Addition

Addition is the arithmetic operation of combining two or more quantities to get a single result, called the sum. If you have two numbers, \( a \) and \( b \), addition is written as \( a + b \), and the result is the sum of \( a \) and \( b \).

In simple terms, addition is the process of putting two or more numbers together to get a result. Sounds easy, right? And it is.

But of course, there are specific terms we need to be familiar with.

• Addends: These are the numbers you're adding. In the equation 2 + 3, both 2 and 3 are addends.
• Sum: This is the result of the addition. When you add 2 and 3, you get 5. That 5 is the sum.

Let’s go through an example to make sure everything is clear.

$$ 2 + 3 = 5 $$

In this case, 2 and 3 are the addends, and 5 is the sum.

Now, you might be thinking, "Is that all there is to it?" Well, not quite. Let’s see what happens when things get a little more complex.

Adding more than two numbers

Sometimes, you’ll need to add more than just two numbers. But don’t worry—the process is the same. You just keep adding them one by one.

For instance, let’s try adding four numbers together.

$$ 1 + 2 + 3 + 4 = 10 $$

Here, you have four addends (1, 2, 3, and 4), and the final result, 10, is the sum. See? Nothing too complicated.

Properties of addition

Addition has some properties you should know about.

• Commutative property: In short, the order in which you add numbers doesn’t matter—the result will always be the same.

  For example, $$ 2 + 3 = 3 + 2 $$ In both cases, the sum is 5.

• Associative property: When adding more than two numbers, you can group them however you like, and the sum won’t change.

  For example, $$ (1 + 2) + 3 = 1 + (2 + 3) $$ In both cases, the sum is 6.

• Additive identity: If you add zero to a number, the number remains unchanged. Zero is the additive identity.

  For example, $$ 5 + 0 = 0 + 5 = 5 $$

• Additive Inverse: Within the set of integers \( \mathbb{Z} \), every integer \( a \) has an opposite \( -a \). Adding a number to its opposite always results in zero, the additive identity: \( a + (-a) = 0 \).

  For example, $$ 5 + (-5) = 5 - 5 = 0 $$

Adding Positive and Negative Numbers

When adding numbers with positive or negative signs, follow these rules:

• Adding two numbers with the same sign (like-signed numbers)
  The result takes the same sign as the numbers being added. The absolute value of the result is the sum of their absolute values.

  For example, the sum of \( +2 \) and \( +3 \) is: $$ 2 + 3 = +(|2| + |3|) = +5 $$ The sum of \( -2 \) and \( -3 \) is:
  $$ (-2) + (-3) = -(|-2| + |-3|) = -(2 + 3) = -5 $$

• Adding two numbers with different signs (unlike-signed numbers)
  The result takes the sign of the number with the larger absolute value. The absolute value of the result is the difference between the larger and smaller absolute values.

  For example, the sum of \( -2 \) and \( +3 \) is: $$ -2 + 3 = +(|3| - |2|) = +(3 - 2) = +1 $$ The sum of \( +2 \) and \( -3 \) is: $$ 2 + (-3) = -(|3| - |2|) = -(3 - 2) = -1 $$

• Adding two opposite numbers
  The sum of two opposite numbers is always zero.

  For example, the sum of \( -2 \) and \( +2 \) is zero: $$ -2 + 2 = 0 $$

Adding negative numbers

Now, this can be a little tricky. When you add negative numbers, it’s like subtracting.

For example, if you add 5 and -3, the result is the difference.

$$ 5 + (-3) = 2 $$

Why does this happen? In these cases, it’s as if there’s a hidden "1" after the plus sign that multiplies the negative number (-3).

$$ 5 + 1 \cdot (-3) $$

Multiplying a positive number by a negative number always gives a negative result.

$$ 5 - 3 = 2 $$

Pretty simple, right? It’s just subtraction disguised as addition. So, don’t let those minus signs confuse you.

Adding decimals and fractions

Ah, now you want to step it up and add decimals or fractions? No problem. You just need to be a little more careful:

For instance, you can add two decimal numbers by lining them up vertically.

$$ 2.5 + 1.75 = 4.25 $$

Start adding from the rightmost digit. If the sum of a column exceeds 9, carry the extra value to the next column on the left.

When adding fractions, the key is finding a common denominator, and then it’s straightforward.

$$ \frac{1}{2} + \frac{1}{4} = \frac{2+1}{4} = \frac{3}{4} $$

Addition is a closed operation within the set of natural numbers

This means that when you add two natural numbers, the result will always be another natural number.

For example, if you add 3 and 5, the result is 8, which is also a natural number.

$$ 3 + 5 = 8 $$

Regardless of which natural numbers you choose, their sum will always be a natural number.

This property is called the "closure of the set of natural numbers under addition."

Remember, natural numbers are the numbers you commonly use for counting. For instance, 0, 1, 2, 3, 4, 5, and so on. $$ \mathbb{N} = \{ 0, 1, 2, 3, 4, 5, 6, ... \} $$

In conclusion, addition may be the simplest operation, but it’s also the foundation for everything else. Mastering this means you’re already halfway there.