Jason Karpeles · Illustrated companion

Mathematical Rules of Thumb

See what the equations mean. Change a value, watch the figure respond, and check the arithmetic yourself. Every chapter also comes as a runnable notebook and an AI skill.

25chapters
117demonstrations
400numbered rules
348precomputed states

Examples use constructed inputs or the book's own values. Readers work in any browser; notebooks need Python with NumPy and Matplotlib. Use the math-thumb-guide skill to bring a question of your own.

Start with a question

What do you want to know?

Is this approximation accurate enough?Chapter 1 · Algebra →What does zero failures tell me?Chapter 13 · Statistics →How uncertain is my result?Chapter 16 · Measurement →Can I trust this matrix solution?Chapter 7 · Linear algebra →Does my numerical model converge?Chapter 23 · PDEs →Are these spectral peaks resolved?Chapter 24 · Signals →

Part I · Chapters 1 to 7

Foundations of Mathematical Judgment

015 demos · 15 states

Algebra

See the Structure Before You Solve

  1. Watch a small change become a large response
  2. Price the omitted geometric tail
  3. See roots appear at a discriminant boundary
  4. Check the rule of 70 for doubling time
  5. Check the denominator sign before cross-multiplying
Open reader NotebookSkill
024 demos · 12 states

Geometry

Estimate Shape Before Measuring It

  1. Separate length, area, and volume scaling
  2. Check whether an arc is shallow enough
  3. Refine a polygon toward a circle
  4. Check the triangle inequality before triangle algebra
Open reader NotebookSkill
034 demos · 12 states

Trigonometry

Control Angles, Directions, and Oscillations

  1. Put an error budget on a small angle
  2. Keep direction when a ratio loses the quadrant
  3. Avoid subtracting nearly equal floating numbers
  4. Count the triangles in the sine-law ambiguous case
Open reader NotebookSkill
045 demos · 15 states

Calculus

Turn Local Change into Global Control

  1. Check a tangent estimate against a remainder
  2. Read a Newton step from its tangent
  3. Distinguish a finite sum from an infinite tail
  4. Use the first omitted Taylor term as an error scale
  5. Confirm 0/0 before using L'Hôpital
Open reader NotebookSkill
054 demos · 12 states

Analysis

Know When Limits and Proofs Are Safe

  1. Turn convergence speed into a stopping certificate
  2. Watch a boundary layer defeat uniform convergence
  3. Control a whole function series at once
  4. Test absolute convergence before reordering a series
Open reader NotebookSkill
064 demos · 12 states

Complex Analysis

Turn Analytic Structure into Bounds and Counts

  1. See powers multiply complex arguments
  2. Use an analytic disk to bound derivatives
  3. Check strict boundary domination before counting zeros
  4. Count only the poles the contour encloses
Open reader NotebookSkill
075 demos · 14 states

Linear Algebra

Choose Stable Matrix Methods Before Computing

  1. Solve a sensitive system and inspect both errors
  2. Fit data and read the residual pattern
  3. Reconstruct a matrix at different ranks
  4. Watch the eigenvalue ratio set the speed of power iteration
  5. See the normal equations square the condition number
Open reader NotebookSkill

Part II · Chapters 8 to 11

Discrete Structure and Scale

085 demos · 15 states

Number Theory

Reduce Large Arithmetic to Small Structure

  1. Find an inverse only when arithmetic permits it
  2. Intersect congruences in a small exact search
  3. Compare counted primes with their asymptotic scale
  4. Reduce a huge exponent by the period of the powers
  5. See why trial division can stop at the square root
Open reader NotebookSkill
094 demos · 12 states

Combinatorics

Counting Without Exhaustion

  1. Choose whether order changes the outcome
  2. Check counts by exhaustive enumeration
  3. Expose unequal symmetry orbits
  4. See the no-match probability lock onto 1/e
Open reader NotebookSkill
105 demos · 15 states

Graph Theory

Structural Tests for Networks

  1. Test a tree with connectivity and edge count
  2. Close a cycle with two colors
  3. Count odd degrees before planning an Euler trail
  4. Break Dijkstra with one negative edge
  5. See the planar edge bound reject but never certify
Open reader NotebookSkill
115 demos · 15 states

Asymptotics

Finding What Matters at Scale

  1. Improve factorial scale on the log axis
  2. Find the scale where competing terms exchange roles
  3. Preserve the term left after cancellation
  4. Watch an exponential overtake a power
  5. Stop a divergent series at its smallest term
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Part III · Chapters 12 to 17

