Show that the results of Exercises 26,27,28, and 29 are...

Show that the results of Exercises $26,27,28$, and 29 are valid with the number 10 replaced by any integer $d>1$ and the number 9 by $d-1$. [Note: The corresponding limiting expression of a real number $x$ is called the $d$ -ary expansion of $x$. When $d=2$, it is called the binary expansion and when $d=3$, it is called the ternary expansion.]

Key Concepts

Positional Number System

A positional number system represents numbers using a base (or radix) and a set of digits. In such systems, the value of a digit is determined both by its face value and its position in the number. This principle underlies familiar numeral systems like decimal, binary, and ternary, and it allows for a systematic way to encode any real number through appropriate sequences of digits.

d-ary Expansion

A d-ary expansion generalizes the idea of expressing a number in a specific base by representing a real number as a series where each term is a fraction with a denominator that is a power of d. This expansion is obtained by successively dividing the number by d and extracting digits, and it formally converges to the original number. The concept underscores the flexibility of numeral representation beyond the common decimal system.

Infinite Series Representation for Real Numbers

Real numbers can be expressed as the limit of an infinite series, where each term corresponds to a digit in the number’s expansion multiplied by an appropriate power of the inverse of the base. This technique ensures that every real number in a given interval has a corresponding expansion that converges to its precise value, thereby linking discrete digit sequences with continuous real numbers.

Convergence and Uniqueness in Numeral Representations

For d-ary expansions, the convergence of the infinite series is guaranteed by the properties of geometric series, as each successive term becomes progressively smaller. Additionally, under typical conditions (with some exceptions related to non-unique representations like terminating versus repeating expansions), the d-ary expansion of a number is unique. These properties are crucial for validating results that extend across different bases.

Transcript

00:01 Okay, so in this question we want to show how octal expansions can be written as binary expansions.

00:08 So if we write this octal number in terms of decimal, so in base 10, we have h k times by eight or eight k plus by h k minus one, eight to the k minus one plus by so on so on plus by h1, 8 to the 1 plus by h 08 to the 0.

00:39 Okay, now each of these h's is between 0 is less than equal to h, i is less than equal to 7.

00:49 For i is equal to 0 ,1, 2, blah blah, blah to k.

00:58 Now because it's less than equal to 7, we could write this in terms of binary, in particular, we can write h i as being a to the three i plus by two times of two squared plus by a times by three i plus by one uh times by two times by two plus by a times by three i times by two to the zero so if you perform this step for all of these hks why you'll have is first n in base ten is going to be a times by 3k plus by 2, 2 squared, plus by a times by 3k, plus by 1, 2, plus by a times by 3k, and this will be times by 8k, okay, and you plus by blah, blah, blah, and then taking the last two terms, h1 would be a to the 3 times 1 gives you 3 plus 2.

02:08 2 gives you 5 times by 2 squared plus by a 4 times by 2 plus by a 3 and there's 8 here and finally plus by a 2 2 squared plus by a 1 2 plus a 0 so now these 8s can be written in 2 in terms of 2 to the power so in turn it's 2 to the 3 to the k so this is 2 to the 3k.

02:42 Same thing here.

02:43 This is just 2 thirds.

02:46 So now, if we push this number inside the brackets, what you'll see is that we have 2 to the 0, 2 to the 1, 2 to the 2.

02:54 Then this will give 2 to the 3...

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