Number and Structure

What you will learn

• Use divisibility and clock arithmetic to reason about whole numbers
• Explain how algebra moved from solving equations to studying structures such as groups
• Describe vectors and matrices, and how they represent systems of equations and transformations
• Explain how these three branches underpin encryption, symmetry, and machine learning

Contents

1. Module 1 Number Theory 2 lessons

   This module covers the branch of mathematics that studies whole numbers: which ones divide which, how they factor into primes, and what happens when you count in a loop the way a clock does. You'll be able to reason about divisibility and remainders, follow Euclid's 2,300-year-old proof that the primes never run out, and see how the same ideas now protect every online payment. Number theory is the clearest case in mathematics of questions asked for their own sake turning out to matter enormously.

   Module overview and further reading

   1. 1.1Number Theory: Primes, divisibility, and Euclid's proof that the primes never run out1h 3m
   2. 1.2Number Theory: Codes, cryptography, and the Riemann Hypothesis1h 33m
2. Module 2 Algebra 2 lessons

   This module covers the branch of mathematics that began with solving for an unknown and became the study of structure. You'll see how a 9th-century Baghdad scholar's methods gave algebra its name, why the search for a formula to solve every equation ended in a proof that none exists, and how that failure led to groups, the mathematics of symmetry. School algebra is the first step in this story; this module shows you where it leads.

   Module overview and further reading

   1. 2.1Algebra: From solving for an unknown to studying symmetry and structure1h 3m
   2. 2.2Algebra: Symmetry in crystals, codes, and puzzles, and what algebra still can't classify1h 33m
3. Module 3 Linear Algebra 2 lessons

   This module covers the branch of mathematics that handles many unknowns at once. You'll see how a method for solving systems of equations, recorded in an ancient Chinese text, became the machinery of vectors and matrices, and how that machinery now ranks search results, moves images on screens, and powers machine learning. It's the branch most likely to be running quietly under the technology you use today.

   Module overview and further reading

   1. 3.1Linear Algebra: Systems of equations, vectors, and matrices from the Nine Chapters to Gauss1h 3m
   2. 3.2Linear Algebra: Search, graphics, and machine learning: linear algebra at scale1h 33m

Course project