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Brilliant

Number Bases

via Brilliant

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Overview

It's a traditional choice to use base ten by default. You see the numerical digits 0-9 every day and you probably find it most natural to use base 10, even if you already know about binary and hexadecimal. However, many concepts in math and applications in computer science are more simply and elegantly expressed in non-decimal bases.

This course introduces a variety of powerful tools for problem-solving that take advantage of knowing and controlling what number base you're working in. You'll learn techniques for doing math in many different bases and explore applications to computer science, magic card tricks, and advanced, abstract math.

Syllabus

  • Introduction: Learn about the positional base system and the impact it has on computer systems.
    • The Invention of Number Bases: What's the difference between an additive system and a positional system?
    • Introducing Binary: Learn how to write any number using only 1s and 0s.
    • Binary on Computers: Apply your skill with binary to real scenarios in computer science.
  • Binary and Other Bases: Study base 2, base 16, and one other base that is at the frontier of computer technology.
    • Exploding Dots: Connect the game of exploding dots with the number base system.
    • Binary: Convert between base 2 and base 10.
    • Binary Operations: Add, subtract, and multiply directly in binary.
    • Perfect Shuffles: Apply binary to decipher the magic behind card shuffling.
    • Hexadecimal: Roll four binary bits into base 16 and learn how hexadecimal works.
    • Hexadecimal Operations: Solve arithmetic problems directly in hexadecimal.
    • An Unusual Computer Base: Is binary the most efficient method of storing computer information, or is there a better way?
  • Digits and Divisibility: Establish and explore patterns related to digit placement and divisibility.
    • Divisibility: Apply the divisibility rules for 1, 2, 5, and 10, and summarize why this set is special.
    • Last Digits Rules: What's different about the divisibility rules for 4 and 8?
    • More Divisibility Rules: The rules for 3, 6, 7, and 9, are a fair bit stranger than what you've seen so far.
    • Cryptograms Solved By Divisibility: Apply your understanding of divisibility rules to cracking cryptogram puzzles!
    • Cryptogram Addition Puzzles: Creatively extend your cryptogram solving strategies to handle a different operation.
    • Cryptogram Variety Pack: Each of these challenges will require the skillful and elegant application of logic, number theory, and algebra.
  • Decimal Expansions in Base 10: The standard base 10 system itself holds some mysteries that you'll unlock here.
    • Factorial Refresher: Check your knowledge of the ! symbol in mathematics.
    • Calculation Tricks: Perform arithmetic in situations where there's a shortcut method.
    • Digital Roots: Investigate surprising patterns that surface when you calculate digital roots.
    • Terminating Decimals: When do fractions terminate when changed into decimal form?
    • Repeating Decimals: Study cases of fractions where the decimal representation continues forever.
    • Repeating Patterns: What patterns can arise when converting fractions to decimal?
    • Problem Solving: Use your general knowledge of base 10 to solve this variety problem set.
  • Divisibility in Other Bases: What happens to divisibility tests when the base is no longer 10?
    • Hexadecimal Divisibility Shortcuts (I): Learn some number theory tricks that work in hexadecimal.
    • Hexadecimal Divisibility Shortcuts (II): Continue adding to your repertoire of hexadecimal divisibility tricks.
    • Hexadecimal Divisibility Shortcuts (III): Discover a divisibility test that uses an alternating sum.
    • Divisibility Shortcuts in Other Bases: Apply what you've learned about divisibility tricks to unusual bases like 6 and 12.
    • Hexadecimal Last Digits: Explore big calculations where it is nonetheless possible to answer: what is the last digit?
    • Last Digits in Other Bases: Combine strange bases with last digit problems for this last round of puzzles.

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