Binary operation

In mathematics, a binary operation is a rule for combining two elements (called operands) to make another element. Imagine it like a recipe that uses two ingredients to create something new. It is an operation of arity two, meaning it needs exactly two inputs.

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

Binary operations are found in many parts of math. You can see them in vector addition, matrix multiplication, and in how groups and other structures work. For example, scalar multiplication in vector spaces joins a number and a vector to make a new vector.

These operations are key in algebra. They help define many structures like semigroups, monoids, groups, rings, fields, and vector spaces. Learning about binary operations shows how different parts of math fit together.

Terminology

A binary operation is a way to take two items and combine them to get another item. 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.

Properties and examples

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

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.

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 with three parts: the two numbers we start with and the result we get after combining them.

Other binary operations

Some parts of math use binary operations. For example, in 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.