Addition is an operation within natural numbers

Addition is an internal operation on the set of natural numbers because when you add any two natural numbers, the result is always another natural number.

In other words, the set of natural numbers is closed under addition, meaning you never leave the set when you add two natural numbers together.

Let’s simplify this concept without overcomplicating things.

First of all, what are natural numbers?

Natural numbers are the familiar numbers that start at zero and continue infinitely: 0, 1, 2, 3, 4, and so on.

These are the numbers we use for counting. "I have 3 apples", "There is 1 dog", "There are 7 days in a week." These are all examples of natural numbers.

$$ \mathbb{N} = \{ 0, 1, 2, 3, 4, 5, 6, ... \}  $$

Addition is the operation that lets us combine two numbers to get a new one.

Take a classic example: you have 2 apples, and someone gives you 3 more. How many apples do you have now? Well, 2 + 3 = 5! You now have 5 apples.

$$ 2+3 = 5 $$

Not only have we performed an addition, but we’ve also demonstrated something important: whenever you add two natural numbers, you always get another natural number.

You’ll never end up with something outside the set of natural numbers.

For instance, you won’t get a fraction or a negative number by adding natural numbers together.

Addition works the same way no matter which pair of natural numbers you choose: you start with natural numbers, and the result is always another natural number. There’s no escaping this. You can’t leave the set when you’re adding. Whether you add 0 and 4 to get 4, or 10 and 25 to get 35, the outcome is always a natural number.

Let’s try another example. Imagine you have a box with 5 balls and another box with 8 balls.

If you combine all the balls, how many do you have? Add 5 + 8 = 13. Once again, 13 is a natural number.

$$ 5+ 8 = 13 $$

Addition has kept the "natural" quality of the numbers intact!

This is the key point we’re making. That’s why we say addition is an internal operation on the set of natural numbers.

It’s like natural numbers are a family: when two members of this family come together (are added), the result is always another member of the same family!

But if you think about it, this isn’t as obvious as it might seem. Not all operations follow this "rule". For example, if you take natural numbers and try subtracting them, you could leave the set. If you try 5 - 7, you won’t get a natural number—unless, of course, you’re in some strange universe where "negative apples" exist! And I don’t think anyone wants to end up with -2 apples. After all, you can count to 2 on your fingers, but you can’t do that with -2.

So, to sum up: when you add two natural numbers, you can be sure that the result will always be another natural number.

And that’s what we mean when we say "addition is an operation within the set of natural numbers".

No matter which numbers you pick, the result of the addition stays within the set of natural numbers!

Simple, right?