# Addition

**Addition** is one of the oldest and most basic mathematical operations. It is also one of the first mathematical operations people came into contact in early childhood.

The “definition” of addition is simple: *combining two or more groups of objects into a single, larger group. The number of objects in that larger group is equal to the number of objects in all the other groups put together. Numbers used in the operation are called the terms, the addends, or the summands.*

The sign for addition is called “the **plus sign**” and it looks like this:”$+$”.

**The history of behind it**

Although counting can be thought of as a form of addition, the earliest discovered artifacts suggest addition has been used between 20.000 and 18.000 years BC. The first written evidence though, indicates that Egyptians and Babylonians were using it as early as 2.000 BC. Due to the complexity of the numeral systems in use in that time, performing basic mathematical operations was also a complicated task. With the development of positional numeral systems, especially the *Hindu-Arabic* system, and the use of a digit representing zero, enabled the use of simpler methods of calculation, like the ones we use in modern times.

## Properties of addition

Addition displays several distinct properties, such as commutativity and associativity, as well as having an *identity element*. The identity element is defined as the element in a set of numbers that, when used in a mathematical operation with another number, leaves that number unchanged. In the case of addition, that element is the number 0 (zero). That means that when you add zero to another number, you get that same number as the result.

It is also worth mentioning that if you wish to add physical quantities with units (such as square feet, yards, meters, pounds, kilograms etc.), they first have to be expressed with common units. In other words, add meters to meters and not to square meters or kilograms.

**Commutativity**

Addition is considered commutative, which means that you can exchange places of the terms without changing the result.

For $a$ and $b$, natural numbers:

$a+b=b+a$.

**Associativity**

Associativity is the property of addition that comes into play when more than two numbers are being added and it tells us that the order of operations is not important when we are adding more than two numbers. That means it doesn’t matter whether you’re going to add the third and fourth number first or the first and second – the result will not change.

For $a$ and $b$, natural numbers:

$a+(b+c)=(a+b)+c$.

## Natural numbers

The simplest examples of addition are those of the addition of natural numbers. Now, we are going to look at the addition of two three-digit numbers. Let’s take a look at the next example:

The process of adding two numbers from the previous example should look like this:

-> $1 + 4 = 5$

-> $9 + 7 = 16$. Write the $6$, carry the $1$

-> $5 + 2 = 7$, “$+$” the carried $1$ it’s $8$

-> The result is number $865$.

In our next example we are going to look at addition of 3 three-digit numbers:

The process of adding the 3 numbers in the example above should look like this:

-> $1 + 6 + 8 = 15$. Write $5$ carry $1$.

-> $3 + 0 + 3 + (carried) 1 = 7$. Write number $7$.

-> $3 + 1 + 7 = 11$. Write number $11$.

The final result is number $1175$.