Discrete valuation
Okay, imagine you have 10 marbles and you want to count how many you have. You can count them one by one, starting with 1, then 2, then 3, and so on until you reach 10. This is a discrete way of counting because you are only counting whole marbles and not any parts of marbles.
Now, let's say you want to pick one marble as your favorite. You might want to give it a special name or value. This is kind of like how some people have a favorite toy or stuffed animal.
In math, we can do something similar with numbers. We can choose a special number and give it a special name or value. This is called a discrete valuation.
When we do a discrete valuation, we start with a set of numbers, like the set of all whole numbers (1, 2, 3, 4, 5, ...), and we choose a special number to be our "favorite." This special number is usually called "p" and it's called the "prime" number in our set.
We then use our "favorite" prime number to classify all the other numbers in our set into two groups: those that can be divided evenly by p, and those that can't. For example, if our "favorite" prime number is 2, then we can split the set of whole numbers into even numbers (like 2, 4, 6, 8, ...) and odd numbers (like 1, 3, 5, 7, ...).
This idea of splitting numbers into two groups based on our "favorite" prime number can be really helpful in solving math problems, kind of like having a favorite toy helps you feel happy and comfortable.
So, in summary, a discrete valuation is a way of choosing a special prime number to split a set of numbers into two groups based on whether they can be divided evenly by that prime number.
Now, let's say you want to pick one marble as your favorite. You might want to give it a special name or value. This is kind of like how some people have a favorite toy or stuffed animal.
In math, we can do something similar with numbers. We can choose a special number and give it a special name or value. This is called a discrete valuation.
When we do a discrete valuation, we start with a set of numbers, like the set of all whole numbers (1, 2, 3, 4, 5, ...), and we choose a special number to be our "favorite." This special number is usually called "p" and it's called the "prime" number in our set.
We then use our "favorite" prime number to classify all the other numbers in our set into two groups: those that can be divided evenly by p, and those that can't. For example, if our "favorite" prime number is 2, then we can split the set of whole numbers into even numbers (like 2, 4, 6, 8, ...) and odd numbers (like 1, 3, 5, 7, ...).
This idea of splitting numbers into two groups based on our "favorite" prime number can be really helpful in solving math problems, kind of like having a favorite toy helps you feel happy and comfortable.
So, in summary, a discrete valuation is a way of choosing a special prime number to split a set of numbers into two groups based on whether they can be divided evenly by that prime number.