Applications of Bases: Part I

WARNING: I'm assuming you know how to work with the standard bases (base 2, base 3, etc.). By work with, I mean know what bases are and how to convert between them, mostly.

Alright, so you've learned about bases, and you think "hmm... I don't really see the use... (except for bases problems). What are they good for?"

The immediate answer is coding, but we want something better than that. Let's start simple. Unary. Unary is another name for base one. Some people will argue that it doesn't exist, but they're wrong (NOTE: do not quote me on this. The statements about unary that follow are about unary as I define it.). I'll illustrate how to count in unary, so those unfamiliar with it can see just how fundamental it is. We begin: 0, 1, 11, 111, 1111, 11111, 111111, ... By now, you should have noticed the pattern. Unary is just tallies!

So what good is unary? (And what brilliant technique am I about to introduce?)

Consider the following example (It's easy enough I'm not going to put it as a problem) PROBLEM: What is the tenth positive integer with digits all equal to 1?

Clearly, the answer is just 1111111111 (there are 10 1s in there). So why am I posing this question? It has nothing to do with bases, right???

Actually, it has everything to do with bases. Suppose I replaced the word "tenth" with "n-th." Then, you retort that you get n 1s strung together, and that's that. But, upon further inspection, we note that that is just n in unary! Coincidence, you may claim, but it actually isn't. Now try the problems for Part I. They will teach you the fundamentals of this technique.

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2 \times 3

2^{34}

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a_{i-1}

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Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

@Matthew Lipman Can you add this to the Brilliant Wiki under a suitable skill in Number Bases? Thanks!

Calvin Lin Staff - 6 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$