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Brilliant

Mathematical Fundamentals

via Brilliant

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Overview

In this course, we'll introduce the foundational ideas of algebra, number theory, and logic that come up in nearly every topic across STEM.

This course is ideal for anyone who's either starting or re-starting their math education. You'll learn many essential problem solving techniques and you'll need to think creatively and strategically to solve each challenge. Each exploration is designed to push your mathematical skills to the next level.
Math isn't about memorizing formulas, it's about problem solving. It's about looking at patterns and predicting the future of those patterns, and it's about solving complex problems by using deductive reasoning to turn them into simple ones.

Syllabus

  • Introduction: Explore the foundations of algebra and logic without any rote memorization.
    • Generalize: Algebra was never meant to be memorized. Learn a new way to see.
    • Rewrite, Rethink, Redraw: Explore fractal patterns and compound interest in the mathematics of fractions.
    • Thinking Forwards and Backwards: To solve new problems, you need to be able to think in several directions.
  • Logical Reasoning: Logic shows you how to win!
    • Reasoning from Results to Cause: You can't trust everything you hear, but you'll always have logic.
    • Werewolves of London: Werewolves are predictably treacherous creatures. But they make for great puzzles.
    • Eliminating the Impossible: Once you eliminate the impossible, whatever remains must be the truth.
    • A Treasure Hunt: The real treasures are the truths we find along the way.
    • Mad Hatter Puzzles: You'll need to keep track of what other people might think you are thinking.
    • Repairing Broken Puzzles: These puzzles can't be solved — until you fix them.
  • Numerical Reasoning: Use the basics of arithmetic and algebra to uncover properties of numbers.
    • Strategic Arithmetic: Learn to sum 100 numbers at a glance with this one simple trick.
    • Dissecting Numbers: See the composition of numbers in a brand new light.
    • Divisible by 3: Dividing numbers in clever ways is the first step towards number theory.
    • Building Numbers to Spec: Bring your divisibility rules, because not just any numbers will do.
    • Numerical Logic: Learn how to track down all the information you'll need from a problem.
  • Contextual Reasoning: Mathematical reasoning about the real world.
    • Compare the Pair: Find the best deals in town by breaking down these problems.
    • Understanding Fractions: A high-stakes Segway race puts you on the fast lane to comparing rates and ratios.
    • Probability: The Lumps has hit Flatland. Can you use probabilities to separate fact from fiction?
    • Compound Interest: Questionable business practices and multiplying fractions greater than one are quick ways to get exponential growth.
  • Algebraic Reasoning: Build and practice problem solving strategies with these algebra puzzles.
    • Magical Variables: Writing down equations is often the easiest way to organize information and solve problems.
    • Reverse Engineering Arithmetic: Find the patterns and you'll be able to solve these problems.
    • Sums and Symmetry: Algebra is all about maintaining symmetry and balance.
    • How Much Altogether?: Mix, match, and compare unknown values – then put them to work.
    • Cryptogram Challenges: These problems are tough, but you have all the skills you'll need.
  • Geometric Reasoning: A new way of learning the algebra and identities that you thought you knew.
    • Pythagoras' Theorem: A visual proof of the most famous theorem in all of mathematics.
    • The Distributive Property: If you replace algebra with geometry, you'll never need to factor again.
    • Difference of Squares: This isn't how you learned this identity at school.
    • Difference of Squares Challenges: Put all your learnings to the test.
    • The Quadratic Formula: Use visual intuition and algebra to solve every quadratic ever.

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