864 ( = 23,958,233 × 800)

• 119791165 ( = 23,958,233 × 5,000) ——————————————— 139676498390 ( = 139,676,498,390 )

In places like Germany, people might write the same problem in a single line and add up the parts:

23958233 · 5830 ——————————————— 119791165 191665864 71874699 00000000 ——————————————— 139676498390

Multiplying big numbers by hand can be hard and mistakes can happen easily. Tools like common logarithms, slide rules, and modern computers and calculators make multiplication much faster and easier.

Historical algorithms

People in ancient Egypt, Greece, India, and China all had their own ways to multiply numbers.

The Egyptian way to multiply was by adding and doubling numbers. For example, to multiply 13 by 21, they would double 21 three times to get 42, 84, and 168, then add those numbers together.

The Babylonians used a system based on groups of 60, similar to how we use groups of 10 today. They used tables to help with multiplication.

In China, early mathematicians used special methods and tables to multiply numbers.

Modern methods

Today, most people learn multiplication using the Hindu–Arabic numeral system, which was first described by the mathematician Brahmagupta. This system was later spread to other parts of the world.

One popular way to teach multiplication in school is the grid method, or box method. This helps students see how multiplying each digit works. For example, to multiply 34 by 13, you can lay the numbers out in a grid and then add the results.

Computer algorithms

When computers need to multiply very large numbers, they use special methods that are much faster than the usual way. These methods can handle numbers with millions of digits quickly. In 2019, mathematicians found a new way to multiply numbers that is even faster for extremely large numbers.

Products of measurements

Main article: Dimensional analysis

When we multiply two measurements together, the result has a new type. This new type depends on the types of the measurements used. For example, multiplying speed by time gives distance. If you travel at 50 kilometers per hour for 3 hours, you will have gone 150 kilometers.

Other examples include multiplying length by length to get area. This is like 2.5 meters times 4.5 meters equals 11.25 square meters. Multiplying speed by time can also give distance, like 11 meters per second times 9 seconds equals 99 meters. In everyday life, multiplying the number of residents per house by the number of houses gives the total number of residents. For example, 4.5 residents per house times 20 houses equals 90 residents.

Product of a sequence

Further information: Iterated binary operation § Notation

The product of a sequence of numbers can be shown using a special symbol, called the capital pi (∏), from the Greek alphabet. This symbol works much like the summation symbol (∑).

For example, the product of numbers from 1 to 4 can be written as:

∏ from i=1 to 4 of (i + 1) = (1 + 1) × (2 + 1) × (3 + 1) × (4 + 1) = 120

In this notation, the small "i" is called the multiplication index. It runs from a starting number to an ending number. We replace "i" with each whole number between these two values, and multiply all the results together.

When all the numbers we are multiplying are the same, the product is the same as raising that number to a power. For example, multiplying the number 3 four times (3 × 3 × 3 × 3) is the same as 3 raised to the 4th power (3⁴).

Exponentiation

Main article: Exponentiation

When we multiply a number by itself many times, we use a special shorthand called exponentiation. For example, instead of writing 2 × 2 × 2, we can write 2³. Here, the number 2 is called the base, and the number 3, written as a superscript, is called the exponent. The exponent tells us how many times to use the base in the multiplication. So, 2³ means we multiply 2 by itself three times.

Properties

Multiplication has some important properties that make it easier to work with. One key property is the commutative property. This means the order of numbers does not change the result. For example, multiplying 3 by 4 is the same as multiplying 4 by 3.

Another important property is the identity element. When any number is multiplied by 1, the result is the original number itself. This shows that 1 acts like a "do-nothing" number in multiplication. Also, any number multiplied by 0 always equals 0. This is known as the zero property of multiplication. These properties help us understand how multiplication works and are useful in solving many math problems.

Main article: Peano axioms

Multiplication with set theory

The product of non-negative whole numbers can be explained using set theory, which deals with groups of items. This method uses special counting systems, like cardinal numbers or the Peano axioms. We can use these ideas to learn how to multiply whole numbers, fractions, and even more complex numbers. To understand how this works for all real numbers, we build on what we know about multiplying fractions. See below for more details, and construction of the real numbers for further explanation.

Multiplication in group theory

Multiplication can be used with special sets of numbers and objects that follow certain rules, called groups. These rules include having a special number that does nothing when multiplied (called the identity), and every number having a matching partner (called an inverse).

For example, the set of all non-zero rational numbers forms a group under multiplication, with 1 as the identity.

Another example is the set of special square matrices, which also form a group under multiplication. However, unlike simple numbers, these matrices do not always multiply in the same order, meaning that a × b can be different from b × a. This shows that not all groups made with multiplication are simple or "commutative."

Multiplication of different kinds of numbers

Numbers help us count, order, and measure things. Math has grown, and multiplication now works with many types of numbers and even with things that aren’t numbers, like matrices and quaternions.

When we multiply whole numbers, we add one number to itself many times. For example, multiplying 3 by 4 means adding 3 four times (3 + 3 + 3 + 3 = 12). This works for negative numbers too. A negative times a positive (or a positive times a negative) gives a negative result. A negative times a negative gives a positive result.

We can multiply fractions by multiplying their numerators (top numbers) together and their denominators (bottom numbers) together. For example, multiplying 1/2 by 1/3 gives 1/6. Real numbers, which include whole numbers and decimals, follow similar rules. For more complex numbers, like those that include the imaginary unit i (where i squared equals -1), multiplication follows special rules that combine real and imaginary parts in unique ways.