Show that any binary integer is at most four times as long...

Key Concepts

Positional Number Systems

A positional number system represents numbers using digits and a base. Each digit’s value depends on its position as a power of the base. This concept allows us to compare the number of digits needed to represent the same number in different bases, such as binary (base 2) and decimal (base 10).

Logarithms

Logarithms provide a way to relate the magnitude of a number to its number of digits in any base. The number of digits in a base b representation of a positive integer n is approximately given by floor(log_b(n)) + 1. This technique is essential to convert between representations in different bases by using logarithms with different bases.

Digit Count Estimation

This concept involves estimating the length (i.e., number of digits) of a number when written in a particular base. By using the relationship between logarithms and digit counts, one can approximate the ratio of the lengths of numbers in different bases, revealing that while the lengths differ, they do so by a constant multiplicative factor.

Asymptotic Analysis

For very large numbers, the constant factors in the logarithmic estimates dominate, and the ratio of the length of a number in binary to its length in decimal approaches a constant value (log?10, approximately 3.32). This analysis shows that despite the worst-case factor being four, the typical asymptotic ratio is lower.

Transcript

00:01 Okay, so in this question we want to show how a hexadecimal number can be converted into a binary expansion so here we have a just a regular number in hexadecimal so first thing that we'll do is convert it into base 10 so n in base 10 is going to be h k 16 to the k plus by h k minus h k minus 1 16 to the k minus 1 plus blah blah blah plus by h1 16 plus by h zero so each of these h's are between zero h i and it's between 15 or i is equal to zero one two blah blah blah to k so because it's between 0 and 15, we can convert this into a binary expansion.

01:00 In particular, we can write it as h .i is equal to a times by 4i plus 1, plus 3, 2 times by 2 cubed, plus by a 4i plus by 2 squared, plus by a times by 4i plus 1 times by 2 plus by a times by 4i times like 2 to 0.

01:31 So the reason why it's 4i will become clearer later, but for right now, just to recognize that we can do this.

01:38 So there is like some sort of binary expansion.

01:40 So if we write this out completely for this n10, so yeah n10, we can write this as h.

01:50 No, it's not h.

01:52 A times by 4k plus by 3, 2 cubed, plus by a times by 4k plus by 2, 2 squared, plus by a times by 4i plus by 1, plus by 2, plus a times by 4i plus 1 times by 16 to the k.

02:16 And then continues on, then taking these last two terms.

02:20 H1.

02:21 So we a times by four times by one plus three so that gives you seven two cubed plus by a to the six two squared plus by a to the five two plus by a to the four times by sixteen and finally plus by a three two cubed plus by a two two squared plus by a two to plus a zero.

02:55 So the reason why we had to have the four in front is because these numbers don't overlap...

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