Course Homepage Preface On Learning
1. 1-Minute Summary 2. X-Ray Vision 3. 3d Intuition
Learn The Lingo
4. Integrals, Derivatives 5. Computer Notation
Basic Understanding
6. Improved Algebra 7. Linear Changes 8. Squared Changes
Deeper Understanding
9. Infinity 10. Derivatives 11. Fundamental Theorem
Figure Out The Rules
12. Add, Multiply, Invert 13. Patterns In The Rules 14. Take Powers, Divide
Put It To Use
15. Archimedes' Formulas Summary
Welcome to an intuition-first calculus course.
Read online, or buy the complete edition for videos, PDF, and discussion.

## 6. Improving Arithmetic And Algebra

We've intuitively seen how calculus dissects problems with a step-by-step viewpoint. Now that we have the official symbols, let's see how to bring arithmetic and algebra to the next level.

## Better Multiplication And Division

2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2

we can rewrite it as:

2 × 13

Boomshakala. If you wanted 13 copies of a number, just write it like that!

Multiplication makes repeated addition easier (likewise for division and subtraction)[^1]. But there's a big limitation: we must use identical, average-sized pieces.

• What's 2 × 13? It's 13 copies of the same element.
• What's 100 / 5? It's 100 split into 5 equal parts.

Identical parts are fine for textbook scenarios ("Drive an unwavering 30mph for exactly 3 hours"). The real world isn't so smooth. Calculus lets us accumulate or separate shapes according to their actual, not average, amount:

• Derivatives are better division that splits a shape along a path (into possibly different-sized slices)
• Integrals are better multiplication that accumulates a sequence of steps (which could be different sizes)
Operation Example Notes
Division $$\frac{y}{x}$$ Split whole into identical parts
Differentiation $$\frac{d}{dx} y$$ Split whole into (possibly different) parts
Multiplication $$y \cdot x$$ Accumulate identical steps
Integration $$\int y \ dx$$ Accumulate (possibly different) steps

Let's analyze our circle-to-ring example again. How does arithmetic/algebra compare to calculus?

Operation Formula Diagram
Division $$average \ step = \frac{Area}{radius} = \frac{\pi r^2}{r} = \pi r$$
Differentiation $$actual \ steps = \frac{d}{dr} \pi r^2 = 2 \pi r$$
Multiplication $$Area = Average \ step \cdot amount = \pi r \cdot r = \pi r^2$$
Integration $$Area = \int actual \ steps = \int 2 \pi r = \pi r^2$$

## Get Free Lessons And Updates

Join 250k monthly readers and Learn Right, Not Rote.

## Class Discussion

Guided class discussions are available in the complete edition.