Natural Numbers

Natural numbers are the numbers we use every day to count. They’re simply the numbers 0, 1, 2, 3, 4… and so on. But here’s a heads-up: you’ll never reach the last one.

If you start counting, you’ll never stop, because the set of natural numbers is infinite.

That’s right—there’s no limit. You can keep counting for as long as you like, and guess what? There will always be a larger number waiting for you.

Each natural number has a successor. For example, the successor of 3 is 4, and the successor of 100 is 101. So if you were hoping to finish counting, think again.

But a quick note: while there’s always a successor, the predecessor only exists if we’re not talking about 0. For example, the predecessor of 4 is 3, but for 0… there isn’t one. So if you’re wondering what comes before 0, I’d suggest not going down that road unless you’re ready to switch from math to philosophy.

The set that includes all natural numbers is represented by the symbol $ \mathbb{N} $.

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

This set has some interesting properties:

1. It’s ordered because we can always tell if one natural number is larger, smaller, or equal to another. For example, 3 is obviously smaller than 7.
2. It has a minimum element. Zero is the smallest number. But there’s no largest number, since the set is infinite. You’ll never find the “last” one. So, the set of natural numbers is bounded below but unbounded above.
3. It’s discrete, meaning that there aren’t any natural numbers hidden between two consecutive numbers (like 3 and 4). For example, between 3 and 4, there are no natural numbers. You could find rational numbers like 3.5 or irrational numbers like π, but definitely no natural numbers.

Now, imagine you had to line these numbers up.

Picture a ray – a line that starts at one point and extends infinitely in one direction. Yes, infinitely.

The ray starts at 0, and each point moving right represents a natural number in increasing order. You can follow the ray and count, but don’t tire yourself out—remember, there’s no end to it.

Oh, and naturally, you can compare these numbers! If you place two natural numbers on the ray, the one on the left will always be smaller than the one on the right.

So, if you place 3 on the left and 7 on the right, that’s right—you’re correct—3 is smaller than 7.

And yes, you can also have fun with comparison symbols like <, >, ≤, ≥.

• < means “less than.” For example: 2 < 5.
• > means “greater than.” For example: 5 > 2.
• ≤ means “less than or equal to,” which includes the possibility of equality. For example, both 4 ≤ 5 and 4 ≤ 4 are true.
• ≥ means “greater than or equal to.” For example, 6 ≥ 5 and 6 ≥ 6 are both true.

If you’re feeling a bit overwhelmed by all these numbers, don’t worry. Take a deep breath and remember: natural numbers are the foundation of everything, from math to everyday life.