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As you may know, along with being an interior design student, I am also majoring in mathematics! I roughly explain why I chose these two polar opposite subjects in my first post.
I would also like to use my blog as a place to post mathematical ideas that I think are really interesting, in a way that is as easy as possible to understand. Right now I am taking a class called "Real Analysis," which examines the properties of numbers and their structures. It is the second-to-last math class I am taking to complete my undergraduate major, so from my point of view... it's a challenge. Largely proof based, this class tests one based on their ability to prove broad statements that can be true for infinite numbers of cases. A few days ago in class, we proved a statement that I thought was really cool: "The real numbers are uncountable."
To prove this statement true, let's first define what the terms are within the statement to gain a better understanding of what we are in for:
The Real Numbers: The real numbers, commonly just called R, are all of the numbers that are, simply put, real. They include three different types of numbers:
Uncountable: A set of numbers is uncountable if there is no way to organize the list of numbers in such a way to include them all. To understand this term, let's first take a look at the whole numbers, which are countable.
The whole numbers, listed above, are all of the numbers that do not have a decimal after them (other than .0). To list them out, we can simply put them in order starting with zero, like this: {0, 1, 2, 3, 4, 5, 6, ...}. But, there is a problem with this. We are not including all of the negative whole numbers! Sure, we could come back once we are done listing all of the positive whole numbers, but we will never finish writing all of the positive whole numbers. No one will ever finish writing them, not even the most powerful machine in the universe, simply because there are an infinite number of them. An infinite number of numbers will take an infinite amount of time to write, so we would never be able to write down all of the negative whole numbers. To do this, we need to order our list in a different way. We need to think of an orderly way to arrange these numbers to make sure we are writing down each of them, not missing any along the way. Consider this: {0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, ...}. And there it is! By starting with zero and then going back and forth, we have found a countable way to list all of the whole numbers. Therefore, the whole numbers are countable- the exact opposite of uncountable. Simply put, there is no way to write down in an orderly way a set of uncountable numbers.
Now that we understand what the "real numbers" are and what "uncountable" means, we can start our proof. We will be using a common method to prove the statement "the real numbers are uncountable," called proof by contradiction. To start off the proof, we are going to state the exact opposite:
Suppose the real numbers are countable.
Let's say we meet someone who claims that they can write down every real number in an orderly way. They give us a sheet of paper that has the numbers on them. Since right now it doesn't matter what these number are, we are going to give them a letter name instead. So instead of writing 6 or 9.43239 or -7.71038, we will be writing a, b, and c etc. instead. Here is what they give us:
The real numbers = {a, b, c, d, e, f, g, h...} where each letter represents the following:
Now let us claim that we can find a number that this person hasn't included (even if their list is infinite!) To do this, we want to create a number that is different from every single number in the list by at least one digit. Here is how we do this:
By finding a number that was not in the person's infinite list of numbers, we have shown that there is a contradiction, just what we needed to prove that the real numbers are uncountable. Once a contradiction is found using the proof by contradiction method, our proof is done!
One can also show that all other continuous sets are also uncountable.
<3
I would also like to use my blog as a place to post mathematical ideas that I think are really interesting, in a way that is as easy as possible to understand. Right now I am taking a class called "Real Analysis," which examines the properties of numbers and their structures. It is the second-to-last math class I am taking to complete my undergraduate major, so from my point of view... it's a challenge. Largely proof based, this class tests one based on their ability to prove broad statements that can be true for infinite numbers of cases. A few days ago in class, we proved a statement that I thought was really cool: "The real numbers are uncountable."
To prove this statement true, let's first define what the terms are within the statement to gain a better understanding of what we are in for:
The Real Numbers: The real numbers, commonly just called R, are all of the numbers that are, simply put, real. They include three different types of numbers:
- Whole numbers - numbers such as 11, -5, 8, 0, 34, -50...
- Rational numbers - numbers that can be expressed in fraction form. For example 0.5, which can be shown in fraction form as 1/2, or 0.666666... which can be shown as 2/3.
- Irrational numbers - numbers that cannot be expressed in fraction form, such as pi (3.14159265...), e (2.718281828459...), and all of the other decimal numbers that cannot be shown as two whole numbers in fraction form.
- Imaginary numbers - these numbers do not often show up in daily life. They are the numbers you get when you take the square root of a negative number. For example, the square root of -1 (this special imaginary number is called i in the mathematical world). For most calculators, if you try to take the square root of a negative number, it will come back as an error message. These are the imaginary numbers.
- Infinity and negative infinity - often called the smallest and biggest number, I'm not going to go far into the logic and conundrums one can get into when merely thinking about these two terms. Just note that these numbers are not included in the real numbers.
Uncountable: A set of numbers is uncountable if there is no way to organize the list of numbers in such a way to include them all. To understand this term, let's first take a look at the whole numbers, which are countable.
The whole numbers, listed above, are all of the numbers that do not have a decimal after them (other than .0). To list them out, we can simply put them in order starting with zero, like this: {0, 1, 2, 3, 4, 5, 6, ...}. But, there is a problem with this. We are not including all of the negative whole numbers! Sure, we could come back once we are done listing all of the positive whole numbers, but we will never finish writing all of the positive whole numbers. No one will ever finish writing them, not even the most powerful machine in the universe, simply because there are an infinite number of them. An infinite number of numbers will take an infinite amount of time to write, so we would never be able to write down all of the negative whole numbers. To do this, we need to order our list in a different way. We need to think of an orderly way to arrange these numbers to make sure we are writing down each of them, not missing any along the way. Consider this: {0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, ...}. And there it is! By starting with zero and then going back and forth, we have found a countable way to list all of the whole numbers. Therefore, the whole numbers are countable- the exact opposite of uncountable. Simply put, there is no way to write down in an orderly way a set of uncountable numbers.
Now that we understand what the "real numbers" are and what "uncountable" means, we can start our proof. We will be using a common method to prove the statement "the real numbers are uncountable," called proof by contradiction. To start off the proof, we are going to state the exact opposite:
Suppose the real numbers are countable.
Let's say we meet someone who claims that they can write down every real number in an orderly way. They give us a sheet of paper that has the numbers on them. Since right now it doesn't matter what these number are, we are going to give them a letter name instead. So instead of writing 6 or 9.43239 or -7.71038, we will be writing a, b, and c etc. instead. Here is what they give us:
The real numbers = {a, b, c, d, e, f, g, h...} where each letter represents the following:
- a = a.aaaaaaaa...
- b = b.bbbbbbbb...
- c = c.cccccccccc...
- d = d.dddddddd...
and so on.
Now let us claim that we can find a number that this person hasn't included (even if their list is infinite!) To do this, we want to create a number that is different from every single number in the list by at least one digit. Here is how we do this:
- a = a.aaaaaaaa... example: a = 1.4739575...
- b = b.bbbbbbbb... example: b = 6.6482746...
- c = c.cccccccccc... example: c = 3.56293643...
- d = d.dddddddd... example d = 2.56284624...
and so on.
By finding a number that was not in the person's infinite list of numbers, we have shown that there is a contradiction, just what we needed to prove that the real numbers are uncountable. Once a contradiction is found using the proof by contradiction method, our proof is done!
One can also show that all other continuous sets are also uncountable.
<3