Uncertainty, Information, and Value

125 demos · 15 states

Probability

Fast Reasoning Under Uncertainty

  1. Accumulate a rare risk across opportunities
  2. Make the base rate visible in an update
  3. Run a finite Monte Carlo estimate
  4. See how soon random labels collide
  5. See why the last coupons take longest
Open reader NotebookSkill
134 demos · 12 states

Statistics

From Data to Defensible Decisions

  1. Count independent information rather than readings
  2. Put an upper bound on zero observed events
  3. Compare interval behavior near zero and one
  4. Count the cost of peeking at p-values
Open reader NotebookSkill
145 demos · 15 states

Stochastic Processes

Rates, Waiting, and Dependence

  1. Approach the queue capacity boundary
  2. Follow a correlated shock through time
  3. Turn exposure into a Poisson count
  4. Watch a random walk spread like a square root
  5. Check whether waiting makes the end nearer
Open reader NotebookSkill
155 demos · 15 states

Information Theory

Measuring Information and Its Limits

  1. Separate rare-event surprisal from average entropy
  2. Price a probability model mismatch
  3. Read a channel limit as a limit
  4. Price signal power in bits
  5. Pick the number of hashes for a Bloom filter
Open reader NotebookSkill
165 demos · 15 states

Measurement and Uncertainty

Building Trustworthy Results

  1. Keep the covariance in an uncertainty budget
  2. Watch repetition meet a bias floor
  3. Propagate a nonlinear transformation by simulation
  4. Spend effort on the biggest contributor
  5. See when k=2 really means 95%
Open reader NotebookSkill
175 demos · 15 states

Financial Mathematics

Comparing Money Across Time

  1. Change the rate and recheck present value
  2. Use the correct base for recovering a loss
  3. Compare compounding conventions fairly
  4. Check the rule of 72
  5. Watch a growing perpetuity blow up
Open reader NotebookSkill

Part IV · Chapters 18 to 21

Modeling, Optimization, and Computation

184 demos · 12 states

Applied Mathematics

Scaling, Regimes, and Model Sanity

  1. Screen a lumped thermal model
  2. Compare transport time scales
  3. Construct a dimensionless flow comparison
  4. Feel the square law of diffusion time
Open reader NotebookSkill
195 demos · 15 states

Optimization

Scale, Search, and Certify

  1. Run gradient descent on a stiff quadratic
  2. Shift exponentials without changing softmax
  3. Use a feasible stopping test at a boundary
  4. See why condition number sets the pace
  5. See constant-step noise set a floor
Open reader NotebookSkill
205 demos · 15 states

Numerical Methods

Approximation You Can Trust

  1. Turn a bracket into a numerical certificate
  2. Find where a smaller derivative step stops helping
  3. Measure trapezoid convergence
  4. Watch equal spacing blow up
  5. Recover small terms with compensated summation
Open reader NotebookSkill
215 demos · 15 states

Scientific Computing

Precision, Performance, and Reproducibility

  1. Preserve a small logarithmic change
  2. Bound speedup before buying more processors
  3. Balance checkpoint overhead against lost work
  4. See why parallel sums disagree in the last digits
  5. Find out whether memory or arithmetic sets the speed limit
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Part V · Chapters 22 to 25

Dynamics, Signals, and Feedback

225 demos · 15 states

Ordinary Differential Equations

Choosing and Checking Time Integrators

  1. Solve a decay problem and watch the stability limit
  2. Verify RK4 by step halving
  3. Check an oscillator through its energy
  4. Watch stiffness force tiny explicit steps
  5. Count steps per period before trusting an oscillation
Open reader NotebookSkill
235 demos · 15 states

Partial Differential Equations

Discretize, Resolve, and Verify

  1. Evolve transport with a chosen CFL number
  2. Execute diffusion against a known sine solution
  3. Measure solution error on successive meshes
  4. See waves drift on a coarse grid
  5. See centered differences wiggle when flow outruns diffusion
Open reader NotebookSkill
244 demos · 11 states

Signal Processing

Sampling, Resolution, and Spectral Judgment

  1. Create an alias by sampling above Nyquist
  2. Compare longer acquisition with zero padding
  3. Expose leakage and the window tradeoff
  4. Filter before throwing samples away
Open reader NotebookSkill
255 demos · 14 states

Control Theory

Performance, Robustness, and Implementation Reality

  1. Connect a real step response with bandwidth
  2. Budget latency at an assumed crossover
  3. Simulate integral windup and a remedy
  4. Remove steady error with integral action
  5. See what phase margin looks like in a step response
Open reader NotebookSkill

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