Natural Numbers and Integers: The First Place to Start
Episode #1 of the course Foundations of mathematics by John Robin
Do you want to brush up on your math? Or do you feel like you missed out on some important information early on in school, after which it all became frustrating and didn’t make sense? Whatever the case, if you’re taking this course, then I expect it’s because you want to figure out just what the foundation of mathematics is.
My name is John Robin, and over the next ten days, I will be your instructor. My goal is that by the end of the course, you’ll walk away feeling like you now have a starting point with your new self-paced math education.
Math, no matter how advanced, boils down to numbers. Understanding numbers, their properties, and their laws is necessary before you can move any further. So, let’s begin our journey.
These are the simplest kinds of numbers: 0, 1, 2, 3, 4, … (whenever you see “…” that means the trend continues in this manner forever). The natural numbers are also called the counting numbers or whole numbers. In math, these are given the symbol N.
Within these numbers, we can label special groups of numbers:
• even numbers, or every second natural number: 0, 2, 4, 6, … (you can think of these as all multiples of 2)
• odd numbers, or every number left when you remove the even numbers: 1, 3, 5, 7, …
• prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, …
We could talk a long time about prime numbers. In fact, there are whole fields of math devoted to them. But for this course, you just need to understand two facts about them:
• Prime numbers are natural numbers greater than 1 that are divisible only by themselves and 1.
• Every non-prime number (called a composite number) is a product of prime numbers.
Let’s see this in action. Take the natural numbers greater than 1:
2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, …
If we go through each one and try dividing by every number less than it, you will find this breaks down into:
2, 3, 2×2, 5, 2×3, 7, 3×3, 2×5, 11, 2x2x3, 13, 2×7, 3×5, 2x2x2x2, 17, 2x3x3, 19, …
Do you see what happened there? 4 is 2×2. 6 is 2×3. 9 is 3×3. And so on. We say that numbers like 4, 6, 9, 12, etc. are composite because they are composed of prime numbers multiplied together. The numbers 2, 3, 5, 7, 11, 13, … (i.e., numbers left over that couldn’t be broken down), are all prime.
Imagine you could count backward from 0. This gives rise to the negative numbers: -1, -2, -3, and so on. In other words, count in the same order as ordinary counting, except put a negative sign in front of each. This might have made no sense as a kid, but when you get your first bank account, you start to appreciate that negative numbers are very real.
If we want to put these all in order from least to greatest, then the most negative numbers should be on the far left, while the most positive are on the far right. It looks like this:
… -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, …
Note here that I’ve used “…” on the left as well. When we use it here, this means we can imagine going backward as far as we want.
This new set of numbers is called the integers. They have the symbol Z. That might seem weird, but it’s because originally, this symbol was used by German mathematicians. It’s short for “zuhlen,” or translated, “numbers.”
Notice that the natural numbers are contained in the right half of the integers: 0, 1, 2, 3, 4, 5, …
I’ll let this sink in for today. Tomorrow, we’re going to head a bit deeper into the world of numbers by exploring how integers give rise to rational numbers, then how these in turn lead us to irrational and real numbers.
A Friendly Introduction to Number Theory by Joseph Silverman
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