Prove q-ary representation of n*q

In summary, Q-ary representation of n*q is a method of representing a number in a base-q system, where each digit has a value based on its position. It is important because it allows for efficient representation and manipulation of large numbers. The advantages include efficient storage, easier conversion between number systems, and better understanding of numbers. It differs from decimal representation in terms of the number of digits and their values. Any number can be represented in q-ary form, as long as the base is greater than 1, with some cases resulting in an infinite representation.
  • #1
Robb
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Homework Statement
Let ##q \geq 2## be an integer. Let ##n = (d_k d_{k-1} \dots d_1 )_(q)## be a q-ary representation of n. Prove that ##nq = (d_k d_{k-1} \dots d_1 0)_(q)##.
Relevant Equations
as above for n and nq.
##nq = q(d_1 + d_2 q + d_3 q^2 + \dots + d_k q^{k-1})##
##= d_1 q + d_2 q^2 + d_3 q^3 + \dots + d_k q^k##
##= d_k d_{k-1} \dots d_1 0##
Can someone please explain how to get from line two to line three. This is instructors solution and not sure I understand. Thanks!
 
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  • #2
In a q-ary representation, we have the lowest digit for ##q^0## and we have to note it. You can always make easy examples with ##q=10## and any numbers you want.
 
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What is a q-ary representation?

A q-ary representation is a way of representing a number using a base of q, where q is any integer greater than or equal to 2. This means that each digit in the number can take on q different values, from 0 to q-1.

How is a number represented in q-ary form?

In q-ary representation, a number is written as a series of digits, with each digit representing a certain power of q. For example, in base 2 (binary), the number 13 would be represented as 1101, where each digit represents a power of 2 (2^3, 2^2, 2^1, and 2^0).

What is the purpose of using q-ary representation?

Q-ary representation is useful in computer science and mathematics because it allows for efficient and compact representation of numbers. It is also helpful in performing operations such as addition, subtraction, and multiplication.

How does q-ary representation relate to modular arithmetic?

Q-ary representation is closely related to modular arithmetic, as both involve working with numbers in a finite set. In q-ary representation, the numbers are represented using a finite set of digits, while in modular arithmetic, the numbers are represented using a finite set of remainders.

Can any number be represented in q-ary form?

Yes, any positive integer can be represented in q-ary form, as long as q is greater than or equal to 2. However, some numbers may have more digits in their q-ary representation than others, depending on the value of q.

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