Binary operation

In mathematics, a binary operation is a rule for combining two elements (called operands) to produce another element. Think of it like a recipe that takes two ingredients and mixes them to make something new. Formally, it is an operation of arity two, meaning it works with exactly two inputs.

A binary operation on a set is a special kind of function. It takes any two items from that set and gives back another item that is also in the set. Common examples you already know include adding or multiplying numbers, which are familiar arithmetic operations like addition and multiplication.

Binary operations appear in many areas of math. You can find them in vector addition, matrix multiplication, and even in how groups and other structures behave. For instance, scalar multiplication in vector spaces combines a number and a vector to create a new vector.

These operations are important building blocks in algebra. They help define many structures like semigroups, monoids, groups, rings, fields, and vector spaces. Understanding binary operations gives insight into how different mathematical systems work together.

Terminology

A binary operation is a way to take two items from a group and combine them to get another item from the same group. Think of it like adding numbers: you take two numbers, add them, and get another number. This idea can apply to many different groups and ways of combining them.

Sometimes, a binary operation might not work for every pair of items. For example, in math, you can't divide by zero. When this happens, it's called a partial binary operation. In some areas of study, binary operations are always required to work for every pair, but other areas allow these special cases.

Properties and examples

Some common binary operations are adding and multiplying numbers and matrices. Another example is combining functions in a certain way.

For example:

• Adding two real numbers gives another real number.
• Adding two natural numbers gives another natural number.
• Adding two 2x2 matrices with real numbers gives another 2x2 matrix.
• Multiplying two 2x2 matrices with real numbers gives another 2x2 matrix.
• Combining two functions in a specific way gives another function.

Many binary operations are commutative, meaning the order does not matter, and associative, meaning grouping does not matter. Some have special elements that act like a "zero" in operations.

Subtraction of real numbers is not commutative or associative. Exponentiation of natural numbers is not commutative or associative either. Division and a operation called tetration also have these properties.

Notation

Binary operations are often shown between two numbers, like a ∗ b, a + b, or a ⋅ b. Sometimes, we just write the numbers together without a symbol, like ab. We also use special ways to show powers, like writing the small number up high as a superscript.

Binary operations can also be written in other styles, such as putting the operation before the numbers, like ∗ ab, or after the numbers, like ab ∗. These are known as Polish notation and reverse Polish notation.

Binary operations as ternary relations

A binary operation is a way to combine two numbers or items to get another number or item. Think of it like adding two numbers together to get a sum. We can also think of this operation as a special kind of relationship involving three parts: the two numbers we start with and the result we get after combining them.

Other binary operations

Some math ideas use binary operations. For example, in a part of math called linear algebra, multiplying a number by a vector is a binary operation. Another example is the dot product, where two vectors are combined to make a number. Whether the dot product counts as a binary operation can depend on the writer.