Whether you liked algebra class or not, there’s no denying that algebra is one of the most vital components of all of mathematics. Given this extreme importance, algebra tends to get pretty complex. As is often the case with this area of math, a phrase like “binary operator” sounds a lot more complicated in theory than it is in practice. In truth, a binary operator isn’t nearly as difficult to grasp as you might think. Here are the facts.

## What is a Binary Operator: Complete Explanation

In mathematics, a binary operator is a concept of abstract algebra that sees a symbol or phrase placed between two numbers to compare and contrast them. The symbols are known as the binary operators, while the equation as a whole is what’s known as a binary operation.

Binary operations are presented in the following form:

- [Operand 1]
- [Operator]
- [Operand 2]

Common binary operators include — but aren’t limited to — the following:

- Addition (+)
- Subtraction (-)
- Multiplication (*)
- Division (/)
- Greater than (>)
- …or equal to (>=)

- Less than (<)
- …or equal to (<=)

- Equal (==)
- Not equal (!=)
- Logical AND (&&)
- Logical OR (||)

Of all these operation symbols, “equal” and “not equal” are known as equality operators. They result in one of two answers: true (1) or false (0). If both operands are the same, it’s true. If they aren’t, it’s false. It’s as simple as this: 1 == 1 would be true (1), while 1 != 1 would be false (0).

Greater than, greater than or equal to, less than, and less than or equal to are what are known as relation operators. They also result in an answer of true (1) or false (0). When you compare two operands with one of these operators, the answer will rely on the location of the two operands. For instance, 1 > 2 would be false (0), meaning 2 > 1 would be true (1).

Logical AND and logical OR are also known as logical operators. Like the other operators, they also compare operands and result in properties of true (1) or false (0). Looking at an example of a logical operator, you can see this is where the abstract algebra shines: (5>3)&&(5<3) would be false (0), meaning (5<3)||(5>3) would be true (1).

If you attempt to perform a binary operation with anything that isn’t a number in either operand place, you will create what’s called a non-numeric argument. A non-numeric argument is an error, and to resolve this error, you need to make sure both operands are a number and not a phrase. For instance, properties of “1 + one” would be a non-numeric argument, meaning properties of “1 + 1” would be a standard binary operation.

## Binary Operator: An Exact Definition

In mathematics, a binary operator is any designated character or phrase that compares two numbers or strings of characters to one another. The former is also known as the operation symbol, while the latter is also known as an operand. The character or phrase — whether it be +, -, =, >, or any other operation symbol — effectively operates on the two operands, thus creating a result.

## How does a Binary Operator Work?

A binary operator works quite easily by comparing two numbers or strings of characters to one another, then determines whether the equation is true or false. At its core, it’s basic math: addition, subtraction, multiplication, division, and so on.

## How Do You Create a Binary Operation?

A binary operation is created by placing an operand on either side of an operation symbol. A binary operator is created by taking any one of the many symbols in mathematics or abstract algebra that you can think of.

## Who Created the Binary Operator?

While there’s no telling just how far back our modern binary number system dates back, many scholars credit the creation to Gottfried Leibniz. In his article “Explication de l’Arithmétique Binaire,” published in the 17th century, Leibniz outlines the modern binary number system — including binary operation — as we know it today. He effectively set the standard with the publication of “Explication de l’Arithmétique Binaire,” right down to the creation of the binary operator.

## What Is the Application of a Binary Operator?

The application of a binary operator involves simply placing one of the many symbols or phrases between two operands. That’s all it takes. As the foundation of almost every aspect of mathematics, the application of a binary operator is about as basic as math can get. It honestly is as easy as adding 1 and 1 together.

## Examples of a Binary Operator In the Real World

Binary operations are the foundation of almost every structure studied in algebra. As such, they’re incredibly significant both in mathematics and in the real world. Here are just a few examples of binary operators outside of a mathematical equation.

### Budgeting

Tracking and managing your finances requires quite a few binary operations. Whether you’re adding up your expenses for the month, subtracting what you’ve already spent, multiplying your monthly earnings by 12 to estimate your annual income, comparing costs between two months, or some other equation entirely, these are all instances of the binary operation.

### C++ Operators

In the C++ programming language, there are six binary operators: Binary AND (&), binary OR (|), binary XOR (^), binary one’s complement (~), binary shift left (<<), and binary shift right (>>). These C++ operators are used to perform operations on integer data on an individual bit-level. These operations could include anything from testing to set to an actual shifting of the actual bits.

### Cooking

Believe it or not, there’s even a use for binary operators in something as routine as cooking. Say you don’t have a certain measuring cup and need to determine how to get the exact measurement with other cups instead. Is half a cup the same as doubling a fourth of a cup? Is one pound the same as four ounces times three? These questions are all answered through binary operation.