Every rational number can be written in one way

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In summary, every positive rational number can be written in one unique way in the form x = a1 + a2/2! + a3/3! + ... + ak/k!, where a1, a2, ..., ak are integers and 0 <= a1, 0 <= a2 < 2, ..., 0 <= ak < k. This can be proven using induction or by constructing the fractions. Furthermore, every rational number can be written in this form by choosing the largest integer less than or equal to x for each term.
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Homework Statement


Prove that every positive rational number x can be written in ONE way in form
x=a1+ a2/2! + a3/3! + ... + ak/k!
where a1,a2,...,ak are integers and 0<=a1, 0<=a2<2,... ,0<=ak<k
I wrote my solution below. Please check if it is correct and rewrite it for me in a neater way. Thank you!

Homework Equations


None

The Attempt at a Solution

I proved by induction a1 > a2/2! > a3/3! > a4/4!>... ak>k! when taking a1,a2,a3,..., ak not= 0 ( I started with 2 as base case since a1 has different interval from them). Can I prove it without induction since I don't use the term before?

Thus I notice that every fraction has a unique interval for a1,a2,a3,...,ak not = 0
and when it is 0, the fraction is 0 and doesn't add up. Thus it is unique

I also need to prove that every rational number can be written in this form. I take x=p/q and I add up the fractions to get that q=k! and p=a1 k!+ a2 k!/2! + ... + ak .
 
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Looks to me like you can just construct it. For any rational number, x, let [itex]a_1[/itex] be the largest integer less than or equal to x. Then [itex]x- a_1< 1[/itex] so that [itex]2!(x- a_1)= 2(x- a_1< 2[/itex]. Let [itex]a_2[/itex] be the largest integer less than or equal to [itex]x- a_1[/itex], etc.
 

FAQ: Every rational number can be written in one way

What does it mean for a rational number to be written in one way?

When we say a rational number can be written in one way, it means that there is only one way to express that number as a ratio of two integers. For example, the rational number 1/2 can only be written as 1/2 and not as 2/4 or any other equivalent fraction.

Why is it important to be able to write rational numbers in one way?

Being able to write rational numbers in one way ensures consistency and avoids confusion when dealing with calculations and comparisons involving fractions. It also allows for a unique representation of each rational number, making it easier to identify and work with them.

Is every rational number guaranteed to be written in one way?

Yes, every rational number can be written in one way. This is because any rational number can be reduced to its simplest form by dividing both the numerator and denominator by their greatest common factor. This process results in a unique representation of the rational number.

Can irrational numbers be written in one way?

No, irrational numbers cannot be written in one way. Unlike rational numbers, irrational numbers cannot be expressed as a ratio of two integers, and therefore cannot be reduced to a simpler form. They have an infinite and non-repeating decimal representation.

How does the concept of writing rational numbers in one way relate to equivalent fractions?

The concept of writing rational numbers in one way is closely related to equivalent fractions. Equivalent fractions represent the same rational number but are written in different ways. By writing rational numbers in one way, we can avoid confusion and ensure that different fractions are not mistakenly thought to represent different numbers.

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