MILO / Mathematical Field Notes / Volume 01
Learning to read mathematics.
A formula is an instruction in a compact language. These six short notes slow that language down: name the objects, follow the addresses, work the arithmetic, and only then return to the shorthand.
01—06 / Six short notes
Begin anywhere
For a suggested path, read 001 through 006 in order.
- 001
What Is This Formula Asking Me to Do?
Read subscripts and sums as directions for a calculation.
cij = ∑z=13 aizbzjWhich symbols stay put, and which ones move through a sum?
Read and try - 002
Rows, Columns, and Why You Don't Multiply Everything
Use one row and one column to find a named entry of a product.
(4, 2, 1) · (1, 3, 2) = 12How does one address select exactly one row and one column?
Read and explore - 003
One Number Acting on an Entire Matrix
See how a single scalar changes every entry of a matrix.
3 × 1234 = 36912If we put 3 beside a grid of numbers, which entries does it multiply?
Read and practice - 004
The Directions a Transformation Doesn't Turn
Find directions a square matrix scales without turning.
Av = λvWhen a matrix moves an arrow, is there a line that the arrow stays on?
Read and practice - 005
What Happens When You Run the Machine Again?
Track one direction through repeated matrix applications.
v → −v/2 → v/4 → −v/8If one application of a matrix reverses an arrow and halves its length, what happens after two, three, or more applications?
Read and practice - 006
When a Mathematical System Stops Changing
Distinguish a fixed state, a cycle, and an approaching state.
x → f(x) → f(f(x))If we keep feeding a rule its own output, what would it mean for the state to stop changing, and how can we tell whether it actually gets there?
Read and practice
After reading
Test the move, then check the reasoning.
Each note has practice questions with worked answers. The cards collect all 38 prompts for quick recall across the volume.
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Guides and answer keys
Each guide is paired with a separate worked answer key. The card deck is laid out for actual-size, long-edge duplex printing.
- 001What Is This Formula Asking Me to Do?
- 002Rows, Columns, and Why You Don't Multiply Everything
- 003One Number Acting on an Entire Matrix
- 004The Directions a Transformation Doesn't Turn
- 005What Happens When You Run the Machine Again?
- 006When a Mathematical System Stops Changing