We can do that by wrapping it into another set as we decided earlier. Note: the case class means that this will be an immutable object (and some other things but this is a topic of a different article).Īnd since it is a member of the Natural numbers it should have the property of “having a number after it”. have the option to be referred to as a 'NaturalNumber' We can now create the first integer and call it Zero.Īs promised, let’s write this down in Scala // in this case trait means that every number we describe will So a class that creates immutable objects and takes no arguments in its constructor will create our zero! The constructor will create an immutable object with no side effects that contains all the arguments that we’ve passed as its elements. We can describe every single number like this, by just wrapping the previous number in another set.īut how do we describe a set programmatically? Well, we can use a class, that takes arguments in its constructor. You can imagine a set as a container of unique elements.įor example ’ and so on and so on. We will describe numbers using only sets (and sets that contain sets). So having all of this info we can start writing code.

Our assumption is therefor incorrect and this proves the theorem. ‘N+1’ is also a Natural number so it is indeed in the Natural numbers set. We take the next number after ‘N’ That’s ‘N+1’ which is larger than the assumed “largest” number. Going back to the definition, we know that we can add ‘one’ to every natural number in order to represent the next natural number. We take that integer and call it ‘N’ for example. If there are finite number of integers, there is an integer bigger than all of them. The proof: Suppose that the statement in this theorem is incorrect and there are finite number of integers. There’s a theorem that states that Natural numbers are infinitely many. Having this in mind we can conclude that no matter what number we have, we can always add one to represent the next. Each of these numbers can be individually described by adding one to the number before. Positive integer numbers or just The numbers we use to count. Describing Numbersįor starters, let’s see what the definition of the Natural Numbers Set is That way we can get a sense of what the number object is in real life, rather than just what we can do with a number. Another way to put it: We want to create the entire infinite set of Natural numbers using only objects. TL DR: We want to create numbers and simple arithmetic functions (addition, subtraction, multiplication, division) without writing any number in our program (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). So what do we want anyway? We want to be able to say “this object is 1”, “this object is 5234” without using the numbers or their representation in the computer. We will use only the knowledge we received in primary school. And we will do this without a super solid background in Maths.

Why? Well, because today we want to create numbers using only objects. There’s a lot of history to be told behind the decision to use binary systems but we are not interested in binary numbers today. Since there is a way to represent any number in binary, having 0 and 1 is sufficient for any computation.

$example of natural numbers$

We are taught that computers work in binary numbers, and the reason is that “modern” computers use the absence of electricity to represent 0 and presence of such (roughly speaking) to represent 1. What is the number 3 for example? We know how to write it down, we know what word to use in order refer to it, we know how to use it when counting, but what is it really? Does it have a physical form? Is it only imaginary? Can you explain to the computer what numbers mean to you? We have an intuition of what a number is, but if you have to give a formal definition what would you say? Schools teach us how to write them down, how to add, subtract, multiply and divide them. That is to say, natural number n is even if there exists another natural number m such that n = 2m 2ℕ = Composite Number Ĭomposite numbers are natural numbers that are the product of some prime numbers.Įvery natural number, except prime numbers and 1, is composite.ġ is neither a prime nor composite number, as the number is only divisible by itself.From a young age, we are taught what numbers are. Even numbers are natural numbers that are divisible by 2.

Example of natural numbers