If you want to talk to the Universe, you need to speak its language. There is one language that the Universe speaks — Math. But we won’t just be learning to speak to the Universe, we will be learning to speak to the Universe through a computer. So, we must learn two new languages — a computer language and the math language. We must take what we say in our language and translate it in a language the computer can understand and subsequently, one that the Universe can understand. And this is all powered by mathematics. So, let’s begin!
It all boils down to numbers. Now, I won’t go into detail about the origin of numbers or why we count things the way we do, but there are two viewpoints we need to understand about numbers. Those viewpoints are the math view of numbers herein referred to as math numbers and the computer view of numbers, herein referred to as computer numbers. But, do know that the computer numbers are math numbers, but they just have slightly different names. But it might help to view them as separate, at least for now. So, let’s start with math numbers. Math numbers are like layers of an onion- or the layers of a planet. A differentiated planet is more fun to use, so let’s use that model! (P.S. differentiated just means that the planet is separated into different layers chemically- just like how the Earth has a core, a mantle, and a crust or like how water and oil separate into different layers!)
This article will be a very basic breakdown of numbers, so I won’t go into too much detail about every single number set out there. First, the quick facts. All numbers are either real numbers or imaginary numbers. And real numbers and imaginary numbers can form complex numbers. But we will worry about complex and imaginary numbers later. For now, just know that they exist. For Universe communication purposes, we are mostly interested in the real numbers. For coding purposes, we are mostly interested in decimals and the term integer, which we will discuss shortly.
Now, to visualize this number planet, we will start in the very middle of the planet, the inner core, and work our way out. When you think of a number, what are the first values that come to mind? Probably 1, 2, 3, and so on. The numbers 1, 2, 3, 4, and so on through infinity (∞) are considered the natural numbers. They only include the positive values, no negative numbers, decimals, or even 0 (because 0 is not natural, I guess). Bump up a layer into the outer core, and include 0 to form the whole numbers. So now we have 0, 1, 2, 3, 4, and so on through infinity. Still no negative numbers or decimals. Seems a bit pedantic, but these number rules were made a long time ago and by people far smarter than me. But what about those pesky negative numbers?
Well, we will go up another layer and into the mantle and add that layer as integers. Integers include the set of all negative numbers and all positive numbers, such as -3, -2, -1, 0, 1, 2, 3, and so on from negative infinity to infinity. Now this does not include decimals just yet. In order to add the decimals in we must step up and into the next layer, our upper mantle. This layer has all of the rational and irrational numbers. Rational numbers are numbers that can be expressed as a fraction, or in other words, a decimal. Finally, we found those decimals! But what about the irrational numbers? Those are all of the numbers that cannot be expressed as a fraction. What? How can a number not be expressed as a fraction? Well, turns out there are some very special numbers like this. Famous examples include pi (𝝅), and the square root of 2. These irrational numbers are infinite decimals that don’t ever repeat or end. You might be thinking, okay, 𝝅 never stops. But neither does a repeating decimal, such as ⅓ or 0.3333333333333… Does that make it irrational?
Let's take a brief pause on our journey through the planet's layers and talk about fractions. It might help to understand just a bit about what a fraction is while we consider these irrational numbers. A fraction is any number a over b (a/b) where a and b are both integers and b can never be 0 (Because we cannot divide by zero. Ever. Don't ask me why. Just accept it.). But 0.3333333333333, while repeating presumably for infinity, can be represented as ⅓ where 1 and 3 are both integers. Remember, an integer is the set of all negative to positive numbers from negative infinity to positive infinity, just no decimals. So ⅓ works as a fraction because 1 and 3 are not decimals. They just create a decimal through division. But 𝝅 cannot be expressed as a fraction with integer numbers. Now, you might Google the fraction for 𝝅 and find 22/7 and think to yourself, but 22 and 7 are integers. But 22/7 is just an approximation of the fraction for pi and not the actual true fraction. The true fraction would never be able to be expressed as an integer over an integer. But I said I wasn’t going to go deep into this, and so that’s about as far as we’ll go into this whole irrational number thing. Just know that 𝝅 is mostly important for things involving circles. Which we will eventually get to. But how 𝝅 exists…well, that’s up to the Universe to truly understand!
Now, back to our planet journey! We’ve made it all the way to the upper mantle with our rational and irrational numbers, but we’ve got one more layer to go — the crust! And here, on the top of our beautiful number planet, we have the real numbers! Real numbers include all of the numbers from negative infinity to infinity. And yes, decimals are included. And yes, even pesky 𝝅 is a real number. Now, there are still more complicated things about numbers. Things called transcendental numbers, complex numbers, imaginary numbers, and so on. But we will leave those alone for now. I encourage you to do some Googling if you are curious! So to sum up, we have our natural numbers in the core, our whole numbers in the outer core, our integers in the mantle, our rational and irrational numbers in the upper mantle, and our real numbers in the crust that surround our whole planet. Check out the diagram below to see a visual of this!
The number planet! The differentiated layers of this planet make it easy to understand what is included in the different number sets.
Alright, so we’ve made it through the math numbers. Phew, that was a wild ride from the core of our number planet to the crust! (Also, fun fact, the layers of the number planet correspond to the actual layers of our own planet Earth) These numbers are the basis of our foundation for talking to computers. And talking to computers is very important. Because unlike a human, who can get bored or tired or hungry or distracted, a computer cannot. So when we want to do a lot of calculations or a lot of tedious, repetitive tasks, the computer is perfect for it! And a computer can help immensely with talking to the Universe! It can help us see images of the cosmos, it can visualize the spectrograph of a star, it can visualize simulations for how galaxies form or black holes collide, and so much more. Thus, we need to get a computer to help us out. I mean, you could try and calculate all of this stuff by hand, but if you’re anything like me, you will get bored or tired or distracted and definitely hungry. So, computer it is!
I had mentioned earlier that coding uses integers. Coding uses all of the negative and positive whole numbers from negative infinity to infinity, including zero. Well, that’s great and all, but an integer does not include decimal numbers. So naturally, we need those as well! And here is where math and computers diverge. An integer means the same thing in both languages —a whole number without a decimal. But in the math world, where we call a decimal a rational or irrational number, in the computer world, a decimal is called a float. Each programming language will use these two types of numbers, integer and decimal. Some might have other names for numbers, such as in Java where there are bytes and longs and doubles. But all of these names really are words for the same thing, an integer or a float. Below are examples of integers and floats in both Java and Python, two popular programming languages:
Java example of an integer and float:
int integerNumber = 1;
double decimalNumber = 1.2345;Python example of an integer and float:
integer_number = 1
decimal_number = 1.2345But we will work through just what the code above means and what the different names are for numbers in programming as we learn to communicate with computers. For now, just know that a whole number in a programming language is an integer, and that a decimal in a programming language is a float. That is all we need to know about numbers for this first lesson in communicating with our beloved Universe!
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