Coefficients on the p-adic expansions

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In summary, coefficients on the p-adic expansions are numbers that make up the unique p-adic number representation, multiplied by powers of p. They differ from ordinary decimal numbers in their base and rules for multiplication and addition. The significance of p in p-adic expansions is that it refers to a prime number, which determines the properties of the numbers. Not all real numbers can be represented by a p-adic expansion, and they are used in various areas of mathematics to study patterns and properties of numbers and solve equations involving infinite sequences.
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MostlyHarmless
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So a p-adic expansion of a rational number was presented to me as an analogue of a Laurent-series expansion and defined as:
$$\sum\limits_{n=-{\infty}}^{\infty}a_np^n$$
Can you find the coefficients for these the same way you would for a Laurent series? I've not gotten to that part of this book, but it mentions calculus on the p-adics.
 
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I've also resolved this question. Derivatives aren't defined on the p-adic numbers. And there is no "formula" for finding the coefficients.
 

Related to Coefficients on the p-adic expansions

1. What are coefficients on the p-adic expansions?

Coefficients on the p-adic expansions refer to the numbers that make up the p-adic number representation. These numbers are multiplied by powers of p to create a unique p-adic number.

2. How do coefficients on the p-adic expansions differ from ordinary decimal numbers?

Coefficients on the p-adic expansions differ from ordinary decimal numbers in that they are based on a different number system and follow different rules for multiplication and addition. In p-adic numbers, the value of a number is determined by its position, rather than the size of its digits.

3. What is the significance of p in p-adic expansions?

The p in p-adic expansions refers to a prime number, which is the base used in the p-adic number system. Different values of p will result in different p-adic expansions and different properties for the numbers.

4. Can any real number be represented by a p-adic expansion?

No, not all real numbers can be represented by a p-adic expansion. For example, numbers that have a decimal representation that repeats infinitely, such as 1/3, cannot be accurately represented in the p-adic number system.

5. How are coefficients on the p-adic expansions used in mathematics?

Coefficients on the p-adic expansions are used in various areas of mathematics, such as number theory, algebra, and analysis. They are particularly useful in studying patterns and properties of numbers, as well as in solving equations and problems that involve infinite sequences.

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