Chapter 1
Foundations of Arithmetic
Count out loud from one and something odd happens at ten. You run out of fresh symbols and start reusing the old ones, first in pairs, then in threes, and the trick never breaks down however large the number gets. Everything you were later taught to do with numbers was built on top of that system for writing them, and none of it was chosen at random. So where did the familiar rules come from, and what is holding them up?
What You'll Explore
4 lessons.
- Place Value and the Number System
Write 40, then write 400. The same digit sits at the front of both, and it does not stand for the same amount. You will look at where a digit's value actually comes from, and what changes about it each time it moves one place to the left.
- Properties of Addition and Multiplication
Swapping the two numbers in an addition never seems to change the answer, and neither does grouping a long sum differently. Would the same freedom survive if the operation were subtraction or division? This lesson asks which rearrangements a calculation tolerates, which ones it refuses, and what separates the two.
- Order of Operations
Hand the expression 2 plus 3 times 4 to two people and you can get two different answers, 20 and 14. Only one of them can be what the expression means. This lesson is about how that ambiguity gets settled, and whether the settlement is a convention or something more.
- The Distributive Property
Six friends each order a burger and a drink. You could work out one person's total and multiply by six, or add up all the burgers and all the drinks separately. Both routes feel like they should land on the same bill. Whether they always do, and what follows if they do, is the question here.