Multiplication is an operation within the set of natural numbers

Multiplication is considered an internal operation within the set of natural numbers because when you multiply two natural numbers, the result is always another natural number.

When we say that "multiplication is an internal operation within the set of natural numbers," we’re stating something simple yet fundamental.

Before we break it down, let’s clarify two things: what natural numbers are and what we mean by an internal operation.

• What are natural numbers? Natural numbers are the numbers we use for counting. These include numbers like 1, 2, 3, 4, and so on. Picture them as an infinite row of Lego blocks, with each natural number being a block you can add to the row. They have a lower limit (zero) but no upper limit, so they go on forever. $$ \mathbb{N} = \{ 0,1,2,3,4,5,6,... \}  $$ Natural numbers form the foundation for much of the math we use in everyday life.
• What is an internal operation? When we talk about an internal operation, we’re asking: "If I multiply two natural numbers, will I still get a natural number?" The answer is yes, and this is what makes multiplication an internal operation in the set of natural numbers.

Let’s look at a real-world example.

Take two natural numbers, like 3 and 4.

If you multiply them, you get 12.

$$ 4 \times 3 = 12 $$

And there you go—12 is still a natural number.

So, you haven’t stepped outside the set of natural numbers. It’s not like you suddenly end up with something unusual like a negative number or a fraction.

This holds true for any pair of natural numbers. Their product will always be a natural number. For instance, $ 5 \times 2 = 10 $, $ 1 \times 4 = 4 $, $ 0 \times 7 = 0 $, and so on.

This is why multiplication is classified as an "internal" operation within the set of natural numbers.

But why does it work this way?

Well, there’s a deeper reason behind it. Multiplication, as we know it, is essentially an extension of addition.

When we say "4 times 3," we’re really saying "take 4 and add it to itself 3 times":

$$ 4 \times 3 = 4 + 4 + 4 = 12 $$

Since addition of natural numbers is also an internal operation (adding two natural numbers always results in another natural number), multiplication works the same way—it’s just a more efficient way to perform repeated addition.

Here’s a fun fact: What happens when you bring zero into the mix? Introducing zero makes things even more interesting. Multiplying any number by zero always results in zero. It’s like saying, "I want zero groups of something!" If you don’t have any groups, you’re left with nothing, which is why the result is zero. But zero is still a natural number, so the operation remains internal.

This tight connection to addition helps us avoid the chaos that could arise from operations leading us outside the set of natural numbers.

If this weren’t the case, math would become a lot more complicated!